Maths Is Interesting!

Caution on Mixed Number

2011-12-21T22:52:00.002+08:00

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Fractions are a necessary part of maths.
They come in many forms; improper, proper and mixed number.

Though improper and proper forms are direct in its presentation and interpretation, mixed number form may pose a potential mistake for young learners.

Example:

Is this 2 + (3/4) or 2 x (3/4) ?

Caution has to be taken to stress it as 2 + (3/4).

Some students have taken it to mean 2 pieces of (3/4) !
Dangerous isn't it.

But rest assure.
If you understand the language of maths and its "grammar", all will be well and interesting.

:-)

Tips On Using Substitution

2011-11-18T17:42:00.002+08:00

Maths entails the usage of our brain juice in solving problems. It is a good platform for stretching our imagination and creativity by using simple concepts learned to handle seemingly complex maths questions.

Let's look at a "complex" simultaneous equations maths problem, and its way of solving (suggested).

Question:

---- (A)

--- (B)

Solve for y and x.

How do you go about it?

Look scary, right?

But like what I said, looks can be deceiving. Use the brain to go around the issue!

Tips: The structure of the simultaneous equations looks similar to the conventional type.
(Conventional type:-
Ax + By = nn
Cx + Dy = kk )

So what we have to do can be to simply substitute by m, and by h (or any variable name).

What we thus convert to is:
---- (A)

--- (B)

Will this simultaneous equations be more comfortable to solve?

Hence, a simple twist to the former mathematical questions can result in a totally familiar situations where we have solve many a times.

Thus, the technique and usefulness of substitution cannot be under-estimated.

It can be powerful at times to reveal a beautiful mathematical expression for user to resolve.

Maths Is Interesting!

Treasure our brain and our thinking.

:-)

Zippy Graphical Maths

2011-10-30T23:23:00.001+08:00

Trigonometry is a fun topic in maths.

It generates curves more than many other topics.

By combining various trigonometrical functions, you can get interesting patterns on a graph.

Putting these functions on an algebraic expression produces even exciting diagram.

Below is one I created and an array of zips appears.

Enjoy maths.
maths is interesting!

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Math Challenge 24

2011-09-09T23:47:00.000+08:00

Math does not purely involve writing mathematical expression .
Sometime what you need is some logically deduction base on, of course, some mathematical principles.

Below is one good example of "deduction" type of math solving.

Let start the challenge, and have some fun!

Above you will find 3 squares.

Do note that the 2 yellows are of the same area and 1 blue of area bigger than the yellow ones.

If the total area of the 3 squares are 57 sq cm, determine the area of the bigger blue square.

I believe you will enjoy this math question.

Using Units to Deduce Maths Formula

2011-05-25T23:55:00.000+08:00

There are times when we cannot remember some simple formula for a maths application.

Or we have doubts to the some maths working especially when many parameters got involved.

I ave a simple tip.

Look at the units for the numerical item.

Example:

To calculate distance travelled by a vehicle, given the speed it goes and time taken,

we look at the speed's units.

Unit: m / s

What does it tell?

Yes, it gave an indirect answer that speed = distance / time.

Thus if time is given, we are able to know that we just need to multiple speed by time in order to retain only the distance.

(m / s) x s = m (only) ==> Distance

The above allow us to use units to deduce the working (and formula).

Hence, we should not overlook the power of knowing units.

It is simply disappointing to sometimes see people missing out on writing the units for certain parameters. Maths loses its value simply by ignoring this step.

Therefore treasure this little but powerful "units".

:-)
Maths is interesting!

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Watery Art using Maths Expression

2011-03-14T00:07:00.000+08:00

Graphs are wonderful thing in the learning of maths.

Not only does it reflects visual symptom or trend in data collected, it displays, if allowed, beautiful images.

This is possible if you allow you maths juice to go free and create mathematical expressions to your fancy and view them on a graph.

Below I have created one. I visual it as water rippling through a surface (on the top view).

Hope you like this maths art of mine.

NOTE: It is created using trigonometry of circulatory expression.

Here I view a water droplet going down into the centre. It then produces ripples or waves spreading outwards in a circular manner.

Imagination ....
Maths expressing ......

:-)

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Explanation of the Elimination Method

2011-03-05T23:42:00.001+08:00

Solving of Simultaneous equations may require one common technique called "Elimination" method.

From the name, we know that it has to eliminate or remove something from the equations.

The target is one selected variable or unknown in the mathematical equations.

However, when approaching this method, you noticed that it involved the subtraction (or addition) of equations.

The question is "Can equations be subtracted?".
And "What is the real meaning of subtracting equations?"

My answers:-

Yes, equations can of course be subtracted. Equations are like other items, e.g. apples, chairs.

The real meaning of subtracting equations is not that apparent.

The true and desired wish to subtract equations boils down to commonising a certain coefficient of a variable.
With this common coefficient, it will then be able to remove this mathematical unknown.

(It is not really the direct processing of equations, and the magical removal of variable as a result!)

We commonise the coefficient of the selected variable first before subtracting the equations in order that same items are eliminated.

Hope this clarify some doubts of new learners to simultaneous equations solvers.

Concepts have to be learned upfront without pending questions for complete understanding and smooth follow-up learning in the later stage. Seek to clarify any doubts as far as possible.

It will reduce maths anxiety and allow you to enjoy maths as a result. The reward of clearing any doubts cannot be spelled out in words but through actual working and practice with proper analysis.

I believe you support this notion.

Cheers to maths.

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Simultaneous Equations - Decimal Numbered

2011-01-16T00:01:00.000+08:00

In the learning of maths, questions grow challenging as one progress upwards.

One such example is the solving of simultaneous equations.

The easy type:

Solve for x and y using elimination method.
4x + 3y = 10 ---- (1)
3x + 4y = 11 ---- (2)

For the above problem can be solved easily by selecting a coefficient to be commonised.
The post of elimination method is reference here for review.

But moving on (higher) with more challenging maths question ... we may get the below.

0.4x + 0.3y = 1 ---- (A)
3x + 4y = 11 ---(B)

What should we do next?

Equation (A) may seems unusual. It is in decimal form!

But as the blog title claims "Maths Is Interesting!", we should not be worried.

This type of question is actually not new in concept or tricky as it seems.
It is there to test you understanding by being "different".

We have to remove the "catch", which is to change the decimated number to integer.
How we do it here is simply multiplying the coefficients by 10.

This makes equation (A) to be 4x + 3y = 10 (back to the original first set of simultaneous equations at the start of this post.

The above example serves to illustrate the simplicity of changing numbers to suit the condition for easy solving. (Other questions may be multiply by another decimal number, or integer).

Just have a clear mind and a confidence attitude will be enough to allow you to solve most of the maths questions.

Try it and you will believe what I say (or write).

Maths is interesting!

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Decimal Number Simplification

2010-12-08T23:48:00.000+08:00

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Algebraic expressions and equations normally come in integer or fraction form.

Examples:
1) 4x - 3 = x
2) (3/4)x - 3x = 1/(3x)

Simplification of the above examples will not pose much of a problem except maybe in the challenge of bringing the numbers and unknowns over the "equal" sign.

But algebraic equations can come in decimal form too.

Example:
0.3(0.2x - 1) = 0.1x

How do we go about solving the above "decimated" algebraic equation easily?

A simple trick that I can think of (or maybe too simply a technique to call it 'trick").

What I would do is to multiply the expression on both sides by 10.
The idea is to bring the decimal number into the integer range.

BUT do note that the expression on the left side has two decimal numbers.

As such I would have to "x 10" twice.
This means that there is a "x 100" on the left and right side.

The new equation will thus be:

3 (2x - 10) = 10 x
==> 6x - 30 = 10x
==> -30 = 10x - 6x = 4x
==> x = -30 / 4 = -7.5

Conclusion:
Decimal can be seen to be intimidating when in the decimal form. However, it can be elevated to the familiar integer form through simple multiplication.
However, do take note of how many decimal number has been multiplied.
Left and right sides of the equation has to have the same number of multiplication (or division) to stay equal and valid.

Maths is not that frightening.
It can be interesting, if the method to "attack" it is properly done.

:-)

Basic Decimal Conversion

2010-11-22T23:26:00.000+08:00

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Every thing falls back to basic.

If the fundamentals are weak, any maths learners will have a hard time moving forward in their maths learning journey.

Let me quote an example.

How do we change 4.75 to fraction.

We can use 475 / 100 and reduce it through long division. This will give 4 and 3/4.

However, if we know that 4.75 is actually consisting of 4 add to 0.75, the conversion will be simpler.

4 and 0.75 means 4 + (3/4) which leads directly to 4 whole and 3/4. Same as answer of above.
(NOTE: 0.75 is a quarter which equates to 3/4).

Simple isn't it?

Maths is interesting.

:-)

Pointers in Teaching, Learning Speed

2010-10-03T00:59:00.001+08:00

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At elementary level in maths education, speed is always a challenging topic for learners.

It caught my attention and I started wondering why?

Many mistakes can be made when dealing with these types of questions.

After studying the various mistakes made by learners, I came to a few conclusion that I like to share here.

How to avoid confusion in doing Speed questions in maths:-

1) Speed involves two parameters, namely, distance and time.
This is the key issue. Dealing with one parameter is already a challenge, and dealng with two is always a "headache".
The concept, has thus to be clearly addressed upon, before the ratio of distance and time leading to speed can be fully understood.
What is distance?
What is time?
These 2 items are variable in nature. They change in value.
They causes confusion when lumped together!

Examples of daily activities will help in this case.

Quote cases like running in a race, where the champion came back in the shortest time covering the same distance as all others.

Get the concept of distance versus time into them.
Also FAST and SLOW relation to speed.

2) Error in units:-
Break up the tasks of calculating km, m or cm and sec, hours, minutes separately.
In other words,deal with one item at a time.
Use basic unit if possible to reduce chances of making costly errors.

The learners have to handle the logical part of the question, and also the mechanical part of unit manipulation in speed problems.
Tell them to find one thing at a time, and the need for doing that. Be patience is the message.

3) Draw out a pictorial image of the question.
This method will help some kids to visualise the real issue.

By having drawn the length for distance to be covered (or covered), they will have a better idea of what distance is about in the maths question. They will not have to "keep" this disatnce in their mind together with the problematic "time" condition.

Use the seeing method helps them clear any doubts and can also reduce mistakes in interpreting the question.

There will definitely be more pointers to be added to my three above.
But with these 3 basic issues settled, most of the queries about speed and its maths problems should be clearer.

If you have any other pointers, you may share in the comment space.
Cheers :-)

Maths is interesting, I suppose you cannot agree more.

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Simple Logarithm Tip

2010-06-13T21:56:00.002+08:00

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Maths expression may at times look challenging, but a bit of a thought may make it otherwise.

Logarithm is always an exciting topics to new learners.

With the "log" coming into the maths expression, one will be confused.
Definitely!

But do rest assure, as the tip below shows.

Maths Tip

e^{ln y} = y

Proving this:
"Natural log" both sides will give ln e^{ln y} = ln y

Applying the law that ln aⁿ = n ln a, and that ln e = 1, you will notice that the above mathematical expressions are true and equal.

NOTE:
This tip applies to "log" too.

10^{log y} = y

Hope this helps.

:-)

Model versus Variable Technique

2010-06-02T00:01:00.001+08:00

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In using Model method of solving maths question, we are using visual blocks to scope our thinking. This is followed by analysis through the models.

Models become a link to our thinking process.

The demerit is when we did not create the Model properly, or miss out some details that cause the model to be represented wrongly.

The merit is that it can be simple and straight forward when drawn properly. It reflects outright the relationship between many unknowns.
Less workings is thus needed, as visual que sets in.

For algebraic variable technique, the unknowns are pre-defined and booked as "letters". A space, mentally, has been reserved for the answer.

The working is just simply to accept that the answer is already there but only not numerical. Following through the working steps will ultimately reveal the letter of its numerical data which is what we want.

Variable as letter is good in the sense that we need less analysis, but just mechanically following the rules and steps leading to the final step, of course with some logic and mathematical strategy.

Each has its own advantages and weakness. It is up to us to make use of them in the correct way.
Experience is the only way to overcome the proper selection of which technique.

Thus practice to gain experience in maths is one good way to master maths.
Skiving is a no-no.

Through practice, you will sooner or later find that maths is interesting.

:-)

Purpose of Variables in Algebra

2010-05-27T23:52:00.000+08:00

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Unknowns are literally unknowns.

In maths, these unknowns are a cuause formaths anxiety.
When you are in unfamiliar territory, you will naturally be uncomfortable and unease.

This is the same feelingwhen dealing with unknowns in maths.

Algebra came to the rescue for this problems.

Here, you will find unknowns named as "variables".

They served as "parking lots" for the final answers or unknowns.

In this algebra, you replace the variables for final numbers and work with them as though you already know them.

You simply go through the motion of solving the question with any given condition and numbers / data.

Upon finally reaching the last step, the numerical answers for the problem will be revealed.

This is the power of the variables in algebra.

Just simply work along, and not be fearful of the unknowns.
The steps will align you to the final answers.

Cheers!
Maths is interesting!

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A Mathematical Waterfall

2010-04-24T18:21:00.001+08:00

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Mathematics equation can be fun.
It is not only used as a problem-solving tool, it can be used to create visual image simulating scene.

By trying a few equations, anyone with patience and basic maths knowledge can do it.
Simply create an expression or equation in a graph and tweet it to form any image.

Here you will see an image formed up to look like a waterfall.

Enjoy yourself.

This was done with both logarithm and trigonometry functions.

:-)

Maths Symbol in Our Applications

2010-04-14T23:30:00.000+08:00

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There are many symbols in maths.
To learn and understand maths, we need to know the meaning of the symbols.

This is very much like talking to a foreigner. Without understanding each other's language, no communication can be carried out (other than the international body language!)

Hence, knowing the usage of the symbols in a mathematical expression helps.

But is it really so?

Partially.

Why do I say that?

Yes, you may know the symbol while doing maths, but if the same symbol is used elsewhere, do you still understand?

One example is:

y = x + 1
This means x is added by one and their total is represented by the variable "y".
This is for the maths operator "+".

But what about the expression x++ ?
This looks odd, isn't it?

To the maths learner, this may be a typo error, or something is missing.

"x++" is actually commonly used in C programming.

What it means is x = x + 1.
It is a short-cut way of writing the addition of x and replacing it by the same variable "x".

Thus this example showed the use of "+" in another application.

It is still maths in some sense, but written in another form.

Maths is therefore always around us. It is a matter of us applying them and understanding them.

Only by learning their "language", can we communicate with them.

Interesting? I bet you agree!

Other applications can be " += ", " :-) " and " x>>4 ".
Can you find their meaning?

:D

Number of Answers | Common mistake

2010-04-10T15:41:00.000+08:00

Maths can be tricky when you are not careful.
This is not to frighten you, though.

This post is just to remind you of the wonderful aspect of maths in covering all areas.

Below is an example of what I meant.

Let's take the quadratic eqaution solving as a starting point

x² = 5x

x = 5x / x   = 5 (Answer)

At first, this looks pretty fine. The answer, when substituted back, produces match of equation.

But this is actually not complete.

Those doing quadratic equation will know 2nd order (x2) equation evaluates to 2 answsers.
The answers may be the same though.

Now, if we approach it using another method, let's see the different.

x² - 5x = 0

==> x (x - 5) = 0   , after factorising

==> x = 0 and (x - 5) = 0
==> x = 0 and   x = 5

There are two answers now.

We had the x = 5 initially, but what about this new x = 0.

We have missed out on the x = 0 with the first mehtod. It looks OK then.

What happen?
It may be due to lack of experience handling this form of maths question.

The concept in solving quadratic equation is actually not limited to second order.
The hidden message is depending on the order, the number of answers will follow suit.

What I meant is :
2nd order gives 2 answers,
3rd order gives 3 answers,
4th order gives 4 answers, etc.

It is this verry message that maths learner should capture. Otherwise you will be tricked to give only one answer which leads you to "mistakes" of being incomplete.

I agree that this is tricky, but within reasonable argument.
If a student practice hard (and smart), he will not fall prey to this type of simple math problem.

Do not get con again.

Enjoy maths. It's fun and interesting.

:D

Simultaneous Equations | Re-write equations

2010-04-05T23:34:00.001+08:00

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Simple simultaneous equation problem comes as 2 straight forward mathematical expressions.

Example 1:

3x + y = 4
x + 2y = 3

But some may come in odd expressions (since life is always the case, which makes learning maths more exciting!)

Example 2:

(12 - x)(1 + y) = 15
(8 - x) (1 + y) = -15

Here you will notice that the unknowns are biased towards one side.

Approach 1:
Multiply the 2 factors to get something like example 1.
Using elimination method, remove one of the unknown.
Solve for the only one unknown left.
Using the result found, compute the other unknown.

Approach 2: (The focus of this post)
Re-write the expression to make it look simpler.

The example 2 can be re-written into below simpler form:
(12 - x) = 15 / (1 + y) ===> (A)
(8 - x) = -15 / (1 + y) ===> (B)

Equation (B) can then be seen to be the negative of equation (A).

With the re-writing, we will be visually aided to see another form, a simpler one, of the simultaneous equations.

Moving forward with the solution...

12 - x = -(8 - x) = -8 + x
12 + 8 = 2x
x = 20 / 2 = 10 (ANSWER)
Putting x = 10 back into either equation (A) or (B),

We will get (12 - 10) (1 + y) = 15, if we select equation (A)
1 + y = 15 / 2 = 7.5
y = 7.5 - 1 = 6.5 (ANSWER)

The solution is not the issue in this post.

The key message here is the technique of "re-writing" the equations to reveal the simplicity of the question.

Maths is not that difficult if you look and think to make it easy.

Cheers!
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Tips on Avoiding Mistakes (Unit writing)

2010-04-02T22:01:00.000+08:00

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Maths involves many traps.

Any one of this traps will make the solution looks odd or even to the extent of wrong answer.

What are this traps ?

Mathematical operators, symbols, units, transferring of numbers, size of the written symbols, decimal points are some of the examples of traps contributing to the error.

Here I would like to mention about "unit".

In maths, calculation of items are aplenty. One of them is the study of speed.

In the topic of speed, students are dealing with three basic elements.
They are the distance, time and their ratio (speed).

All these three elements have different units all to themselve.

Distance == metre
Time == second
Speed == metre / sec

There are variations of the above.
km, mintues, hours, km / h, m / min, etc

Do you now see the danger?

If you are dealing with so many units in one maths question, what are the chance of making mistakes?
If you are careful, the chance is low, but it does not mean zero.

You still have to be careful.

How to avoid having mistakes due to this undesired slip?

One tip is to write down the units in the working steps.
Do not leave the numerical answer (in the working) without any unit indicated.

Make clear the item of interest, whether it is distance or time by reflecting the unit besides the number.

Example: 5 km, 40 sec.

A complete maths example will push the message across, thus ....

Example :
Alan travelled at a speed of 60 km / h for 2 h. After that, he slowed down by 20 km / h and travelled the last quarter of the journey at this new speed. How long did he take to travel?

Working:
60 x20 = 120
120 / 3 = 40
60 - 20 = 40
40 / 40 = 1
2 + 1 = 3
Answer: 3 hrs.

What is your comment on the working?

I personally feel uncomfortable. What about you?

The danger in that sort of working is the lack of showing the actual item in the calculation.
It does not allow a good way for checking after completing the worksheet (if many maths problems are within).

Clearly writing the units will, at least, make checking later an easier task.

It also allows the marker (teacher) a clearer picture instead of guessing what you intend to show.

Along the way, during the working, you will also have a lesser chance of getting confuse as the items are listed with the proper message (through the units).

So are you convince proper unit presentation is worth the while?

A pointer for your thoughts.....

Cheers :-D

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Using Equation To Create A Square Graphically

2010-03-30T23:04:00.001+08:00

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While studying maths, I have been exposed to equation that forms a circle.

We know that x^2 + y^2 = 1 creates a circle.

But I have been wondering what is an equation to form a square.

I had tried a few mathematical expressions till today.

And finally I found the interesting and mysteries equation.
It utilises the same concept as the circle except that hyperbolic trigonometry is applied.

Below is a graph plotted with that equation.

The corners are rounded though. Any one has any try with a more sharper corner?

Graph is a wonderful tool as it can present results visually with one view.
Appreciating maths and using it appropriately can reduce many complex problems.

Maths is interesting.

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Area Displacement Theory

2010-03-27T13:25:00.001+08:00

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Maths is not all about calculation.

There are always more to it than meet the eyes.

This is especially true when you are doing geometrical questions where you are involved with area, perimeter and so on.

Displacement theory or its equivalent is always done without the knowledge of many people.

What is this theory about?

Let's look at one example below.

In the diagram above, you will see a path (white coloured) going across a blue platform.
If you are asked to find the area of this path, what can you do to obtain this area?

If no data of dimension is given, it is definitely not possible.

Now if the width of the path and the vertical length of the blue platform is given, can you compute the answer?

Again , this need a bit of thinking.

Displacement theory kicks in here. Look at the diagram on the right.

It is the displaced or closed up portion of the blue platform that does the trick.
Here you will notice the dashed line forming a white rectangluar area on the right-most side of the white blue platform.

Are you able to find the area of this white rectangular piece?
The width of this rectangle piece is ACTUAL the width of the white path!

You should now be able to calculate the area of this rectangular piece since the path width and length of the rectangular block is known or deduced now.

How this is possibe is through the "hidden" clue or step of closing up the path revealing the simpler rectangular area that any decent maths student can calculate.

Hence, maths is wonderful in that it tests you not only about applcations of maths tools, but your other "intelligence".

Having known displacement theory here, I believe you are really for the Math Challenge 23.
Go there and answer the question, and be quick before others grap the position one ...

:-D
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Percentage Increase in Area

2010-03-24T15:57:00.002+08:00

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Is there any formula or maths expression showing the ncrease in area when its length and its breadth are increase by m% ?

If you cannot find one, it does not matter. You can easily derive one!

Let us work on this and show the others how simple maths can help us solve daily issue.

Let the length be x and breadth be y.

If x and y increase by m%,
length becomes x + x(m/100), and breadth becomes y + y(m/100).

Area is length x breadth.

Thus new area becomes [ x + x(m/100)] [ y + y(m/100)]

This gives us an area of xy + (m/100)xy + (m/100)xy + (m/100)(m/100)xy.

From the above maths expression, we can deduce that increase in area is:
2 x m% + (m% x m%)/100

Example with numbers will convince readers better, therefore ......

Example

If the increase in perimeter is 10%, what is the increase in area?

Answer is 2 (10%) + (10% x 10%) /100 = 20% + 1% = 21%

Easy isn't it?

For other post related to this concept in percentage increase, see the post Percentage Increase in Perimeter.

Maths is interesting.

;-D

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Percentage Increase in Perimeter

2010-03-23T23:10:00.003+08:00

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Percentage is a nice and mystery word in maths.

Why do I say that?

Look at the example below:

If the perimeter has increased by 30%, does the length also increases by the same amount?

The answer is obviously YES.

Next,

If the perimeter is increased by 30%, does the area covered by it also increases by the same amount?

??? The answer needs some pondering, right?

Answer to this:
If the perimeter is increased by 30%, the length and width will both increase by 30%.
This makes the area increase by 2(30%) + (30% x 30%) = ?

(I will explain this maths calculation in a later post.)
For now, let's concentrate on the maths operation.

What do you get from 30% x 30%?
30% = 0.3
Thus 30% x 30% = 0.3 x 0.3 = 0.09 = 9%

This is a potential mathematical mistake.
Error: 30% x 30% = 900% !

So, increase in area becomes 60% + 9% = 69%

Interesting how the mind works.

If the mind is not clear when doing maths, common mistakes do occur.
With more practice, however, this form of mistakes will be lesser.

Hence, be careful when dealing with parameter such as perimeter, length and AREA.
Know their relation and be aware of the "catch" when this type of maths question is being asked.

Do not fall for the maths trick.

:-)
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Hidden Clues in Maths Questions

2010-03-17T10:36:00.001+08:00

There are different levels in any educational system.
This goes with the learning of mathematics too.

At various level of learning, you will be presented with different level of complexity.

At the elementary stage, you will be shown maths questions that are real straight forward type.
At intermediate, a bit of mind twisting has to be done to resolve any challenge.
At the highest level, the questions come embedded with hidden clues to be discovered by learners and used to continue with the solving process.

But hidden clues are now becoming the norm among intermediate level due to its benefits to prevent pure memorising of mathematical technique.

A example of this interesting "hidden clue" can be seen in my Math Challenge 23.

There anyone taking up the challenge needs another step in order to "see" through the simple trick of solving the issue.
(Note: The challenge requires only one step to calculate the area of the path).

Multi-discipline is thus needed for merit of helping get the answer.
Knowledge in utilising maths tools and technique are not sufficient these days.

Maths students have to know some basic theory of motional replacement to understand Math Challenge 23.

Hence, to master mathematics, it will be good to read more, especially, topics outside maths.
This enlarge your understanding of real-life cases roped into maths questions.

Maths is interesting in this manner since it involves not only one learning discipline but encompasses more.

Enjoy maths. It widens your perspective of the world.

:-)

Math Challenge 23

2010-03-14T22:32:00.000+08:00

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Albert needed to create a path through a garden of his.

The garden has a size of a rectangle with length of 20m and width of 15m.

He intend to have a path of 3m wide.

His design is shown below.

But he has a problem.

He wanted to know what is the area of this path he is going to lay across the garden.

Can anyone help him calculate that area?

Basic geometry knowledge may helps.

Math Challenge 22

2010-02-24T00:20:00.000+08:00

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Given the diagram of boxes below, determine, in the fastest possible way, the area of the dark blue region.

Assume the individual boxes to be 1 unit square in area.

Give your answer in the comment space, please.

Maths does not involve plain counting.

It involves some form of intelligence to get things going.

:-)

Artistic Mathematical Lantern

2010-02-03T22:29:00.000+08:00

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Have you wondered what can maths do to art?

Below is a lantern made by a mathematical expression.

With maths, you can be assured of producing wonderful images when you have the appropriate expression.

Do create some for enjoyment.

Maths is interesting.

:-)

Maths Solution Presentation

2010-01-29T23:20:00.000+08:00

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To get good marks for a maths test requires understanding of how teacher marks the paper.

"Why do I not get full marks when I have the correct numerical answers?".
This is a common question at the back of any maths students when they see marks deducted "illogically".

Explanation:
When maths teacher give a maths question, she will like to know how is the answer obtained.
She wants to know whether the "thinking" part of solving the problem existed.

With the objectives in mind, the marking schemes are sometimes created to have marks for every steps involved in getting the answer.
Thus getting the answer without the required steps, even though it is mental, is a no-no.

Let me give an example.

Solve (x + 1)(x - 4) = 0

Solution A:
x = -1
x = 4

Solution B:
x + 1 = 0 ===> x = 1
x - 4 = 0 ===> x = 4

Comparing the two solutions presented above, you will notice clearly that Solution B is a better presented solution with proper steps reflecting the "thinking" process of the students.

Though the student of Solution A has the answer correct, he did not reveal the steps and demonstrate his understanding.

With that lack of presentation, he lost precious marks.

However, do note that not every time, we need to write down every steps.
It depends on which educational level you are in.

For the above example of presentation, the level is that of elementary, where foundational understanding is a necessity.

Upon graduating to high school, less detailed steps are needed. This is because it is assumed that the students had obtained a certain level of mathematical computing skill to that level of studies.

As such, reflection of the internal thinking to show minor details can be ignored and "by-passed" to shorten solution time.
However, the marks will still be given for steps needed at high-school level.

This goes for university level too.
By then the marking scheme will access advance thinking steps rather the minor calculations.
When errors do occurs in the calculations, it will normally be taken as "human" error as opposed to conceptual error.

In summary, do know the necessary solution steps to present during test or important assignment. Do understand the requirement and objectives of the test.
Do know what is being tested.

Writing too little can be detrimental at a lower educational level.
And writing too much can be disastrous at higher level, since you will be left with little time to complete the paper.

Hence doing maths is not simply completing the paper and getting correct answers.
It is a total strategic plan involving a lot of soft skills besides the computational abilities.

Cheers to maths, and
Cheers to it being interesting!

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Proper Way Of Writing Maths Expression

2010-01-08T23:24:00.000+08:00

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Maths expression tells certain message. When it is not written properly, or written in such a way that it causes wrong interpretation, then you will expect marks to be deducted.

Examples:

1)     y = cos (A + B)
2)    g = x + log K
3)    y / x + 2

Let's look at the above examples one by one.

Example 1:
If the brackets are taken out, y = cos A + B.
Does it also mean B + cos A?

Example 2:
If the sequence is swapped, y = log K + x
Does it mean y = log (K + x)?

Example 3:
Is the denominator just x or (x + 2)?
Or is the correct expression 2 + (y /x) ?

From the above 3 maths expressions, you will observe and sense that something will go wrong when you did not write "properly".

This need practice and does need some "maths" sense to go along with the practice.
You need to know the different form of expression and its implications.

Questions like:
- one term or two terms in the desired expression?
- which is the actual denominator?
- will anyone mis-interpret the logging of term?
- If the words or symbols are too small, will they be able to see clearly?

To save time and marks, write with the reader or marker at heart.
Write as though they are reading them.
Think and write like they will be.

Maths is afterall, a language that has to be shared and used to solve certain objectives.
Do write clearly and appropriately.

The practice and skill mastered will do you and everyone one good.
Strive to make less unnecessary mistakes and reduce the chance of your marks being subtracted off through improper writing.

Cheers!   ^.^

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Math Challenge 21

2009-12-25T21:50:00.000+08:00

Some math questions are simple and can be answered easily.

See this math challenge 20 and its answer in the comment.

But a twist of the questioning will and can make it more challenging without changing the math expression.

Below is one:

Base on math challenge 20, if all unknowns CANNOT be repeated, what are they?
Again they are integers and below 10.

Enjoy the math thrill answering this.

Merry Christmas!
^.^

Math Challenge 20

2009-12-19T18:23:00.000+08:00

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Can anyone come out the answers for A, B, C and D in the below math expression?

A² + B² + 2C² = D²

The rule is that the unknowns are all integers and below 10.

Happy trying, and
don't forget that maths is interesting!

:-)

Percentage | Common Mistake

2009-12-07T00:18:00.000+08:00

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Percentage problems can be tricky at times when you are careless.

Let's us look at one maths word problem related to it.

Question:
There are 3 persons, John, Mary and Jane.
John is richer than Mary by 10%, and Mary is richer than Jane by 20%.
Is John richer than Jane by (10 + 20)% = 30% ?

Most students, upon quick thinking, will acknowledge that 30% is the correct answer.

Is it so?

To verify the answer, let us assume that Jane has $1000.
As such, Mary will have (100+20)% of $1000 = 1.2 x $1000 = $1200.
John is then 1.1 x $1200 = $1320 richer than Jane ==> By 32%.

If 30% is correct, we should get 1.3 X $1000 = $1300.

The latter number (dollar) is not the same as the first worked out solution.
Why?

Mistake in understanding what is percentage:
To assume that John is 10% + 20% richer than Jane is incorrect.
This is due to the fact that percentage has to take a common reference for this to be correct.

In the word problem, the percentages of comparison are not to a common reference.
The first one is to Mary, while the next is to Jane.

These made the denominator of the ratio different.
Thus adding the percentage up is a mistake, and an easy one too!

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Square Root | An Exciting Outcome

2009-11-29T22:28:00.000+08:00

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Everyone knows what a square root is for and what it does to a number.

But have you tried multiple square rooting ?

What do I mean?

Let's take an example.

Start with a number, say, 3.

i) Square root this 3.
ii) You will get a number after step one.
iii) Square root this new number again.
iv) Continue the steps above and look closely at the number.

Findings:
You will notice that the number after multiple square rootings, will give you closer and closer to or approaching the number "1".

Now let's start with another number. This time, let's choose, 0.4

After doing the same procedures, you will again notice that the answer is getting and approaching the number "1"!

Amazing isn't it?

What other wonders can you find from this Square Root maths operator?
Share with me in the comment section.

Maths is interesting.........

:D

Logarithm Operation Explained (2)

2009-11-23T22:33:00.000+08:00

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Logarithm operator exists as a mathematical tool to allow us to convert a number to another form in terms of base and power.

Can you make 0.5 in terms of 10^y or 5^y?

The above is a maths question that can use logarithm to solve.

The application requires the Power rule that states that log_kD^m is equivalent to mlog_kD.
With that knowledge, to convert the above question of 0.5 to various base, we simply log the 0.5 to its respecive base.

Let's go for the base 10.

Original:   0.5 = 10^y

Performing "log" on both sides:
log₁₀ 0.5 = log₁₀10^y
-0.301      = y log₁₀10
-0.301      = y

or another way to put the outcome is 0.5 = 10^-0.301

We have managed to convert the original number of 0.5 to one with base 10 and a power (index) of "-0.301".

We can also similarly do the same for a base of 5. => 0.5 = 5^y
Here we just "log" to base 5.

log₅ (0.5) = log₅ 5^y= y
-0.431       = y

Thus 0.5 = 5^-0.431

From the 2 examples above, you can see the wonders of having logarithm as a conversion tool.
Once you understand this principles, you will appreciate logarithm.

Cheers!
:D

Logarithm Operation Explained

2009-11-20T22:46:00.000+08:00

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Everyone knows what happen when we add a number to another.

We also know what happen when we do subtraction.

There is also no problem with multplication or division of numbers.

These are all simple mathematical operations that are basic.

What about doing a logarithmic operation in math?
This is a slightly complex but interesting question.

When we do a logarithmic operation on a number, what we actually do want out of the mathematical process is the power or index with reference to a base number.

10⁵ has a base number of 10 and index of 5.

Doing a "log" of the above will reveal an answer of 5.
This is the power after doing a "log" operation.

Thus, when anyone does a logarithmic operation on a number (or expression), he is trying to find the index with respect to a base reference.

log 10⁵ = 5.

Note:
Full written log expression is "log_kP". When the "k" is left out, it implies that k = 10.
And log 10 = 1.

I hope the above can explain why we do logarithmic operation and its significant.

There must be a reason for each math operation, otherwise we will be learning and doing some insane process on earth!

Maths Is Interesting!

:-D

Math Challenge 19

2009-11-17T23:10:00.000+08:00

Math does not involve variables that we can see only.

There are the logical deduction type whereby you have to visualise and come out with an answer.

Below is one good example that I would like it to be a challenge.
(Don't be frightened by this, it is just for fun.....)

Above you will find a stack of cubic boxes. There are the blue and the yellow cubes.

Question:
What is the least quantity of blue boxes must we use in order to hold the yellow boxes in the same place?

You may present your answer and how you arrive at the answer in the comment section.

Thanks for trying. And I am waiting for that interesting mathematical deduction...

Maths Is Interesting!

Cheers :-D

What Does "of" Means In Mathematics?

2009-10-26T23:26:00.000+08:00

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Reading maths question is a critical skill to learn.

Without understanding the question, you will not be able to solve the challenge correctly.

A typical maths problem uses the word "of" to express ratio.

What really does this simple word means?

To an adult this is the understanding of English.
But to a kid, this is not English but a maths question!

What then is "of" in maths?

Answer: It means MULTIPLY.

Examples:
half of the time ==> 0.5 x time

2/3 of the class ==> (2/3) x (class total)

If 40% of the apples are rotten, how many are left? ==> 0.4 x apples are rotten

Learn English well to handle maths.

This simple word "of" may make you happy or sad.
The choice is yours.

But remember, "Maths Is Interesting!".
So despair not, enjoy your maths.

Cheers :-D

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Tricky Angles | Be Aware!

2009-10-21T22:15:00.001+08:00

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Geometry in maths can means dealing with angles from a square or a rectangle.

Normally the question is to determine an unknown angle given some shape and angles.

However, mistakes can happen when basic knowledge of relationship between angles and shapes are not proper understood.

Here, I will stress on the square and rectangular matters. This is basic but can pose a tricky problem to the unwarys. Poor thing.....

Let's look at the diagram below.

Here, if sides M and N are the same, that is, if the box is a square, angle A will be 45 degree.

This is so since the corner where angle A lies is 90 degree divided EQUALLY by half due to the diagonal lines reaching to the opposite side. (symmetrical sides).

However, if the side M and N are not equal in length, then angle A WILL NOT be 45 degree. It will depends on the ratio of side M and N.

Note this message and unnecessary mistake can be avoided.

Sometime it is to test the logical thinkng through maths, by not telling you angle A is 45 degree but stating that the box is a square.

This type of maths problem will require you to calculate another angle but using angle A which is not given.

It is tricky but good to have. Your brain will be stretched to make it "flexible" for future use.

Maths is good in this sense as it twists our mind and makes our life interesting!

Work hard as well as smart.

For more examples on avoiding unnecessary mistakes, visit this time calculation post.

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Solving Maths Visually

2009-10-17T12:56:00.000+08:00

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Maths is interesting!
There are many exciting concepts and techniques one can apply.
Read on ...

There are many ways to solve a maths question.
Normally it involves taking many steps with related sequence.

However, there are also simple ways to handle a maths problem.

One such solving method is simply through visual steps.
This has no working at all.

What do I mean?

Let's look at an example.

Example:
Determine the angle B from the diagram below. Angle A is 40 degree.

Solution:

Angle B is same as angle A = 40 degree.

There is no working at all. Just the visual determination.

Concept of this trigonometrical question in geometry:
When 2 straight lines cross and meet at an angle to each other, the angle opposite to any one is the same.

As such, angle C is also equal to angle D.

This is visual maths.

Interesting?

:-D

Math Challenge 18

2009-10-14T21:46:00.004+08:00

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From the logarithm equation below, can you determine what is the expression for y?

log y = x + log x

You may give your answer in the comment section and also help explain how you get it (for the sake of sharing).

Thanks ^^

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Time Mathematics | Tips On Avoiding Mistakes

2009-10-06T22:43:00.002+08:00

Dealing with time and its calculation can be a bit tricky sometimes.

There are the hours, the minutes and the seconds to handle.

We can add or subtract them.
We do not operate in the hundreds, or ones, like any simple arithmetric.

We are calculating in terms of sixties.
1 hour = 60 minutes
1 minute = 60 seconds

Hence when given a maths question on subtracting two times, how do we go about to avoid mistakes?

Let's take an example to illustrate.

Question:
John start his journey to the market at 09:15am. If he arrived at the market at 10:05am, how long did he travelled?

Approach 1:
We can use the conventional method of carrying back 60 to the minutes, since the ending minute is smaller than the starting minute. And reducing the 10 to 9 (hour).
Next, we can then subtract with the new numbers.
That is 9:65 - 9:15 = 50 minutes.
This answer is fine.

Approach 2:
Change all the times to mintes.
10:05 ==> (10 x 60) + 5 = 605
09:15 ==> ( 9 x 60) + 15 = 555

Subtracting the two new numbers gives 50 minutes directly.
And that is the answer!

Remarks:
Comparing approach 1 with approach 2, you would notice that the latter seems to be simpler.

Why so?
This is because we have simplified the mixture of hours and minutes to only one dimension, that is, the minutes.
Thus, subtracting the newly created numbers involved only the minutes, avoiding the distraction of handling the hours.

To do maths, clear the mind of the unnecessary.

In the example above, we have removed the "hours" factor to focus purely on the "minutes".

Maths is interesting in that it is up to us to "play" with the techniques.
We can work with it or go against it.
The choice is up to us to select.

Hope you pick up some tips to make maths interesting.
Cheers!

:-D

A Simple Way To Reduce Maths Errors

2009-09-30T23:56:00.000+08:00

Anyone doing maths will know that making maths errors is very frustrating as the results obtained will definitely be wrong.

Just a simple slip-of-the-mind type of error can cause havoc.

Once a step is incorrect, the following steps will make use of the wrong numbers to "accumulate" the errors.

If the maths teacher is merciful, she will look for the application of concepts instead of hard numbers or the final outcome.

But maths is maths.
Numbers are always there.
Techniques and systematic approach are almost a must in handling maths questions.

So can we avoid making maths errors if every step is important?

Yes, if you try hard. But note, we are human. Thus to completely eliminate errors everytime is calling for being an angel!

One simple way to reduce maths mistakes is to stay focus.

Being focus and understanding the maths probem is one of the easy method to solve making careless mistakes.

Pay attention to ever steps you write and know the purpose of each working.
Every expression must have a meaning. Otherwise what for write it down as a working.

Simply focus and do not be distracted by the surrounding. This is one of the main cause of making errors.

If you are working in front of the television, switch it off or re-locate to another pleasant place.
If there is too many people around and talking aloud, move away if you do not have any ear plugs.

Just stay away from areas or surroundings that are dysfunctional to your maths learning (and in fact to any learning).

Apply what I wrote and you will find a different.

Happy maths learning and do not forget that "Maths Is Interesting!"

:-)

Maths Symbols versus Real life Applications| Teaching style

2009-09-27T18:42:00.002+08:00

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Is maths interesting and is easy to learn or teach?

There are always argument over how to really capture maths students' attention and make them understand the principles and concepts in maths.

One school of thought is to approach the real life case studies.
Here actual applications are taught to make aware the usefulness of maths.
Problem-based case studies are planned into the curriculum to allow the learners to learn maths.

Another approach that is conventional is to pump in concepts and techniques using symbols and hard formulae.
Here students learn maths consists of symbols and their true meaning in the workings.
No real life indication is mentioned or just skimmed through. The focus is purely the use of maths tools to solve questions.

What are the advantages and disadvantages?

1) For real life approach, there are chances that the learners may couple their maths learning to only that particular application.
If train speed is mentioned or used, the students may only understand that the maths tools apply to train and not areoplane. The scope of aplication is a factor and serve to be a disadvantage.
The advantage is that the learners can relate to the usefulness of maths and will pay more attention and feel more fulfilled.

2) The advantage of pure symbolic approach in the conventional teaching method is that scope of the maths tools are wide. No tying down of the maths techniques to any specific area enables fredom of use.
The disadvantage, of course, is that the students may take a longer time to see that actual useulness of the maths tools and principles.

So what is the best method to learn maths?

I suppose the answer lies with the type of students and the topics to be taught.
No one way is best.
It has to be customised to suit the students or majority of them
Flexibility is thus the best methods and getting a good maths teacher who can read the minds of the students is the better choice.

If the classroom teacher is not up to expectation or has some constraints, it will be good to look for private tutors to brush up the maths learning. Note, classroom teaching does not cater for individual needs and this is a fact.

Happy maths learning.
Maths is interesting!
Don't forget it.

:-)

Simple Tricky Maths Pattern

2009-09-21T22:38:00.000+08:00

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Maths forms interesting patterns if you care to look around.

One such example is highlighted below:

12 11 10 09 08 07 06 <== row 1 (decreasing by 1)
01 02 03 04 05 06 07 <== row 2 (increasing by 1)

If you add the numbers in row 1 to that in row 2, you will get, surprisingly, the same answer throughout!

13 13 13 13 13 13 13

Why is it so?

It is a simple trick to the unwaries, actually.

Maths is not that difficult, to start off.
The answers to the above additions seem to be accidental in having the same number.
It is actually not accidental if you think abit.

The answers were intended to be 13 to begin with.

13 = 12 + 1
13 = 11 + 2
13 = 10 + 3 .... and so on.

Though the additions seems magical with one row in ascending mode while the other row in descending mode, it is the visual maths trick that corrupts and confuses the mind.

If your principles of mathematics is good and strong, you will break through this simple trick in no time.

To excite young minds, this is a good one to try on them.

Happy playing with maths.
Maths is interesting!

:D

Purpose of Graph

2009-09-16T22:48:00.000+08:00

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What is the purpose of graph?
This may be the question every learners first ask when they were exposed to this maths topic.

When do we use graph as opposed to using, for example, Argand diagram or vectors sketch?

Graph by nature is a graphical presentation of data that collectively form into information that reflects the trend of some parameters.

It shows the past, current and possibly the future (prediction).

Graph is a relative as well as an absolute maths tool for people using it.

An example of graph application is that in stock market data prediction.
Using past records, people tends to forecast the future through looking at the graph.

Another example is in engineering work.
Collecting data of a certain electrical system behaviour, engineers can predict the failure or potential life of its operation.

A simple graph is plotted with normally 2 parameters.
But this is not always true.
Graph may come in 3 dimensional. The x, y and z direction.

Knowing graph is an alternative problem solving skill or prediction skill.
It allows users to see an overview of the relation between specific targets.

Graph is wonderful if you let it be.
Enjoy it.

:D

Graph | Length of line

2009-09-13T22:48:00.001+08:00

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In graph plotting, something we need to know the length of a segment of the line plotted.

This may be for the distance to be travelled (like in a field trip).
Or it may be for checking the material to be used in building a slanted pole / support.

Let's take an example to illustrate.

From the plot, if we are to calculate the length of the line between the two red crosses, we can use the well-known Pythagoras' Theorem.

However, we need to know the co0ordinates for the crosses or markres first, to check their positions.
For the lower cross, we will have x1 = 2, and y1 = 3.
For the upper cross, x2 = 6 and y2 = 5.

This allows us to determine that the length in the x-axis direction is 6 - 2 = 4 units.

The length in the y-axis direction will be 5 - 3 = 2 units up.

Using then Pythagoras' Theorem, lenght of targetted line segment will be given as sqrt(4² + 2²) = 4.472 units.

From graph and its application with other maths theorem, you can find answers easily.
It is the choosing of the appropriate maths tools that is is key to having a solution in a proper way.

Many a times, you may find answers or solutions through different techniques and methods. But the number of steps are more. But it is still correct.

It is through practice and gaining experience in maths problem-solving that helps you reach a level that let you handle maths with mental ease and confidence.

Everyone can achieve that. It is the attitude. Do not fear maths. It is just a tools to solve problems.

Maths is interesting! Love maths !

Cheers! :D

Misconception of Percentage

2009-08-30T22:29:00.000+08:00

Maths is interesting!

It catches you nicely when you are not aware of its full implication.

Many of its techniques and concepts has wide boundaries.

Any maths learners has to understand its fundamentals and basic concepts in order to "escape" being caught with wrong usage.

Percentage is one area I wish to mention in this post.

You may find that percentage or percent is a simple term in maths.

Besides being a ratio, it is a comparative element.

It tells how big the target is to the original, or how small it is.

Misconception of Percentage:

The numerator is always smaller than the denominator (which is the total).

Is it true?

A definite NO!

Reason:

If a ratio has its numerator less than the denominator ==> The numerator is relatively smaller in size than the total.

If the numerator is larger than the denominator ==> The numerator is bigger in size than the total.

An example can illustrate the concept.

Example:
If a costume is now priced at 90% of its original, $100, it means that the price is not only $90.
It is lesser than the original, since the ratio is 90 / 100 or 0.9.

If the costume is newly priced at 120% of its original, $100, it means that the price is now at $120!
A price value more than the original.
It is a practical real-life number.

Percentage can be more than 100%.
The numerator can be more than the denominator.

Understanding this concept will serve anyone good. Things in life goes up as well as down. Percentage reflects this sense through its number.

Interesting? You bet.

:) Happy maths learning.
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Usage of Percentage

2009-08-26T21:57:00.002+08:00

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In maths learning, every learners will come across the terms "percent" or "percentage".

Percentage is a ratio.
It is a ratio between two numbers.
The denominator is normally the total of an item.

The unit used is %.

The above may be common knowledge for anyone doing percent maths problems.

However, what is the presentation of the solution that is appropriate?

Let's us do an example.

Example:
Class A has 30 students. If 40% of them are girls, how many are boys?

Solution:
Number of boys in the class = 30 - 40%

Now, is this "30 - 40%" correct?

The idea may be there, but someting is amiss.

What is this "40%"?
Can we just write 40% as it is?

The answer is NO!

Percentage is a ratio of the total (the class size of 30).
We cannot simply write 40% if we want to know the actual number.

The correct way is to present the step or working as: (40/100) x 30 = 12 students

We cannot simply write :   30 - 40%.
The correct way has to be 30 - (40/100) x 30 = 30 - 12 = 28 students.

2 mistakes, in concept, were made:

1) Number cannot subtract a percentage. Their units are different.

2) Percentage is a ratio, an indication of the proportion of a piece to the total. It is a relative term, not an absolute number. Thus, an absolute number cannot operate with a relative term.

Understanding this concept of "percent" and "percentage" will be handy and avoid the unnecessary trouble and anxiety of doing maths.

Tips:
Thinks of a slice of cake when doing "Percent". It has meaning only when compared to the whole cake.

Remember: Maths Is Interesting!  And it WILL be get more interesting.

:D

Average Explained

2009-08-23T22:25:00.001+08:00

When you encounter the word "average", it means that all the items within the consideration has the same number.

Example:
When 3 tests has an average score of 80 marks. All the tests are 80, 80 and 80 marks each.

But nte that the three 80 marks each may not be true.
It is assumed to be to make the average correct.

The actual marks may differ from the 80 marks.

They may be 70, 80 and 90 actually, but their average is 80marks.

You may see the post on averaging if you need more information.

:D

Sinc Function in Trigonometry

2009-08-21T22:50:00.002+08:00

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Trigonometrical graphs reveal many exciting properties of their functions.

One such function is the "Sinc" function.

This "Sinc" function is represented by the equation (sin x) / x.

NOTE: There is no typo error for the word "Sinc".

This special maths function is the trigonometric sine of an angle divided by that particular angle.

Looking at the graphs of various multiple of sinc functions, you will notice some unique properties in the cross-over angles (markings).

Looking closely at the multiples of radian pi, 2pi and 3pi, you will see that the amplitudes of the various sinc functions are zero.

This is a special characteristics of "Sinc" function.

If you sample these functions at interval of pi, you will get nothing or zero amplitude.

Interesting trigonometry, right?

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Character of A Person Revealed Through Maths

2009-08-06T22:49:00.002+08:00

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There are many ways to dig into the true character of a person.

One way is through maths.

By being in a classroom of maths learners, which I suppose everyone went through or is in one, you will notice many types of characters and behaviours.

Some are strong and stubborn, the never-say-die doer.
Some are the "can do, then do" type.
Some are the easy surrenders of maths.
Some cannot even be bordered to try!

Those who attempted the maths question, also revealed some weakness also.
The careless type and the long-winded type.

They made all sort of mistakes due poor handwriting or not reading the questions properly. They may miss a few variable or maths operators in an equation.
They may indirectly simplified the problems given through seeing or copying wrongly.

Those long-winded are the "better" lot with the will to stay on track.
They do and do, even when applying the wrong technique. They, however, do get the result through their hardworking attitude.

Some are the intelligent type who spot the trick just by reading the maths question.
They are the "flexible" ones.
They are the ones that seemed to enjoy most of the maths lessons.

Whatever, the type you are, maths is still an important life-long subject.
It is a practical module that serves us till we leave this world.

Love maths, and maths will love you, whatever your character and feeling towards maths.

Maths is interesting! You have a choice for that.

Cheers.

:D

Is Maths Really Interesting?

2009-08-02T00:38:00.003+08:00

'
One question those who detest maths will ask is "Is maths really interesting?".

It is a very subjective question.

Everyone has likes and dislikes.

However, in the case of maths, it is the gain versus the lack.

Maths is a necessary life skill to have.

Knowing it makes a whole lot of different.

It will speed up your solving to some daily questions.

"What is the time needed if I drive at 60 km/h for a distance of 90km?"
"What is the area of the metal sheet needed to cover this pillar?"

We are weak in maths due to many reasons.
If you do not arrest these reasons, or reduces the obstacles to it, you will always fear maths.
This will create a mental block to your maths learning.

Practice and practice to reveal your weakness. Learn through mistakes.
You will feel the confidence of handling maths problems after that phase.

Like what Mark Twain said "Action speaks louder than words".

I would like to tweet it in the context of maths.

Instead of pure saying that you cannot do maths or you hate maths, practice (action) on it.
You will feel the difference.
You will get the hang of doing maths.
You will realise that maths is not that difficult.
You will find that it is your mindset that is the block, not maths!

Practice.

Do it.

Practice speaks louder than words.

Maths is interesting.
That will be your final conclusion if you take action and do hands-on practice.

Cheers! :D

.

Meaning of "Average"

2009-07-25T00:41:00.003+08:00

'
I managed to see a word problem shown below:

" Mary obtained an average of 80 marks for 2 tests. What marks has she to get for an average of 80 marks for 3 tests?"

If a maths student understand the meaning of "average", she can do this without even working out the mathematical steps.

The first statement in the question stated that Mary scored on both tests an average of 80 marks. This implied that for one test, she obtained 80 marks. Same as for the second test.

For the next test (the third one), to get an average of again 80 marks, it meant that all the test she has to get 80 marks each. This is the power of "average". That is to say, all the test can be concluded as the same score.

The actual differences between the individual marks can be offset from each other to achieve a final "average" of same marks (here, 80 marks).

The maths problem is to test the understanding of the word "average" in maths, and its concept.

Thus, knowing concepts do help in solving maths questions.
It does not mean working out the mechanical computation steps to get answers.

With strong maths concept, anyone can solve simple problems without doing working.
But that is maths also (in the brain, the step-less way).

Interesting. Maths is so. Fear not.

:-)

Maths Is Just Another Language

2009-07-18T08:29:00.004+08:00

'
"I don't understand maths!"

Does this sound familiar?

This statement is not only from young learners, but from adults too.

What actually happened?

Maths is interestingly unique. This is so because it represents a different way of presenting messages.

Maths is a "short form" of written langauge.

It is like the short-hand symbols that many people use to note down minutes of meeting.

What is 1 + 3 = 4?
In the common English language, it simply means "one added to three equals four".
Nothing different.
However, visually, the maths equation is encoded with symbols.

What is this "+" and "="?
If you do not understand the meaning and usage of these two symbols, then you are lost.
Thetefore, learning and understanding maths is knowing the unique symbols presenting the "message".

How about 3y = 24. What is "y"?

The question is finding "y" given the relationship above.
The equation in English means "When we multiply "y" by 3, it gives 24".

The solution, thus, it English, is: If I divide this 24 by 3, I will get the answer that is 24/3 = 8.
In maths concept: 3y = 24 ===> y = 24/3 = 8.

It is the interpretation of the maths symbol and its communication with the learners that is key to doing maths.

It is not difficult if you understand the language of maths.

Maths is interesting!

:D

Amazing Trigonometric Art

2009-07-10T23:46:00.004+08:00

'
Maths does wonders when presented in graphical form.

This is provided the mathematical equation makes it so.

Using the trigonometric relation, x sin x = y cos y, the artistic effect of this equation is shown below.

You can now see that, though maths can be boring at times, it can reflect its beauty through other means.

Don't you agree?

Maths is Interesting! Watch out for it!

.. :D

Application of Boolean OR operation

2009-07-06T22:24:00.003+08:00

'
Besides the Boolean AND operation and its application, Boolean OR operation also serves a special application.

For the AND operation, which is equivalent to the maths multiplication, the use is for it to pull any untied input to a system to zero state.

For the OR operation, the concept is similar, except that now it is the maths addition.
Instead of pulling the untied input to the system to zero, you can set it to the other Boolean state, which is the "1".

In base 2 number system (Boolean), if the input is 000101, and you need the data to be all 1, just OR the input with 111111.

What you will get after this OR (adding) operation is 111111.

This is so due to the fact that 1 + 0 = 1.

Here you will see that option to set the untied input to 0 or 1 can be done using either the AND or OR operation of the Boolean system.

This is maths, if you are aware.
This is the application of maths after understanding the principles of maths operation.

Any comment?

:D

Boolean OR operation | Special Maths Addition

2009-07-05T13:05:00.003+08:00

'
Boolean means 2-state operation.

OR means any one state that is valid will result in a positive outcome.

Example: When James or Mary come, the show will be start.

In maths, this translates to ADD, except only 2 states can occur (that is, on or off only).
1 = on, 0 = off (or vice versa).

What do I mean?

0 + 0 = 0
0 + 1 = 1
1 + 0 = 1
1 + 1 = 1

From the above addition, you will notice that as long as one state is ON, the result will be ON.
The operation is that of an "add".

Note: Since the operation is Boolean, it cannot go above 1 or 2 and above cannot exist. Only base 2 number (0, 1) can happen.

This field of maths is known as Boolean Algebra, a special maths operation used in digital electronics.

Caution:
If you are using this Boolean process, make it clear that the number base system is 2, otherwise 1 + 1 = 2!

1 + 1 = 2 in Boolean means that you go over the ceiling!

Another interesting maths concept, right?

Maths is interesting!

:D

Application of Boolean AND operation

2009-06-30T23:02:00.003+08:00

'
Boolean, though, of 2 states, can still be useful.

The concepts of multiplication in this AND operation can be used to perform cancellation of unknowns.

Anything times zero gives zero.
This is a common knowledge.

Here we are talking about basic fundamental maths. Nothing difficult.

Cancellation of unknowns in the AND operation means that we are able to mask out the unwanted or redefine the unknowns to a definite state.
Here the unknown become a zero, a known state or condition.

A more specific application is in the masking of unconnected inputs to a processing system. If the inputs are scanned and compared for status updates, the inputs have to be accurate or of a known status. Otherwise, comparison results become meaningless.

Here the Boolean multiplication (AND) forces the unconnected input to a zero. This is then an accurate input status for comparison.

Here, maths is applied to technical application, and its usefulness become apparent.
This is the power of understanding maths , and is exactly what makes maths interesting.

.

Boolean AND operation | A Special Multiplication

2009-06-25T22:48:00.002+08:00

'
There is a special field of algebra called the Boolean Algebra.

Here the algebra operates in the base 2 number system.

One special operation it performs is the AND operation.

What AND?

A simple analogy is " Mary AND John went to the park".
The meaning is that BOTH Mary and John moved together as a whole.
If either one is absent, they did not go to the park!

This is a form of multiplication.
0 x 0 = 0
1 x 0 = 0
0 x 1 = 0
1 x 1 = 1

Only when Both are present , the outcome becomes present.

Here you will notice that maths is applied to real life situation, forming into the English word "AND". But in maths, we call this "AND" as multiplication.

Boolean is utilised when the outcome is of 2 states (on or off, present or absent).

Do you see the interesting part of maths here?

Maths is mingled into daily events and is always around us if you keep an eye for it.

... :D

Percentage Humor

2009-06-23T23:38:00.003+08:00

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Father: How many marks did you get for your math test?

Son: I obtained 100 marks!

Father: Great! You have done me proud. You deserve an ice-cream.

Son: Thanks Dad!

Son to brother: Actually my math test is over a total of 200 marks. I almost failed the test! Luckily dad did not master percentage, otherwise I would not get my free ice-cream.

(To understand percentage, click this link).

..

What is Percentage?

2009-06-16T22:09:00.003+08:00

'
Many people do understand what percentage is about.

If you see a sales offer with 50% discount, you will know that it is cheaper by half.
If the offer is with 40% less, you will also realise that is a good deal since it is almost half the original price.

But what is this percentage in detail?

A mark of 4 / 5 is reflected as a fraction.

A learned maths person will understand that it is (4 /5) x 100 = 80%

If you tell a student that he achieved 50 marks. Is that enough?

The information will not be enough as the total score is not known (unless using the default told before hand).

Thus a mark of 50 upon 50, and a mark of 50 upon 100 means different story altogether.

I believe you will agree totally!

A percentage will always reflect better information since 80% means 80 / 100.
The base of 100 is taken as the default.

The actual base number is immaterial in this matter.

An information of 50 marks compared to 50% showed the power of using percentage.

For absolute marks, you need to tell the total marks to form a complete message.
Using percentage, the fraction part of the calculation can be ignored.

Percentage and fraction are related. But percentage used a common base number (100) to commonise the value.

For comparison sake, percentage, thus stand a more proper way to tell the result.

Example:

Is it easy to tell the closeness between 49 / 56 and 46 / 56,
or is it easier to tell between 78% and 81% ?

The answer is obvious, I hope.

Maths is interesting, and mastering simple concept makes maths learning even more interesting.
Happy learning.

Cheers! :D

Meaning of Minus

2009-06-08T21:48:00.002+08:00

'
To the ignorant, a "-" or minus is a strange symbol.

Is this more so when it is expressed as "-6", "-5 km", "-$3", etc.

What is then the true meaning of this "minus" sign?

Answer: It indicates a reversal of action of intention.

Maths is a useful tool to help explain this concept.

Example:

When a car moves forward by 5 km, it moves +5 km. ( By default, no sign means positive)
When it reverses by 5 km, it moves -5 km. The direction of movement reverses!

When a person gain $3, he has +$3 in his pocket.
When he loses $3, he has -$3.

From the 2 examples above, you can see that the "minus" sign is a reversal to the default.

If you "reverse" and then "reverse" again, you find yourself in the positive direction.
(-1) x (-1) = + 1

If you reverse 5 times, 5 x (-1) = -5 . You are facing the reverse to the default starting direction.

Now take note of this coming information.
If you reverse the car by 5m and another 5m, you reversed in total (-5) + (-5) = -10m.

If you reverse the car by 5m, followed by changing the direction and moving by another 5m, you moved (-5) + (+5) = 0m.
(Reverse direction followed by forward direction).

Does all these "reversing" cause a daze in you?

Do not despair.
Message is "When there is a reversal, put a minus in front of the number". Simply that!

I owe you $5 ==> -$5.
I gained $5 but lost $3 ==> + $5 + (- $3) = $2 (Action followed by another using "+").

Understand?

Hope this post on minus sign reduces your anxiety about this little maths symbol.

Cheers :-)

.

Power Law of Logarithm Explained

2009-05-31T21:49:00.002+08:00

Logarithm study has a few formulae that are important and key to solving math questions.

By remembering them , you will be in line to solve logarithmic problems and, maybe, fast too.

However, what if you forget them?
Does it mean that you are not able to solve the question regardless of speed?

Do not despair.
As long as you are able to manage the 2 basic laws in logarithm, you are safe.

The product law and the quotient law are must for any students.

Why do I say that?

Let us take the Power Law and do a review.

n log x = log X ⁿ

Why is it so? What if you forget this law? Any problem?

These are the very queries any new learners exposed to logarithm will ask.

First allow me to go through the product law of logarithm.
log (XY) = log X + log Y

Here you see that the product of "X" and "Y" in logarithmic operation, becomes a "sum"of the individual logarithms.

4 log Y = log Y + log Y + log Y + log Y (adding up 4 of the log Y)

Using product law, you know that these 4 terms can be combined to log (Y x Y x Y x Y).

log (Y x Y x Y x Y) = log Y⁴

Now, you see, through the product law, you are able to equate the 4 log Y into log Y⁴,
meaning, 4 log Y = log Y⁴.

You see that you did not utilise the Power Law here,and yet is able to form this formula!
Amazing isn't it.

What is the message here?
The message is that, when you have the basic understanding in logarithmic principles, you will be able to twist and turn any given problems to come out a solution.

You had used the basic product law to discover this unique Power Law.
It is similar to other laws and also can be expanded to cover other maths topics too.

Do enjoy maths.
Do discover more exciting twists it presents wwith a bit of thinking.

Happy learning :-)

.

Learning Math Topics in Isolation

2009-05-09T10:34:00.003+08:00

Learning encompasses linking with other area and related topics.

Learning thing with disregard for other is alright for the sake of triggering the mind. But does it benefits more if linked to others?

Does indices related to quadratic equation?
Does multiplication relates to addition?
Does complex number relates to algebra?

All the above questions are common in the mind of a math learners.
If you do not have these questions along the learning phase, something is very wrong.

Learning math in isolation is similar to living in an isolated island all by yourself.
You do not know what is happening in the world.
You do not know if there is famine somewhere, or swine flu going round, or a plane crash near you.

Math has to be done with linkage to many other mathematical topics. It cannot be done in isolation.

Math is a tool that solves real-life problems. With mastery of various mathematical concept and relation among them, you will be better prepare to solve more problems.

Many a time, you will come across students who just study topical math without knowing that they can apply what have been taught to them previously.

They start fresh when a new topic is introduced.
Algebra is different from complex number.
The addition in complex number is done differently from that done in algebra. That's what they assumed, since the heading is different!

Interesting learning ways, right?

That is human nature, to be frank. Only when you are told, sometimes, otherwise you will not know it. Adults learn through experience that this assumption is pulling you down.

The ability to link many things together is a very beneficial skill to permanently internalise.
This does not point to math alone. Others apply.

Math can be tough if learned using an improper learning method.
One good technique is the linking technique where you will see yourself happily doing math, being able to apply and solve questions using previous and current taught concepts.
It motivates you.
This is the STARTING point if you are unaware. This is the point where it decides whether you can sustain math learning.

Learn wide and later deep into math. But start with the correct footing. Link as much to previous as possible. It will be a sure way to happy math doing.

:-) I like math!
.

Counting Down With Base Eight

2009-05-02T23:56:00.002+08:00

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Counting up is an easy task for anyone, even when the base is not ten.

You may refer to this link for counting (up) with base 8.

However, counting down may be slightly harder than normal, especially when dealing with another number base other than 10.

Let's try with base 8 for a start.

10
03 -
----
??
---

Here a count down of 3 from 10 is needed.

How do you go about it?

Thinking of the way base 10 subtraction was dealt with, this is similar.
After all maths is the same. It is the technique that is important.

When you encounter a " bring back" from the upper (rightmost) digit to the lower leftmost digit, you will, for base 10, add a 10 to the "ones" digit.

Here, with base 8, you will do likewise, except that now it is adding the number 8 to the leftmost digit.

Thus, 0 + 8 = 8.

And 8 - 3 = 5.

10
03 -
----
05 (Answer in base 8).
---

There is nothing difficult if you look at the technique or concept in counting up or down.

Maths is just playing with methods to resolve numerical issues.

The above is a good example. Hope you agree?

Have fun counting in other number base.
Cheers!

.

Careful With The Use of 2 and Z in Maths

2009-04-22T23:02:00.003+08:00

Mistakes are normal in maths. This is during the learning stages.

It will not be acceptable during tests and examinations. This everyone knows!

But human do make mistakes. This is a fact. Therefore, we have to know our weakness and avoid moving towards it (them).

A very common mistake made while doing maths is to use the variable "Z". From the writing, you can see that it is very similar to the number "2".

During examination, we are tensed up and our mind may not see what the eyes received.

2Z may end up as 22 finally. This "22" will then be used for computation and of course results in a BIG shock!

Thus knowing this danger, avoid using variables that are close to number in writing, unless stated by the maths question itself.

Do not choose a similar looking variable and end up with disappointment.

"b" and "6", "l" and "1" or "S" and "5". These are dangerous combination.

Thus look carefully when dealing with these numbers and variables.

Do not cause unnecessary mistakes. Save your effort to deal with better challenging thinking.

:)

Counting With Base Eight

2009-04-18T22:45:00.002+08:00

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Number system has an important factor attached to it.

It is the base of the number.

The base defines the quantity of item (in this case, the counted number) before the sequence repeats.

Take for example, base 8.
The count is: 0, 1, 2, 3, 4, 5, 6, 7, 10, 11, 12 ....., 17, 20, 21, 22, 23,.......

The number "8" does not appear in the base 8 number system.

Once the 7 is reached, the next number increments to 10.
That is, the range is from 0 to 7 only.

How about the addition?

Example 1:

05
03 +
----
10
----

Since 5 + 3 exceed the maximum count of 8, the final added answer is 10, the start of next sequence.

Example 2:

14
05 +
----
11
----

It can be seen from the 2 examples above that the second digit in the addition is added with "1" after the first (right-most) digit shoots over "8".

This counting technique is nothing different from our normal base 10 (decimal) method of addition.

Thus, knowing the meaning of the base in the number system helps in proper counting, and includes addition and subtraction with the respective base.

It is nothing complex and abstract. Just counting with the correct quantity of number.

.

Time Calculation May Not Be Easy

2009-04-13T23:23:00.002+08:00

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Counting from 1 to 100 is normal for anyone. Just increment by 1. It's that simple.

But counting time may be another story for young math students.

Why is this so?

There is the seconds, the mintes, and the hours to handle. They differ by the umber 60.

The "carry over" through addition is the number SIX!

1 min 30 sec added by 40 sec gives .......

1 min 70 sec?

Here the 70 sec includes 60 sec + 10 sec. You need to understand that for time, 60 sec means one whole minute.
Thus, 70 sec = 60 sec + 10 sec ===> 70 sec = 1 min + 10 sec.

Therefore, 1 min 30 sec add 40 sec gives ==> 2 min 10 sec.

Compare this to adding 30 by 40. Answer = 70. No more analysis.

Time base on 1 hour = 60 min, 1 min = 60 sec concept.

To test true understanding of addition (and subtraction), time is a good gauge and tool to assess learners.

Test it on young kids today to annoy them... *#^&@!
Cheers!

.

Common Mistake Of Gaps and Length

2009-04-11T23:22:00.004+08:00

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There are some maths questions that will be out to catch the careless learner.

A common one is that which require you to calculate the distance or length given the gap of items.

Look at the diagram below for an example.

Here, can you find the length from the left-most pillar to the 8th pillar, given the distance from start to 3rd pillar is 30m?

Solution: (Wrong slip-of-the-mind working)

Since the distance is 30m for 3rd pillar, answer to 8th pillar has to be 80m.

Seems to be right and logical. ==> Careful here!

Why?

Look at the step distance in between pillar. It is 30m / 2 = 15m.

As the gap between pillar from start to 8th pillar is only 7 gaps,
the actual correct distance is 7 x 15m = 105m.

Interestingly tricky question, right?
Be careful and alert for this "step" or "gap" maths problem.
Slamp down this carelessness, and mistake will eventually disappear (for this type).

.

Constants and Variables

2009-04-10T22:10:00.003+08:00

In maths, you will encounter many "letters". This "letters" are the symbols used to form equations and mathematical expressions.

("letters" here may mean symbols like the theta, beta, etc)

Some of these symbols are constants and some represent variables.

Knowing what are constants and variables is crucial for mastery of maths learning.

What are constants?
Constants, as the word literally means, are items that have number that never change.

What are variables?
Variables are symbols that changes in value.

Example A:
y = 3 x + 2

y and x are variables, and 3 and 2 are obviously constants.

Example B:
log x = 5^y

x and y are variables , and 5 is constant.

Example C:
y = mx + c ( for straight line equation)

y and x are variables, and m and c are constants.

This may be confusing to some maths students when they start plotting graphs.
Here, the straight line is continuously moving with the value of x and y.

So why is the "m" identified as constant?

You need to know that "m" represents the "GRADIENT" of the line.
The line has the same slope at any value of x and y.
Thus "m" is a constant.
This is a typical concept that commonly goes wrong.

Therefore when you really understand what changes are considered "variables" and those that remain stable are known as "constants", you are in line for good maths study!

.

Algebra | Moving Forward in Usage

2009-04-02T17:18:00.002+08:00

'
In the study of algebra, symbolic representation of number or unknown is key concept to solving mathematical equations.

The letter "x" or "y" are examples.
Other symbols can also be used as long as the usages are understood.

In the expression, x + 0.5 = 3.
This meant that the unknown "x" added to 0.5 will give a total of 3.

"x" here is nothing other than an unknown item to be solved.
It should be a number that relates to that maths equation. Nothing more, nothing less.

Another example:
x² + 2x - 1 = 0

This "x" again is an unknown number to be found out.

Thus this algebraic expression and its "x" are just mathematical item representing a relationship.

Many students learning maths, when faced with this "x" always look puzzled.
With this post, the queries of this "x" (or "y", etc) should be cleared.

With this knowledge of the symbolic representation of unknowns, other areas of maths can be explained easily.

Topics like the trigonometry and logarithm will be expanded from this symbolic concept.

Cos A and log B will thus be finalised to a number, with this "A" and "B" yet to be solved.

Equation like
cos A + 2 = 2.4
will then be nothing more than to relate this unknown "A" to the expression.

It is also an easier way to explain and express this relationship between the unknown (A) and the other number (2 and 2.4).

Similarly,
log x + log 2 = 3
means that "x" is related to the 2 and 3 according to the given equation.

From the above few examples, the question now of what really is this "letter" doing forms meaning, right?

Maths starts off easy when this concept is clear.
Alot of the maths study involves this simple "trick" of presenting unknowns.

You notice how clever past mathematicians were now?
The use of simple symbol to pass off as number to carry on with maths solving.
Without this algebraic presentation, maths will not be as interesting as now.
Alot of guessing will have to be done and .... guess what? Maths will be HELL then.

Enjoy this symbolic concept in maths.
Enjoy your maths.

:-) (-:

Application of Algebra

2009-03-27T23:16:00.002+08:00

'
I have seen lower primary school kids learning mathematics.
They are exposed to many logic "games" which tested their mathematically analytical skill.

One of them is the math Word problem topic.

Here they are always given a scenario and asked to give an answer.
They are not taught algebra, however.
The expectation is for them to think out logically.

This is good in a way.

But along the journey of learning mathematics, they will sooner or later be told of an exciting area called the "Algebra".

Here, algebra comes in helpful for those who did not do well in the logical word problem questions.

Why?

It is because, in algebra, the unknown can be replaced by a symbol, normally a letter.

This solves the poor kid the time to "guess" the answers, with iteration of checking and re-trying at times.

With the use of algebra, the kid can attach the unknown to a letter and proceed with the calculation.

When this algebra concept is not mastered at a later stage while studying math, the learner will face tremendous obstacles along the way. The meaning of the "letter" will be an alien to him, not knowing the power of its usage, and thus the magic of algebra application.

Thus, in conclusion, any math student has to die-die, managed simple algebra in order to have a good time learning math.

Hope this advise and information helps.

:-)

Not Knowing Maths Is OK

2009-03-16T22:39:00.003+08:00

.
Anxiety is caused by not being able to fulfill your desire but strongly wishing for it.

Maths anxiety is likewise. Wanting to master maths but is unable to grasp the concepts.

Don't worry. It is not the end of the world!

Maths is just a tool for you to solve problems easily.
Maths lets you have a systematic approach in solving questions.

But if you do not understand maths, does it mean that you cannot solve problems?
You still can, but maybe only through more guessing (that is, non-systematical).

While learning or doing your maths homework, forgo the idea to quickly master the topic.
Forcing your learning through at a faster pace than you are able to handle does not serve any purpose.

Learning needs time to digest and analyse information. It is just like eating a meal.
Eating too fast will get you indigestion!
Same to learning maths, as well as other subjects.

Just understand that not knowing maths is OK.
Knowing maths is a privilege, a bonus.

With this mindset, you will find maths interesting.
It is a tool only for helping you find answers in an "education" and impressive way.

Life still goes on without you knowing maths.
(In fact, you actually use it, except that you did not know that it is called maths!).

Does this message make you feel better?
Hope it does.
Let maths be your slave.
And not you being the slave to maths.

Cheers! :D
.

Simple Way to Master Indices Maths Question

2009-03-12T22:57:00.003+08:00

.
Have you seen many mistakes like the below?

Question 1:
3^x = 27

Solution:
3^x = 3⁴ ===> x = 4

Question 2:
9^x = 3³

Solution:
3^3x = 3³ ===> 3x = 3 ===> x = 1

Why the error?

A simple explanation is that the maths learner is not familiar with the basic multiplication of repeated numbers.

3 x 3 = 9
4 x 4 = 16
5 x 5 = 25
6 x 6 = 36
7 x 7 = 49

2 x 2 x 2 = 8
3 x 3 x 3 = 27

2 x 2 x 2 x 2 = 16

etc.....

Once you have mastered this basic repeated multiplications, you can rest assure that indices question will not be there to haunt you.

How about solving "x" in this 9^{x + 1} + 2(3^x) - 3 = 0 ?

I bet that if you understood the above criteria of learning indices, the equation can be easily solved for x (using quadratic formula as a hint).

All complex things start off with simple things.
Do you agree this applies to maths?

:D

Application of Maths - A Surprising One !

2009-03-10T22:41:00.003+08:00

It just came to my mind that maths is a special subject.

Why do I say that?

How do you gauge whether you have mastered a subject or topic?

Your teacher will award you marks for the assessment done to check your understanding, right?

What is this process? It is maths!

Maths learning is being monitored through itself!
Applying maths to learn maths.

Giving marks is counting, logical thinking and judging. These are related to maths.

Thus you see that maths is interesting in that it checks itself like no subject does.

*** : - ) ****

Math Challenge 17

2009-03-06T22:38:00.004+08:00

This is one that I came across and found to be exciting.

You have to solve using the number 1 to 9 only once.

_ _ _ _ _

0 _ _ _ _ (minus)

3 3 3 3

Any one dare to try?

Quadrant Identification for Trigonometrical Questions

2009-03-06T21:50:00.002+08:00

For question regarding trigonometry, quadrant is one of the key parameter to obtain correct answers.

What is this quadrant about?
A complete cycle (360 degree) is divided into 4 quarters.
They are zones defined for specific trigonometric functions.

The first quarant (0 to 90 degree) gives positive sign for ALL trigonometric functions.
The second quarant (90 to 180 degree) allows only "sine" to have positive number.
For the thrid quarant (180 to 270 degree), "tangent" has positive number only.
Lastly, the fourth quarant (270 to 360 degree), "cosine" gives positive number only.

So, you can see that given a sign of a trigonometrical operation, the specific quadrant can be found or identified.

Example:

sin X = - 0.5 ===> Identifies quadrant as 3rd and 4th.

tan X = 0.2 ===> Identifies the 1st and 3rd quadrant.

This is simple, right?

However, do note the below example.
It causes a mistake that is common!

Example of potential error:

sin 2X = -0.5 ====> which quadrants ?

The answer is not that direct!

Why?

Now the math question is not on "X", but on "2X".

To identify the quadrant, you need to start off from the "2X", working as per normal.
But, after identifying the 2 quadrants, you have to compute the "2X" reference angle.
Using the reference angle, you have to obtain the 2 angles.
After which, you need to divide the angles obtained by 2.
The divided angles is then the final angles lying within the quadrants.

Confused? Never mind. See the numerical solution below.....

Solution:
2X = sin^-1 (0.5) = 30⁰
This is the reference angle used to compute the actual answers.

Final answers are (Quad 3)= 180 + 30 = 210⁰
and (Quad 4) = 360- 30 = 330⁰.

Common mistake is to obtain reference "2X" angle and straight away divide it by 2.
Using this newly found "X", you proceed to identify the angles of the quadrant identified using the "2X". THIS IS INCORRECT!

Do not confuse double angle with single angle.
When the problem is "2X", solve all the way using the "2X" first until reaching the end.
After which, you then divide the angles by 2 to get to the final answers.

Maths is simple if you follow the rules accordingly.
If you mess up double angle with single angle while solving, you just literally mess up the workings.

Maths forces you to follow rules set out. It punishes only if you do not obey orders.

Maths is interesting isn't it? Never expect that maths can police your behaviour while practicing it, right?

:-)

Careless Algebraic Mistake

2009-02-25T22:45:00.002+08:00

There are times when simple algebraic operations are confused by introducing trigonometric functions or logarithmic terms.

Example:

13 = 7 - 3x

This can be easily computed to be
13 - 7 = - 3x
==> 6 = - 3x
==> x = -2

But how about 13 = 7 - 3 tan X ?

Solution: 13 = 4 tan X ==> tan X = 13 / 4 , ..... and got into hot soup!

Why?

A careless mistake has been made.

When tan X was substituted into the original equation, the eyes refused to acknowledge this "complicated" tan X.
The eyes can only see the simpler "7 - 3" and thus compute it to be (7 - 3) = 4!

This caused the 7 - 3 tan X to be 4 tan X, which is WRONG.

The correct mathematical process of solving should maintain.

Therefore,

13 = 7 - 3 tan X
==> 13 - 7 = - 3 tan X
==> 6 = - 3 tan X
==> tan X = -2
.......

Maths is not that complicated when you follow the rules closely, even when the terms have changed into a seemingly complex expression / term.

By following what you have known with simple expression / term, any challenging equation can be easily solved.

This is the power of learning maths properly.
Being discipline in the way you handle maths is the key.

With a discipline mind, maths becomes fun , .. and interesting.

.

Algebra Is Useful

2009-02-22T22:04:00.003+08:00

.
There are everyday events that requires the use of algebra.

Solving simple math question with unknowns can be done easily with algebra in mind.

Take the example of the math challenge 15 given by clicking this link.

What the challenge requires is the addition of a pair of 2-digit number obtained from a 4-digit number.
The higher 2-digit number is to be added to the lower 2-digit number to obtain the centre 2-digit number.

Example:
1978

Upper 19 is added to lower 78 to produce centre 97.

In that post, you are to come out with more examples of this type of 4-digit numbers.

Use of Algebra can easily solve this cahllenge.

How?

Here it goes...

As in algebra, let's assign "letter" to each digit of this 4-digit number ==> abcd

The upper pair is then 10a + b, and
the lower pair is 10c + d.

Adding them up gives, 10a + b + 10c + d = 10b + c (this is the requirement)
==> 10a + d = 9b - 9c -----(A)

Also a + c + 1 = b ==> a = b - c - 1 -------(B)
and b + d = c + 10 ==> d = c + b + 10 ----(C)

Here, it is necessary to assume b + d >10, since otherwise negative number relation will appear.
(If you find this statement tough, never mind, and read on..)

From the above 3 equations formed, you will then be able to randomly choose numbers that fit them.

You will now appreciate the usefulness of algebra in solving this math challenge.

Enjoy!
.

Logarithm | Common Mistake

2009-02-17T22:33:00.002+08:00

A common mistake occurs normally during simplification to a single logarithm term.

Question like,

" Simplify log X - log Y + log Z into a single term "

catches many students who are careless.

What is the error or mistake made?
- Doing the solving at one go when not familiar with the logarithmic rules
- Sign interpretation

Wrong answer given: log X/(YZ)

Correct answer: log XZ/Y

"log Z" is commonly taken to follow the previous log term, which is, "- log Y ".
Since "- log Y " causes the "Y" to be a denominator, "Z" is also taken to be a denominator too!

This is a mis-cue. A mental slip, mathematically.

Advise:
Look at the sign carefully before jumping to conclusion.
Go slow in the combination to a single log term.

Remember the idiom: "Slow and steady wins the race"

You can apply this to log simplification when you are new to it.

Cheers! :-D

.

Mathematics Letter S

2009-02-15T14:58:00.004+08:00

Using graph and equation, you can create wonders.

Graph does not only mean lines and curves. It can be "letter" too, as seen below.

With creativity and a bit of trying, you can have surprising images formed through graphs.

With mathematics, you are not limited to equation and solving problem. You can have fun and that makes mathematics interesting.

:-)

Math Challenge 16

2009-02-13T22:41:00.002+08:00

Jane started off a multiplication process with the number 4.

John followed up by multiplying it by 4 also. This led the result to be 16.

Jane, being her turn now, times the current answer by 4 as required. The answer now is 64.

After a number of alternating multiplication by Jane and John, the result became 4 ,194 ,304.

Guess who did the last x4 operation (without using the calculator or equivalent).

.....

The answer is not as difficult as seems to be. Just a simple stare at the alternate answers will solve the mystery.

Happy thinking.....

;->

Complex Number | Common Mistakes (2)

2009-02-08T12:17:00.007+08:00

Complex number consists of 2 parameters.

They are the modulus (length) and argument (direction).

Argand diagram is the pictoral form of representing this complex number.
In the Argand diagram, quadrants define the position of the "complex" line.

Z = a + ib (click for information) is the general form of writing the complex number.
"a" and "b" will define its polar counterparts, modulus and argument.

Having 4 quadrants in the Argand diagram means having 4 combinations of "a" with "b".

They are:
1) Z = a + ib
2) Z = -a + ib
3) Z = -a - ib
4) Z = a - ib

The first case lies in the first quadrant (Q1).
The second case lies in Q2.
Third case lies in Q3, and
fourth case in Q4.

Therefore knowing the sign of "a" and "b" let you know which quadrant the complex line lies.
And this is where the mistake lies!

The argument is always computed wrongly for the Q2, Q3 and Q4.
Only the positive sign of "a"and "b" is taken to get the value of the angle (argument).

Example of error:
Z = 5 + i5 ==> Argument = +45⁰ (Q1)

However, Z = -5 + i5 ==> Also taken as + 45⁰ forcing it to lie in Q1 (wrong!).
It should lie in Q2 since now the real term is negative.

Values of "a" and "b" are not the only parameters needed to find the angle.
Their signs are equally important!

Advice:
Think of the Argand diagram representation before the definition of the angle.
This will ensure that the angle is correctly calculate later on since you have an idea which quadrant the complex line should lies then.

Doing the complex number and its conversion from rectangular form to polar form properly will make you happy and like maths. Proper thinking process will path you into a good habit that leads to confidence in maths.

Maths is Interesting! And fun ...

:D, Smile.

.

Geometrical Errors Can Be Exciting

2009-02-07T22:55:00.004+08:00

The learning of geometry is to allow anyone to have an idea of correct perspective to objects.

Dimensions are important in this area. Angle of view is equally important.

Drawings of object in the fore-ground and background differs because of geometry.

When the concept of geometry is violated in drawing objects, you will get interesting outcome.

This outcome, however, is apparent only in the virtual sense and cannot be physically produced.

Example of links are quoted below:

http://anythinggorgeous.blogspot.com/2009/01/staircase-of-character.html

http://anythinggorgeous.blogspot.com/2009/01/cunning-overlapping-of-objects.html

http://anythinggorgeous.blogspot.com/2008/12/twisted-rods-or-fishy-rods.html

Geometrically deceptive objects are not easily identified, and have to be closely stared at to reveal their "wonders".

Only through learning maths and its relevant topics, you can then appreciate the importance of having done it. This is very obvious in the above few examples.

.

Math Challenge 15

2009-02-07T22:38:00.002+08:00

This is an interesting one.

Given a special four-digit number, 1978,
we can get the centre two digit (97) by adding
the left-most 2 digits (19) and the right-most 2 digits (78).

Another of this special number is 1538.

Can you get 2 more of this special numbers?

Brain-raking isn't it?

To be frank, it is not that difficult.
Hint: Algebra can solve this maths thinker.

:-)

Math Challenge 14

2009-02-01T22:54:00.002+08:00

Sometimes when you go for shopping, you may encounter items on offer.

10% discount, 36% discount, ......

So if a $18 toy is put up at a discount of 25%, and another toy, priced at $25, is at a discount of 18%, which has a bigger reduction in dollars?

Do not use calculator, though, to answer this challenge.
Otherwise all the toys will be gone!

Be quick.....

Answer please.

:D

Math Challenge 13

2009-01-31T23:03:00.003+08:00

Once upon a time, there are three numbers. They like to play with each other mathematically.

One day, they decided to add themselves up to see how big they can become.
They kept the answer for future reference.

Another day when they met, they decided this time to multiply themselves.
They got a huge surprise. The answer remained the same as when they added up.

Question: What are the 3 numbers?

.

Complex Number | Common Mistake (1)

2009-01-29T23:15:00.003+08:00

Multiplication of complex numbers remains the same as done for normal algebraic operation.

However, due to complex number having 2 terms, namely, real and imaginary terms, care has to be taken for the "i"unit.

This is specially so when multiplication of conjugate is involved.

A popular mistake made while doing this form of multiplication is:

(3 + i2)(3 - i2) = 3² + (i2)²

What is wrong?

The concept of conjugate and its multiplication states that:
(a + ib)(a - ib) = a² + b²

The "i" symbol is NOT reflection in the final outcome!

Only the "a" and the "b", the numerical part, are extracted out for computation.

Taking the "i" into account will cause the sign of the last term (i2) to be incorrect.
This is because i² = -1.

Therefore, regardless of the sign in the multiplicands, just pull out the numerical part in the complex number and use them for calculation, that is, the 3 and 2 in the example above.

The correct answer, thus, is (3 + i2)(3 - i2) = 3² + 2².

Looking carefully at the application of the formula, you will notice that this is a simple and easy technique to do conjugate multiplication.

Message: "Touch me not" i said.

Mastery takes place when we do not repeat mistakes.

.

Using "i" Imaginary | 2 key functions

2009-01-28T21:45:00.003+08:00

Complex number consists of both real and imaginary terms.

The imaginary term utilised the letter"i" as an operator.

This "i" is a special element in mathematics.

What does it do?

It has 2 key functions:

1) It can change a real term or value to an imaginary term, and vice versa.

Example:
Given Z = 2. If you perform an "i" multiplication on this Z, you will get i x 2 = i2.

The real number, 2, became an imaginary term, i2 !

Likewise, an imaginary number i4 multiplied by "i", gives you a real number, i4 x i = -4 !

2) It can rotate a target by 90 degree anti-clockwise.

Example:
Given Z = 2 (lying on the horizontal axis at 0 degree). By multiplying an "i" to it, the number changes to i2, which means lying on the vertical axis at 90 degree from the original.

The "i" works as a rotating operator on its target.

Interesting?

The cheeky, little "i" can make number change direction as well as characteristics; a real number into an imaginary one!

:-) (-:

Complex Number Explained (2)

2009-01-27T21:56:00.004+08:00

Complex number consists of real and imaginary terms to cover the concern that not all real number can satisfy any maths question.

An example was provided in this post (click the link for information).

What is then the different between this Real and Imaginary term?

In graphical form, the Real term is presented in the horizontal axis whereas the Imaginary term is along the vertical axis.

This is purposely so to has no impact of the Imaginary term on the Real term.
(Think in term of the cosine aspect of a pure Imaginary axis).
The overlapping portion of the Imaginary vertical part is ZERO on the Real axis.

See the diagram below for understanding.

The concept of this diagram is to allow learners visually see that the Real and Imaginary terms are unique in themselves and have no link to each other.

But another issue appears.

What is it?

Looking at the diagram, you will see that the complex number defined as Z = a + ib,
where "a" is the real term and "ib" is the imaginary,
will create directional value.

Some solution to specific maths problem requires the complete a + ib format.

Thus the introduction of complex number to offset the impossibility of solving equation using only real numbers, forces the angular dimension into the answer.

With this angular dimension coming into the mathematically picture, the principles of quadrant, as in the trigonometry studies, will be utilized to identify the various answers.

Complex number is then made "complex" mainly due to this directional information added to the normal Real numbers.

Do not be frighten off by this new addition, as, if you know very well how it comes about, you will welcome it. This imaginary term helps you solve many interesting maths equation that normal working cannot.

Love this complex number.

.

Complex Number Explained

2009-01-25T22:58:00.002+08:00

Mathematician believes that all expressions in maths have solution.

But there are equations that seems to be out of sort.

No solution looks fitting.

One example is shown below.

Example:
x - (x + 1)² = 2

Here, by pure comparison without touching on mathematics, you can deduce that x is definitely smaller than ( x + 1).

Furthermore, what if the (x + 1) term is squared!

x is surely smaller than (x + 1)² by this logically deduction.

Now the question, can x - (x + 1)² be a POSITIVE number?

You will fully agree that it is impossible.
A smaller number minus a bigger number will give us a NEGATIVE outcome.

Then how do you get the answer to the above expression?
What is the "x" value that produces a positive "2"?

There is no way for any REAL number to satisfy this!

To solve this type of "impossible" equation, you need to venture into the "Complex Number" concept. Since real number cannot meet the criteria to resolve the maths question, you need to imagine a number to meet this task.

"Complex Number" consists of number formed by a REAL term and an IMAGINARY term.

It is this imaginary term that will give you an answer to the challenging question.

With the understanding of "imaginary number", you will be in a better position to appreciate the usefulness of solving any maths problem with complex number.

Maths is interesting, right?

When you cannot get an answer in the normal sense, you imagine a number!
What a way to get an answer.

. .
U

.

Simple Factor Multiplication

2009-01-24T21:36:00.003+08:00

Multiplying is simple.

What is 4 x 3?
Answer is 4 x 3 = 12.

Simple?
Sure it is.

How about y(y - 1)?
Answer is y² - y.

Again simple? Sure.

But how about (y + 1)(2y + 3)?
Many of you may find this simple and basic.

But you may still come across some who did not grasp this factor multiplication.
Mistake still occur for this maths operation involving factors.

What is the mistake commonly seen?

(y + 1)(2y + 3) is given as (y)(2y) + (1)(3).
First term multiply by first term, second one multiply with the second one. That's all.
This is incorrect mathematically.

This is a misconception of what multiplication does.

Let me explain.
(y + 1)(2y + 3) can be interpreted as (y)(2y + 3) plus (1)(2y + 3).
This is key to this form of maths operation.

The second term (2y + 3) is multiplied by the first term "y" of the first factor (y + 1).
(2y + 3) is next multiplied by the second term "1" of the first factor.
The result of these two operations are then added up, since it is y add 1 (as reflected in the first factor).

The correct answer is then:
(y + 1)(2y + 3)
= (y)(2y) + (y)(3) + (1)(2y) + (1)(3)
= 2y² + 3y + 2y + 3
= 2y² + 5y + 3

Learn from the mistake, and do not repeat it.
This is the basic concept in learning from mistakes. They are our teacher.

Remember, maths is interesting!
A twist can be destructive or constructive.
That is where maths is special and challenging.

Cheers! :-)
.

Indices | Interesting Mistakes (3)

2009-01-22T22:48:00.002+08:00

If you are not careful with the base and power in a mathematics index expression, simple mistakes will occur. These simple mistakes will expose the weakness in your basic understanding of indices concepts and principles.

But, although, mistakes do occur, rest assure that learning from them is a good thing to have in the learning process.

What is the common mistake student normally make when doing indices?

Here is an example:

y = 9^{x + 1}

y = (3²)^{x + 1}

y = 3^{2x + 1}

You see the error here?

Yes, the power to the base 3 is wrongly done.

It should be 2(x + 1) = 2x + 2. The last term was left out of the multiplication.

A good way to prevent this mistake, or slip-of-the-mind error, is to use parentheses.

Using parentheses at the (x + 1) index will visually group up the "target" for multiplication by 2.

The correct answer: y = 3^{2x + 2}

Take note of this simple maths error and you are on the way to a happy maths learning journey.

Cheers! Making maths interesting goes a long way.....

:-)

Indices | Interesting Mistakes (2)

2009-01-17T23:14:00.003+08:00

For maths problem related to indices, you do not only look at the power. You have to take care of the base too.

The common mistake is to ignore the sign of the base when doing computation involving index.

A simple example illustrates the error that is very common when learning maths.

Example:
Solve 3x² - 4x + 1 = 0 using the quadratic formula.

Solution:
In solving, we need to extract out the a = 3, b= -4 and c = 1 to fill into the quadratic formula.

However, the quadratic formula requires the utilisation of the b² - 4ac expression.

Here the mistake is to fill in b² as -4² = - 16!

This shouldn't be the case. It should be (-4)² = 16.

It is a difference of the sign (positive versus negative).

Thus to handle question on indices, you will have to be extra careful on the base and its sign. Make full use of parentheses, if possible, to group and focus the targeted term or number.

In summary, index or power affects whatever it includes, base and its sign together.

. .
U

Indices | Interesting Mistakes (1)

2009-01-17T22:57:00.003+08:00

I came across an amazing mistake on the index conversion which made me thinking.

One student wrote: a^-1/2 = a²

What is the going in his brain?

He is not completely wrong. He had applied something related to the properties of indices. But had confused the application through improper usage.

This may be the result of trying to memorise the working instead of fully understanding the mathematics principles.

I suspected he may be using the idea of a^-m = 1/a^m.

However, instead of changing or converting the numerator as a whole, he converted the index only.

Though the mistake made was minor, it created something of a surprise. Many interpretation came out of the simple index law.

Although formula is given at times to aid the solving of mathematics questions, the correct interpretation and understanding of the concepts and writing of the terms has to be digested with clarity.

Otherwise outcomes may appear that amuses the teachers.

:) Maths is fun when looked at from some angle.

.

How Is Maths Interesting ?

2009-01-15T22:25:00.003+08:00

Anyone who do a taxing job will detest the job finally.

Anyone who has trouble learning a subject will lose interest ultimately.

Anyone who find a task difficult will not have interest developed for it.

These are threads of a human being and a fact of life.

When you put in effort to rectify the undesired for a good cause, you will find fulfilment gradually.

This gradual increase of satisfaction will arouse your interest for the subject or task.

The interest will develop, if the process is correct, to a liking.

Learning maths is the same.

Put in the appropriate effort and see interest develop.

Finding maths interesting will lead you to like maths finally.

You will find that maths is not that difficult after all. You will start to challenge yourself when faced with testing questions. You will like the thrill.

If you find that thrill, I can safely say that you have succeeded with maths learning.

. .
U SMILE

.

Seeing Expression As A Block

2009-01-14T23:23:00.003+08:00

In maths, the many expressions and numbers confuse the working mind when you are not alert.

This is so especially when you do subtraction.

Do you look at expression as a block or isolated terms?

Let's take an example.

If Z = 2x + 1, and A = x - 1, perform Z - A.

How do you go about this?

Do you directly work the subtraction out, like this :
2x + 1 - x - 1, or

Do you treat the A as a piece or block, like:
2x + 1 - (x - 1)

Looking at expression or numbers, requires "seeing" skill. You need to see with your mathematical mind.

Always understand that expressions and numbers alike are to be operated as a whole.
The use of parentheses is a good habit.

Parentheses can be used to group the expression or target, and make it visually clear to the mind that you are working on a piece of information.

From the above 2 ways of seeing the Z - A, you will notice that the first work-out will give a mistake that is very popular among math learners. It is always repeated even after tons of corrections.

The true mistake lies in the way you look at numbers or maths expressions.

If you can't and always make careless mistake, apply the parentheses ( or bracket) to the desired target.

Know your strength and weakness while doing maths. It will at least help reduce some careless mistakes along the way.

. .
U
Smile... Maths is interesting!

.

Cranes of Number "Four"

2009-01-10T11:11:00.004+08:00

Numbers are everywhere.

Some are obvious but some are not. They have to be discovered.

Below is one that is nicely presented.

Can you see the "number" that they form?

.

Trigonometric Form of Complex Number (Part 2)

2009-01-09T22:53:00.002+08:00

The trigonometric form of the complex number consists of 2 parts:

i) Modulus (length)
ii) Argument (angle)

An interesting question : Can the argument be different?

Example:

Z = 5 (cos 45⁰ + sin 30⁰)

Is Z possible as the trigonometric form ?

If you comprehend the concept of vector, you will understand that having different angles in an expression carries no meaning.

The complex number Z represents a line of specific length pointing in a specific direction.
(Argand diagram can be drawn to reflect this.)

With a specific direction means only ONE angle is possible with reference to the positive Real horizontal axis.

Thus it is not possible to have the argument of different values in the trigonometric form.

.

Trigonometric Form of Complex Number

2009-01-07T22:00:00.004+08:00

Trigonometric form of the complex number combines algebra with trigonometry.

Z = r(cosA + jsinA)

In Trigonometry:(examples)

cos(45⁰)= 0.707, sin(45⁰) = 0.707, and
cos(-45⁰⁾ = 0.707, sin(-45⁰) = -0.707,

cos(135⁰) = -0.707, sin (135⁰ = 0.707, and
cos(-135⁰) = -0.707, sin (-135⁰) = -0.707

The above characteristics showed that "cosine" has less impact to the overall angular interpretation of the complex number, Z.

What do I mean?

If Z = 3(cos45⁰ + j sin 45⁰), we know the angle to be +45⁰.

However, if Z = 3(cos45⁰ - j sin 45⁰), what is the angle?
"+45⁰" as in the cosine term, or "-45⁰" as for the sine term.

Looking at the trigonometric examples using positive and negative angles, the sign of the computed values having the same angle for the cosine operation remained.

Rather, the imaginary term of sinA has more information regarding the actual value of the angle.
This is so since "sine" can cause sign change to the computed value with positive or negative angles.

Thus to quickly identify the angle from the trigonometric form, we have just to look at the imaginary or sine portion of the complex number.
The cosine part will not reveal the correct answer.

An understanding of trigonometrical principles, especially the quadrant concepts, has to be strong in order to handle complex number studies.

:D Do not be overly fearful of maths. It is interesting, if you follow the ideas behind it.

:) ....