Zippy Graphical Maths

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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Decimal Number Simplification

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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.

:-)

Model versus Variable Technique

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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

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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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Maths Symbol in Our Applications

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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

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

Boolean OR operation | Special Maths Addition

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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

Boolean AND operation | A Special Multiplication

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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

Algebra | Moving Forward in Usage

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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

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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.

:-)

Careless Algebraic Mistake

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.

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Algebra Is Useful

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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!
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Complex Number Explained

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.

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U

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Simple Factor Multiplication

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! :-)
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Indices | Interesting Mistakes (2)

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.

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U

Seeing Expression As A Block

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!

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Mistake in Sign of Reshuffled terms

Look at the difference in algebraic operation for the 2 math examples below:

Example A:
2x + 4y - 3z = 3
after re-shuffling, becomes
-3z +2x + 4y - 3 = 0

Example B:
3x - 4y + z = 2
becomes, after re-shuffling,
-z -4y -3z -2 = 0

Example A can be seen to be correct mathematically, whereas, Example B isn't.

Why so?

Example B, after having the terms re-shuffled, has the signs of those terms changed!

The explanation to this sign change is that since, the terms were moved from left to right, and right to left, the sign must change. A shocking mistake has been made!

This is a mis-understanding and also a mis-conception.

What was missed out here is that the movement of left to right (or vice versa) has to cross over the "equal" symbol.

4x -5y = 1

0 = 1 - 4x + 5y <== This is correct sign change after re-shuffling across the "=" symbol.
-5y + 4x = 1 <== This is correct re-shuffled terms with no change in sign.

Message:
As long as the terms remain on the same side of the "=" symbol, the terms will not have their signs changed, even though their positions may have shifted.

The sign change results only when the term moves across the "equal" symbol, crossing over the opposite side.

Thus, do not confuse re-shuffling of terms within the same side to crossing the "equal" symbol.
This simple mistake can produce a big mistake through wrong understanding of math principles.

Math make us think properly and logically with reasoning to every steps taken. It is a good subject that aids mankind. Treasure the learning.

Cheers!

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Special Product A^2 - B^2 and A^2 + B^2

The product format A² - B² is a special form of algebraic expression.

It is equal to (A + B)(A - B).

Many other expressions, like the cos2A = cos²A - sin²B,
can also be expressed in the special product form, that is,

cos2A = (cos A + sin A)(cos A - sin A).

But, how about A² + B² ?

Can it be expressed in the (A + B)(A - B) format?

Why not.

However we need to deviate a bit from the norm.

We need to know the imaginary "i" in complex number system.
Click here for a review to it.

Since i² = -1, we can make use of this property for the special product A² + B².

Here it goes....

A² + B² = A² - (i²)(B²)

Using the basic of Indices, it can be modified to,

A²-(iB)²

This becomes, therefore, (A + iB)(A - iB).

In summary,

A² - B² = (A + B)(A - B), and

A² + B² = (A + iB)(A - iB).

The principles of the special product still holds regardless of the addition or subtraction operation between A square and B square.

Student: What an interesting twist is mathematics in this matter. Anything seems to be simple if we know the technique!

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Mistake with A^2 - B^2

In algebra, a very common mistake you can see learner making is the below:

5² - 2² = (5 - 2)² = 3² = 9

This is very interesting.

It seems to be correct. That is the problem with this form of mathematical operation.

If you are aware that A² - B² = (A + B)(A - B), then this mistake will not occur.

It is this slip-of-the-mind type of human error.
It occurs when you are not alert or too tired after too many assignment quesions.

The correct answer is 5² - 2² = 25 - 4 = 21.
Or (5 + 2)(5 - 2) = 7 x 3 = 21.
Simple?

This is why maths is interesting. It catches you when you are not alert!

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Math Challenge 12

Although math may seem difficult at times, proper use of its principles will render it simple to handle and use.

Here, a bit of algebraic juice can help in solving the below math challenge.

Following the pattern, you will observe a "trick" that you can make use to solve the challenge later.

2² - 1² = 3

3² - 2² = 5

6² - 5² = 11

11² - 9² = 40

The above equations are done without detailed written working, just simple mental calculation.

Challenge:
16² - 14² = ??

Do it without calculator.

What is the answer to the above? At the comment session, please.


Hmmm........ tick tock tick tock

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