Using Units to Deduce Maths Formula

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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Simultaneous Equations - Decimal Numbered

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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Basic Decimal Conversion

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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.

:-)

Using Equation To Create A Square Graphically

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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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Hidden Clues in Maths Questions

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. 
 
:-)

Tricky Angles | Be Aware!

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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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Purpose of Graph

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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

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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

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

Common Mistake Of Gaps and Length

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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).

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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.

:-)

Application of Maths - A Surprising One !

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.

*** : - ) ****

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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Algebra - Why the need?

In doing math, we are actually trying to use a systematic approach to arrive at an answer. It trains the brain to trouble-shoot problems for a consistent result.

Many a times, problems or questions can be solved through guessing and checking the answers to see if they fit the original questions. It is OK.

But if the answers are guessed wrongly, many iterations of the processes take place. It does sound correct or a better way should be better.

Algebra is therefore developed to handle this issue.

See the below simple question of finding the weight of a cheese.



Here, you can see that by guessing the weight of the cheese and checking it back with the other information, you can finally obtain the correct answer.

Another way is to use algebra. This is a systematic approach that when applied, will give you the correct answer on the first try, as opposed to the guessing method.

In the above example, you can replace the weight of the items by x, y, and z.
By solving the simultaneous equations thus formed, you can easily get the answer to the weight of the cheese.

That is the power of algebra and math. No uneducated guess, and time saving as a merit.

Practice, therefore , with algebra as the need is a necessity in our daily life.

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Inspirations To Learning Maths

Enjoying learning maths is a choice.

Giving your best in maths:
When you put in your best effort to learn and do maths, you will be happier.
You work hard not for your teacher, but for yourself.

When we get casual, learning gets into downward trend, and finally... collapses.
Gear up and energy flows.

Learning starts with a positive mindset. " I love maths". " I can handle maths".

:) :)

Math As A System Modelling Tool

A huge system consists of smaller sub-systems with functions related to the system operation. Sometime the function of the individual sub-systems are known, but the function of the system as a whole is not clear. We just know that given a certain set of inputs, we will get another set of outputs.

But what are the constraints and limits to the system?

What is the strength or weakness of the system?

These are typical questions that system engineer need to know.
How then does he know?

It is through modelling that we can get the answers to the scope of the system operation, its weakness, strength and limits.

Many techniques address this modelling issue, but the simplest is the Math Model.

Here, mathematical function or expression describes the relationship between the input and output. A complex drawing of sub-systems can be simplified into a single block indicated by the Math Model.

Let an example illustrate the usefulness of Math Model.


A negative feedback audio amplifier consists of a main amplifier (A) in the forward path with a feedback circuit (X) that returns a portion of the output signal for control purpose. The main input signal is fed into a circuit that subtracts the feedback signal from the main input signal. (This is the detail operation of the system involving its sub-systems).

To simplify this model, math is used to summarise the overall function. Math is therefore a system modelling tool, in this instance, to describe relationship between the input and output.

Brief explanation on derivation of math model expression:

Using algebra, we can see that the direct input to the amplifier (A) is V1 - XV2, and the output is V2.

The ratio of the (output of A) / (input of A) is V2 / (V1-XV2) = A.

Re-arranging the above expression, we will get V2/V1 = A / (1 + AX) which is the system gain ratio of the complete negative feedback amplifier.

This final ratio is then the Math Model of diagram 1 and is shown in diagram 2.


Diagram 2

After getting this simplified version, system analysis can be carried out to test the extent of its operation and discover its limits. Without this modelling, the testing will be tedious and time consuming.

From the above example, you can see the usefulness of math and its application as a system modelling tool. From an abstract case, math understanding can be used to solve an real-life situation through this specific application.

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How Can Graph Be Used As An Analytical Tool

Graph can be used in many ways. It can be used to show the trend of an event, it can solve unknowns as in simultaneous equations, it can identify the value of minimum or maximum points, and many more usages.

One important purpose of graph, however, is its ability to reflect the constraints of an equation or function. This function can come from a system under the Math Model.

Here, graph can analyse the weakness or strength of the system, or locate the value of a certain parameter that endangers its operation.

Let's show an example of the usefulness of Graph as an analytical tool.

Take the negative feedback amplifier as case-study.

The function or mathematical modelling expression of this amplifier is A / (1 + XA),
where the A is the amplification factor (or gain) of the amplifier and X is the feedback factor.

Let us plot the math equation and see the feature of this amplifier from the graph plotted.



In diagram 1, the math model equation is plotted with A = 1.

From this graph, let us analyse the features it exposes.

1. When the feedback factor (X) increases, the overall gain (y) decreases. This is so since more output signal is feedback resulting in reduction of actual input.


2. At the point when X = -1, the gain (y) becomes infinite! This is a dangerous value in that the amplifier will not operate as normal.

The 2 key features of the amplifier can therefore be revealed through proper analysis of the graph plotted using its model equation. The weakness of the system (amplifier) can then be exposed for caution and care in designing and usage.

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Eye Pattern - An Interesting Analytical Concept

This post talks about what exactly a sine wave is, and a very unique application of this sine wave as an analytical tool.

So what is a sine wave or what does it represents?

The diagram below will explain the concept of this sine wave.




On the left side of the above diagram is a rotor cycling anti-clockwise. What it produces, in vertical measurement, is a variation in height over time. This results in a sine wave.

Therefore, the rotational frequency defines the period of one cycle.

If this frequency changes, the period changes as a result.

This concept is used to test the consistency of rotation. It is an engineering application that has a useful indication.

See an example below for a slowly deviating rotational frequency of the rotor.



The eye pattern shown in the diagram is an indication of how consistent is the rotation.

If the area of the eye pattern is small, it indicates that there is large fluctuation to the rotational speed. The target is to have a large area for this eye pattern.

If the speed does not change, the sine wave will be only one, and the above diagram will not reflect many of the other sine waves.

Therefore, a simple sine wave, if understood properly, can be used as a useful analytical or measurement tool to detect any defect or inconsistency in the motor.

This concept is mainly applied in the rotating disc industries. Compact disc player, hard disc control, DVD player are some examples utilising this eye pattern detection.

Maths, thus, serves many applications besides purely training the mind to think systematically.
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Maths Was Used To Create This PCB

Maths is everywhere, even though it may exist in the subconscious mind. That is what I personally discovered recently. I would like to share one example. And this example lead me to one maths term I called Math-lectronics.

I was going through some of the hardware projects done recently. The project called for the fabrication of printed circuit board (PCB). This board carries electrical current that does certain functions for a system. The board is consist of thin layer of copper glued onto a paper-based support. For functionality, the copper on the PCB is etched to a customised pattern that serves as path for electrical current to flow. I created one as shown below.



The diagram above is a mask that will be used to create the physical circuit board.

Looking at the pattern, I discovered that I had used, sub-consciously, maths to generate it. Why?

1) I need to come up with the physical dimension of the PCB to suit the housing.

2) I need to estimate the density of the components to be packed into the PCB.

3) I need to know the height of the highest component to be used.

4) I need to check the amount of current flowing through each individual copper track. The higher the electric current, the wider the track has to be. This is to offset the resistance of the copper.
Formula: Current = Voltage / Resistance where Resistance relates to length and (1/area).

5) I need to know the frequency of the current in each track. This may create cross-talk between the tracks if placed too closely to each others.

6) I need to decide the bending angle of the tracks to reduce noise within.

7) I need to determine the diameter of some holes to be drilled for component mounting.

8 ) I need to add up the currents for deciding the number of connectors I need.

The words bolded in the above considerations are related to maths, and was done sub-consciously by me.

Had I not learn maths, I would not be able to generate this copper pattern.
Do you think so? :)

Finally I would like to present the fabricate PCB using the pattern created.



So what is the term when maths and electronics merge? I call it Math-lectronics.

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Using Trigonometry To Create Pictures

Mathematics has many purposes. The main use is for calculation. However, if we know its principle and how they relate to other design tools, we can create interesting pictures out of it.

Here I demonstrate how the understanding of the principle of trigonometry can be useful in generating graphics. It is actually another way to present the trigonometry function besides the use of common timeline.

In the first picture, I used the simple expression of sinA to create a flower. With the help of Microsoft Excel drawing tools and the Radar chart function, mathematics is turned into art.

Picture 1 Flower

In the next creation, I used the trigonometric multiplication of 2 sine functions;

"sinA x sin 4A ".



Picture 2: Multiplication Flower

The next picture is created through using the maths operation "2 + (sin A x sin 4A)".


Picture 3: Addition Flower

Did you see the wonders of mathematics now?

By understanding the principle of trigonometry, we can, with the aid of other tools, create something interesting. Thus the merging of maths and other applications poses an unlimited spectrum for human imagination.
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