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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Does Math make you clever?

Why do we need to study math?

Is this question a familiar one?

We know that we need math to do some calculations for our daily necessity. But have we to go to the extent of learning logarithm, calculus, and the more complex mathematical techniques?

Math learning also involves some form of drilling to pump in the key steps and concepts. It is also a form of structured and systematic approach to solving problems.

So does this so-called systematic approach tie us to a rigid way of doing thing?
Will math, therefore, make us a more flexible or rigid, straight-only person?

Compare this math learning to literature learning, which has more room for personal expression?
My guess, is that you will say "literature".

Yes, learning subjects, like the literature and languages , do give us the freedom of expression through writing what comes to the mind, filling in beautiful words for descriptions and the like.

Math, on the other hand, forces us to think only in a structure manner. Any deviation from the "laws" will be deemed inappropriate.

Thus, learning math makes us less clever.

WRONG!

Now let's look at learning math in another angle.

Assume you are the civil engineer having to plan the site layout for constructing a building. You are given a boundary with the area within filled with hard rocks beneath.

Another civil engineer is asked to build a similar building but now in a prairie, without any boundary constraint and the soil is marvellous.

Who has a better time?
Who need to be more experience or knowledgeable?
Who need to be better?

You have to be better.

Link this to learning math, where you are given "boundaries" to operate in. The structured approach ties you within the laws, but still requires you to come out with the answers.
Learning subjects, like the English, is equivalent to the other engineer. He has his challenges but in another form (maybe to optimise budget, appearance, etc).

So, does learning math makes you clever?
Maybe. But it will definitely not make you less clever.
Having to operate and think with more constraints requires building, in fact, more flexibilities, for you to overcome any obstacles.

This is the power of learning math, and the very reason why we need to have math in our education since young.

Interesting ironical concept, right?
(Constraints create more flexibilities).

:D

Scalar Multiplication in Determinant and Matrices

The study of determinants and matrices can be confusing at time. They look almost the same, and are closely related. As such, some mathematical operations on them are similar.

But, there are differ somewhat and have to be taken care of.

Below is one such mistake that is often made by learning math students.



The upper row involves matrices. The scalar multiplication of it by "a" affects ALL the elements within the matrix.

For the lower row, the scalar "a" multiplication affects only one column (or one row) of the determinant.

Just note this different and commit it to memory. This is good for you and can save you a lot of further troubles with matrix and determinant.

So, just simply identifying the differences between matrices and determinant, in term of their operations, you will be enjoying them for a long time.

With less mistake made, you will be more gear towards doing a better job out of maths. This is human nature.

Cheers! and Happy mathematics!
:D

Lines of Wisdom

In an earlier post, the enjoyment of knowing maths was demonstrated in its ability to form pictures.

Here is another example of playing with the trigonometry to create mathematical art.



In this piece of art, 4 graphs of various mathematical expressions were overlapped to form the result.

I name this piece of art the "Lines of Wisdom".

In the graphs, maths functions like the sine and sinh operation were used together with trigonometrical multiplication.

:D

Creating A Maths Picture

Maths can be used in many ways.

It is normally used to solve daily calculations related to work, life, etc.
It can be used to model a system to understand its performance and behaviour.

However, on the fun side, it can be used to create a picture using graph as a means.

Here in this post, a picture formed through merging trigonometry with modulation technique is shown.


What do you think this picture is about?

While creating this maths picture, a kid so happened to have a glimpse of it, and commented that it looked like the front view of an aeroplane.

For me, it seems to be the captured voice of Optimus Prime, the Transformer Autobot leader.

With maths, if you understand the underlying principles of various elementary topics, you can freely come up with any mathematical figures that you wish.

Here, I have simply used the special sinc function modulated (multiplied) by a high frequency trigonometric sine function to get this mixed result.

Using graph is one way to form picture, like the one created here.
It is fun seeing your imagination materialise through using maths and graphs as the tools.

Try it for yourself. You will enjoy maths and its fun picture creation.

:D