Wonders of Numbers - Part II

Welcome back to another part of Wonders of Numbers. Today, we will be discussing about inequality. Don't know what inequality means? It means not equal (literary). So, has anyone got any idea of what I am going to say? Drum please... Applause... AAAAAAAAAAAAND THIS IS THE TOPIC! INEQUALITY! 

(I know it's a lame joke, sorry for that)

Alright. Let's get to the main topic now. I am not going to teach you what inequality is cause what I am going to share now is very basic for those who had taken Add Math and I will make it as simple as possible for those who doesn't know what Maths is all about. But first of all, you have to recognise some symbols.

1. equal to -- =

2. greater than -- >

3. less than -- <

4. greater or equal to -- >= 

5. less or equal to -- <=

6. not equal to -- !=

* I got these symbols from C++, it's faster for me to write in this form

So let's start of with some simple Mathematics. 

1. Learn some number classification.

Integers and Rational numbers.

Integers are whole numbers, positive and negative numbers. NO FRACTIONS NO DECIMAL POINTS!

Rational numbers are fractions of 2 integers. where the denominator is not 0. eg. 1/2, -3/9.

Alright we can start now. Most of us know that fractions are also numbers and decimal are also numbers. So they are actually equal at some point. Say 1/2 = 0.5 and 7/8 = 0.875. All fractions can be changed into decimal form but not all decimal can change into fraction form. I'll give you an example of a decimal number which can't change into fraction form or we say rational numbers. Try changing 1.414213562 into a fraction (rational number). How do we know that this number can be changed into fractions? So there is a basic rule. If your decimal number has some numbers been repeating at the back. it can be changed into fractions.

eg. 0.111111111111... = 1/9 

you can see that 1 is repeating

eg. 0.923076923076923076... = 12/13

notice that 923076 is repeating

So we made a conclusion that all number with some numbers being repeated at the back can be changed into fraction. For those in Form 4 and Form 5, remember changing some stupid numbers into fraction form using the Geomatric Series Rule and the Sum To Infinity of a Geometric Series? S = a/(1-r)? Alright now try changing 0.9999999....9999...99...9999 into a fraction. Now first thing first, answer this question.

Is 

A) 0.9999999999999...9999999...999999....99999 = 1 ? Or you say that 

B) 0.99999999999...99999999...9999999...9999 <>

Don't bluff! I know u chose A. Yeah everyone will choose A base on what they see. 

Now let us work this out and you will not believe what you see! Math is Magic!

So we use the geomatric way i mentioned just now.

So 0.99999999999....999999...99999...9999... = 0.9 + 0.09 + 0.009 +... right?

There for we have a Geometric Series of 

a = 0.9 and r = 0.09/0.9 = 0.009/0.09 = 0.1 or 1/10.

Use the formula of Sum To Infinity. ( This series has no end cause its a convergent series, the limit will goes to 0 (zero) when the number of terms goes to 00 (infinity).

So the formula is S = a/(1-r) --- if you don't believe me, google it.

So S = 0.99999999999999....9999999999...9999999... right?

S = a/(1-r)

S = 0.9/(1-0.1)

There for we put 0.9999999999999.....99999999......999999.... back into the S

0.99999......9999999.....9999999... = 0.9/(1-0.1) = 0.9/0.9 = 9/9 =1

So we make a conclusion that, 

0.99999999...9999999...99999 = 1

Now do u agree? NO? DON'T LIKE IT THIS WAY? Ok, I'll show you and esier way. 

0.9999999.....999999999...99999..99999 = 0.6666666666666...66666... + 0.33333333333...3333333...33333 right?

and 0.6666666666666666666....6666666666....666666... = 2/3

and 0.3333333333333333333...33333..333333...3333..... = 1/3

so 0.999999999........9999999.......999999.... = 2/3 + 1/3

      = 3/3

      = 1

Now you agree? Just accept it. Now Math has once again shown its Magic to you.

Challenge my theory.