Infinity

While studying mathematics, we often come along to the special symbol named infinity, ∞.
But what is infinity ?

Infinity ...
	... it's not big ...
	... it's not huge ...
	... it's not tremendously large ...
	... it's not extremely humongously enormous ...
	... it's ...

Endless!

There are few facts about infinity.

1. Infinity has no end

Infinity is the idea of something that has no end.

In our world we don't have anything like it. So we imagine traveling on and on, trying hard to get there, but that is not actually infinity.

So don't think like that (it just hurts your brain!). Just think "endless", or "boundless".

If there is no reason something should stop, then it is infinite.

2. Infinity does not grow

Infinity is not "getting larger", it is already fully formed.

Sometimes people (including me) say it "goes on and on" which sounds like it is growing somehow. But infinity does not do anything, it just is.

3. Infinity is not a real number

Infinity is not a real number, it is an idea. An idea of something without an end.

Infinity cannot be measured.

Even these faraway galaxies can't compete with infinity.

Using Infinity

We can sometimes use infinity like it is a number, but infinity does not behave like a real number.

To help you understand, think "endless" whenever you see the infinity symbol "∞":

For example: ∞ + 1 = ∞

Which says that infinity plus one is still equal to infinity.

When something is already endless, we can add 1 and it is still endless.

The most important thing about infinity is that:

-∞ < x < ∞ Where x is a real number

Which is mathematical shorthand for
"negative infinity is less than any real number,
and infinity is greater than any real number"

Here are some properties of infinity:

Special Properties of Infinity
∞ + ∞ = ∞
-∞ + -∞ = -∞
∞ × ∞ = ∞
-∞ × -∞ = ∞
-∞ × ∞ = -∞
x + ∞ = ∞
x + (-∞) = -∞
x - ∞ = -∞
x - (-∞) = ∞
For x>0 :
x × ∞ = ∞
x × (-∞) = -∞
For x<0 :
x × ∞ = -∞
x × (-∞) = ∞

Conclusion

Infinity is a simple idea: "endless". Most things we know have an end, but infinity does not.

Reference: https://www.mathsisfun.com/numbers/infinity.html

~Chow Cheng Li

Is 2=1?

Is 2=1?

Now we start the proof.

                     

First,

                                                          let      a = b

 [multiply both side by a]                          a² = ab

[subtract b² from both side]                  a²-b² = ab-b²

[factorise the equation]                (a-b)(a+b) = b(a-b)

[both side divide by (a-b)]                      a+b = b

Since a=b, then we have                           2b = b

[divide both side by b]                                2 = 1

Finally, we will get 2=1. But is this a correct proof? 
        
                 

Does 2 really equals to one? What’s wrong with this proof? Where does the logic break down? 

Actually, the method of getting the solution was not mathematically sound. This was a fake proof. 

This means that there is a mistake or false statement in somewhere in some line. 
      
  
                   

Still remember that we let a=b before we start the proof? 

If  a=b, then a-b=0. 

Hence, the proof above cannot divide by a-b, because it will be divide by 0. 

The division by 0 is also undefined. 

Hence, we cannot conclude that the number 2 is equals to 1. 

                                                                                          ~Ewe Sin Hooi

Square Roots

Before you learn how to do square roots, you'll need to learn a quick, simple trick for squaring two-digit numbers ending in a 5. As long as you know your multiplication tables up to 10 times 10, you'll pick up on this trick instantly.

Squaring 2-Digit Numbers Ending in 5

When given a number ending in 5, simply take the 10s digit, and multiply by a number one higher than itself. Take that answer, take a "25" on the end, and you've got the answer!

For example, let's say you're asked what 35 squared is. Take the 3 (the 10s digit), and multiply it by 4 (which is one higher than 3), and you get 12. Take a 25 on the end.

So, 35 squared (35 x 35) is equal to 12(front) combine with the 25(end), giving the answer 1225. 

Let's try a higher number, like 115 squared. 11 times 12 = 132. Take 25 on the end, gives us 13225!

Square Roots

Now, you can start to learn how to square root a number faster in mind. First, you should know the squares for the numbers 1-9, but here is something you may not have noticed before:

1^2 =  1

2^2 =  4

3^2 =  9

4^2 = 16

5^2 = 25

6^2 = 36

7^2 = 49

8^2 = 64

9^2 = 81

Notice that both 9 squared and 1 squares end in a 1. Both 2 and 8 squared end in 4, 3 and 7 squared both end in 9 and 4 and 6 squared both end in 6, as well! 

So how do we determine the square root?

The process starts out by breaking the number up. With squares, we'll break the number up into 2-digit numbers. For example, we'll use 6889 as an example. This would break up into 68 and 89.

68 is greater than 8 squared (64), but less than 9 squared (81), so we know that the square root is somewhere in the 80s. The other half of the number, 89, ends in 9. Here's where the tricky part comes in. The fact that the number ends in a 9 means that it could either end in a 3 or a 7. So, how do we determine whether the square root is 83 or 87?

This is where that squaring trick ends in 5 comes in handy. Obviously, if 6889 is greater than 85 squared, it can only be 87. If 6889 is less than 85 squared, it can only be 83. So which is it? We know that 85 squared is 7225. 6889 is quite obviously less than 7225, so the square root must be 83.

As a final example, let's take 4356. 43 is greater than 6 squared, but less than 7 squared, so we know right away that the root is in the 60s. The other half of the number, 56, it ends in 6, so the square root is either 64 or 66. Since we can quickly determine that 65 squared is 4225, and that 4356 is greater, then the square root can only be 66.

Obviously, when a number end in 25, you know right away that the ones digit is 5, so these are even simpler. For example, 9025. 90 is greater than 81 (9 squared), so we know the 10s digit is 9. The 25 tells us that the 1s digit is 5, so the square root of 9025 must be 95!

Reference's link: http://gmmentalgym.blogspot.my/2010/10/root-extractions.html#rtsqr

THE DIGIT ZERO

0 may refer to:
-a digit 0
-the integer between 1 and -1
-null
-0 (years) in some calendar  

Some of the properties of 0~
1. it is neither positive or negative

2. it is an even number: by definition, any number can divided by 2 is an even number, since 0 /2 = 0, then 0 is an even number

3. it is a natural number

4. additional property : any number added to 0 equal to itself

example:
1 + 0 = 1
2 + 3 + 0 = 2 + 3 = 5
a + b + 0 = a + b

5. additive inverse property : if the sum of two numbers is zero, then they are additive inverse or opposite of each other

example:
since -3 + 3 = 0, 3 is the additive inverse of -3

6. multiplication property : any number times zero is equal to zero

example:
4 x 0 = 0
53 x 0 = 0
p x 0 = 0
rq x 0 = 0

7. any number divided by 0 is undefined, it is neither zero, infinity or negative infinity

8. 0 divided by any number is equal to 0

9. 0 to the power of any number is equal to 0, but 0 to the power of 0 is equal to 1

example:

• 1⋅03=1⋅0⋅0⋅0=0
• 1⋅02=1⋅0⋅0=0
• 1⋅01=1⋅0=0
• 1⋅00=1=1

10. 0 factorial is equal to 1: 0! = 1

11.0 divided by 0 = indeterminate

if we do multiplication as follows:
 4 x 9 = 36,
the we know that
36 / 9 = 4 or 36 / 4 = 9

if we do the same with a number 0:
7 x 0 = 0
so what we gain from 0 / 7 or 0 / 0?


on the other hands, the associative rule: (a+b)+c=a+(b+c) will not always work, for example:

1+(infinity - infinity) = 1+0=1
(1+infinity) - infinity = infinity - infinity = 0

Black Hole Number in Mathematics

What is the Black Hole Number??

6174 is known as Kaprekar's constant after the Indian mathematician D.R.Kaprekar.

      So, why 6174  is the black hole number??

     1. Simply take 4 digit number.(At least 2 different digit)

     2. Arrange the digits in descending and in ascending order.

     3. Substract the smaller number from the bigger number.

     4. Repeat the step 2 to step 4.

      For example 6767.

      7766 - 6677 = 1089

      9810 - 0189 = 9621

      9621 - 1269 = 8352

      8532 - 2358 = 6174

      7641 - 1467 = 6174

      Not convinced yet ?

      Lets try another number : 1234

      4321 - 1234 = 3087

      8730 - 0378 = 8352

      8532 - 2358 = 6174

      7641 - 1467 = 6174

So, how about 3 digit number??

495 is another black hole number for 3 digit number.

Lets try!!

Assume we put 624 as a lucky number.

642 - 246 = 396

963 - 369 = 594

954 - 459 = 495

954 - 459 = 495

    You can try your lucky 4 digit  and 3 digit number on your own.

When mathematics meet with the Egyptian Pyramid

The most popular ,largest and oldest pyramids in Giza pyramid complex is known as Pyramid of Khufu. It is the oldest of the Seven Wonders of the Ancient World.

1.   The area of the slope of pyramid is almost same with the area of the square with the               length = height of pyramid.


This pyramid is such a miracle.

Egyptologists believe that the pyramid was built as tomb over a 10 to 20 year period concluding around 2560 BC.

Now, let me introduce some “magic” of mathematics in this pyramids.

Let’s consider Khufu’s Horizon as an example where its

Height = 146.6 m 

Base = 231.93 m and
The angle of slope = 51^o50’40.

                            

The area of the slanted area=21500 m²

                                      

          The length of the square = 146.6 m

By applying the formula of length = height,

The height of pyramid = 146.6m

The area of the square = (146.6)² = 21491.26 m²

Note that the slanted area is almost the same as the area of the square.

2.  The height of the pyramid times 1 million is almost same to the closest distance between              earth and sun which is 147 million km

The circumference of the base of the pyramid ( in inches) is almost the same as the number of the days in a year.

The length of the base of pyramid = 231.93 m = 9131.1023 inches

Circumference of the base of pyramid = 4 * 9131.1023

                                                                  = 36524.2092 inches

If we divided it by 100 = 365.24

This number is also equal to the number of days in a year which is 365.25 days.

Is it fantastic?

3.   And lastly you can also build your little pyramid on your own.

Height = 20 cm

Length of slope = 30 cm

Length of base = 31.6 cm

Angle between of slope and base = 52^o

Note: The pyramid must face to the direction as shown in the picture

You will find that the food you put inside the pyramid will not rot but only dehydrate.

~Wong Yee Wei

Vedic Mathematics

What is Vedic Mathematics?

Vedic Mathematics is a super fast way of calculation whereby you can do supposedly complex calculations like 998 x 997 in less than five seconds flat. It is highly beneficial for school and college students and students who are appearing for their entrance examinations. There are 16 sutras or formulae covered in Vedic Maths which solve all known mathematical problems in the branches of Arithmetic, Algebra, Geometry and Calculus. They are easy to understand, easy to apply and easy to remember.

Vertically and Crosswise

One of the sutras is "vertically and crosswise". The sutra "vertically and crosswise" has many uses. One very useful application is helping children who are having trouble with their tables above 5 x 5. For example 7 x 8. 7 is 3 below the base of 10, and 8 is 2 below the base of 10.

The whole approach of Vedic maths is suitable for slow learners, as it is so simple and easy to use. The sutra "vertically and crosswise" is often used in long multiplication. Suppose we wish to multiply 32 by 44. 

Steps

A. First, we multiply vertically 2 x 4 = 8. 

B. Then we multiply crosswise and add the two results: 3 x 4 + 4 x 2 = 20, so put down 0 and carry 2. 

C. Finally we multiply vertically 3 x 4 = 12 and add the carried 2 = 14. Result: 1,408.

We can extend this method to deal with long multiplication of numbers of any size. The great advantage of this system is that the answer can be obtained in one line and mentally. Multiplication can also be carried out starting from the left, which can be better because we write and pronounce numbers from left to right. Here is an example of doing this in a special method for long multiplication of numbers near a base (10, 100, 1,000 etc), for example, 96 by 92.

Steps

A. First, we know that 96 is 4 below the base and 92 is 8 below.

B. Then we can cross-subtract either way: 96 - 8 = 88 or 92 - 4 = 88. This is the first part of the answer.

C. Lastly, multiplying the "differences" vertically 4 x 8 = 32 gives the second part of the answer.

This works equally well for numbers above the base: 105 x 111 = 11,655. Here we add the differences. For 205 x 211 = 43,255, we double the first part of the answer, because 200 is 2 x 100.

Thank you......

Reference's link: http://www.hinduism.co.za/vedic.htm