Rational numbers:
A rational number is the ratio of two integers p and q (p and q are either positive or negative) and are in the form p/q, where q is not equal to zero. Examples of natural numbers are 3/5, -7/3, 4. 4 is a rational number as 4 = 4/1.

Irrational numbers:
The numbers which are not rational are called irrational numbers. √2and pi cannot be expressed as ratio of 2 integers (as fractions) and hence they are irrational numbers. Their values are approximately 1.414… and 3.14159… which are non-terminating decimals.

Among irrational numbers, there are numbers which when converted to decimal form are non-terminating decimals. E.g.,
1/3 = 0.333…
1/7=0.142857142857…

In case of rational numbers, the decimal parts repeat themselves, whereas it is not the case in respect of irrational numbers.

The usual notations are:
Z defines the set of integers like -2, -1, 0,1, 2…
N defines the set of rational numbers like 2/3, -5/6, 5, …
R defines real numbers
C defines complex numbers.
Note: 0 is not taken to be a positive or negative integer, but it is a whole number (integer).

Even numbers:
Numbers that are multiples of 2 are even numbers, i.e. 2, 4, 6, 8, ….are even numbers. Numbers that are not multiples of 2 are odd numbers.

(i) In general 2n is an even number and 2n + 1 and 2n – 1 are odd numbers (where n is a whole number).

(ii) The sum or difference of 2 evem numbers is even,
e.g., 18 + 22 = 40
40 – 24 = 16

(iii) The sum of two odd numbers is even and the difference of two odd numbers is also even.
e.g., 25 + 31 = 56
45 – 29 = 16

(iv) Any power of an even number is even
e.g., 2⁴ = 16; 6³ = 216

(v) Any power of an odd number is odd
e.g., 3² = 9; 5³ = 125

(vi) An even number multiplied by an even number is even
e.g., 6 x 4 = 24

(vii) An odd number multiplied by an odd number is odd
e.g., 3 x 7 = 21

(viii) An odd number multiplied by an even number is even
e.g., 7 x 6 = 42

Example: The sum of a two digit number and the number formed by reversing the digits is a perfect square. Find the numbers.

Given a number 29 with 2 digits, the number got by reversing the digits is 92.
29 could be written as 2×10 + 9
92 could be written as 9×10 + 2
The sum of the two numbers is 29 + 92 = 121 = 11²
In general, if the digit in the ten’s place is x and the unit place is y, then the number is 10x + y.
The number on reversing becomes 10y + x.
Sum of the 2 numbers is
(10x + y) + (10y + x) = 11(x+y)
The maximum value of x or y can be 9.
Therefore, x+y= (9+9)
Implies, x+y = 18
Ii(x+y) must be a perfect square.
This is possible only when x+y = 11
The possible number of x and y are given by
(x,y) = (2,9) or (9,2)
(x,y) = (3,8) or (8,3)
(x,y) = (4,7) or (7,4)
(x,y) = (5,6) or (6,5)
Hence, the required numbers are 29,92; 38,83; 47,74; 56,65;
For example, 38 + 83 = 121 = 11²
Consecutive numbers are those numbers, such that the difference of any number from the previous number is 1.
2,3,4,5,6,7,8,… are consecutive numbers
2,4,6,8,… are consecutive even numbers
3,5,7,9,… are consecutive odd numbers
Example: Convert the repeating decimal expansion 3.3333 a a rational number
Let x = 3.3333
10x = 33.33
Subtracting (1) from (2), we get
10x – x = (33.3333….-3.3333) = 33 – 3 = 30
i.e., 9x = 30
Implies, x = 30/9 = 10/3 = 3-1/3

Example: Express 2.5737373737373…. as a fraction of the form p/q
2.5737373737373… means the decimal part 73 repeats itself
Let x = 2.573737373… (i)
10x = 25.73737373… (ii)
1000x = 2573.737373… (iii)
Subtracting (ii) from (iii), we get
1000x – 10x = (2573.737373) – (2.5737373…)
= 2573 – 25 = 2348
i.e., 990x = 2548
Therefore, x = 2548/990 = 1274/495

What is Divisibility?

What do you understand by divisibility? Well, it is really very simple. If you divide a number by another, and the result is a whole number, the number is divisible by the other. If you divide the two numbers, and get a remainder, then that number is NOT divisible by the other. Here’s an example

Suppose you divide 8 by 2. What is the result? It is 4. Are there any remainders after the division? No. Then, 8 IS divisible by 2.

But what if I divide 8 by 3 instead? You would get 2 as a quotient and 2 as a remainder! So, 8 is NOT divisible by 3!

How Do I Find Out If A Bigger Number Is Divisible By Another?

The examples above consisted of a single number only! But what if you have a number that is like 288? Do you sit and divide them? No, you don’t. There are simple steps that can tell you if a number is divisible by another. Take a look at them:

Let us take up an example. Consider the number 150. Let us find out if it is divisible by the other numbers:

• 150 is divisible by 2 because the last digit is 0.
• 150 is divisible by 3 because when we add the digits (1+5+0), we get 6 which is divisible by 3.
• 150 is NOT divisible by 4, since the last 2 numbers i.e., 5 and 0 are not divisible by 0.
• 150 is divisible by5 because the last digit is 0
• 150 is divisible by 6 because it is divisible by both 2 and 3
• 150 is not divisible by 7 because when we double 0 we get o again. If we subtract that from 15, we get 15, which is not divisible by 7.
  
• 150 is not divisible by 8 because the three digits 1, 5 and 0are not divisible by 8.
• 150 is NOT divisible by 9 because the sum of the digits 1+5+0=6, which is not divisible by 9.
• 150 is divisible by 10 because the last digit is 0.

Divisibility rules are very simple. Just make sure that you practice them for at least 10 minutes every day. You will soon be the new found genius in class and your teachers will be amazed by the speed you’ve developed!

Further Tips on Divisibility:

(i) A number divisible by 2 or 5

Any number ending in 0 or an even number is divisible by 2 e.g. 12, 256, 328, 2060. If the last digit of a number is 0 or 5, that number is divisible by 5 e.g. 150, 2025, 3175.

(ii) A number divisible by 4 or 25

Any number is divisible by 4 if the last two digits are divisible by 4, e.g. 132, 5276, 208. Similarly if the last two digits of a number are divisible by 25, that number is divisible by 25, e.g. 1375, 2500.

(iii) A number divisible by 8 or 125

For a number to be divisible by 8, its last three digits must be divisible by 8, e.g. 864, 1248, 3000. Similarly numbers ending with the last three digits divisible by 125 are divisible by 125, e.g. 4250, 12375, 12000.

(iv) A number divisible by 16 or 625

A number having its last four digits divisible by 16, will be divisible by 16, e.g. 31776, 28528. Similarly numbers having their last four digits divisible by 625 are divisible by 625, e.g. 83125, 125000.

(v) A number divisible by 3 or 9

If a number is divisible by 3, the sum of the digits is divisible by 3, e.g. 38451, 285612.
In the above numbers, sum of digits 3+8+4+5+1 = 21 and 2+8+5+6+1+2 = 24, which are all divisible by 3. Hence, the numbers are divisible by 3.

In a similar manner, a number is divisible by 9, if the sum of all its digits is divisible by 9, e.g. 1548, 653229.

In the above numbers, the sum of digits = 1+5+4+8 = 18, 6+5+3+2+2+9 = 27. Hence, the numbers are divisible by 9

(vi) A number divisible by 11

A number is divisible by 11 if the difference between the sum of the digits in the even places and the sum of the digits in the odd places is either 0 or a number divisible by 11, e.g. 65274, 538472.

In 65274, difference in the sum of numbers in the odd places and the sum of numbers in the even places = (6+2+4) – (5+7) = 0

In 538472, difference in the sum of numbers in the odd places and the sum of numbers in the even places = (5+8+7) – (3+4+2) = 20 – 9 = 11, a multiple of 11.
Hence the two numbers are divisible by 11

• Let us go through a little tip about divisibility. Have you looked at a long number and wondered whether it was divisible by 4? There is an easy way of finding it out.

1. Just take the last digit in the number and

2. Add it to two times the second last number. You have the answer!

3. Let’s take up an example:
Take the number 212334436. Is it divisible by 4? Let’s see. Take the last digit in the number i.e. 6 and add it to twice the second last number i.e. 3. Then, 6+ (2X3) =12. Find out if this is divisible by 4. In the example above, it is. Therefore the number 212334436  IS divisible by 4!

4. You can even use this tiny trick to find out whether a particular year is a leap year or not. Try it out yourself, right now! We TOLD you it’s fun!

• Let’s try out another fun trick! Have you always been confused by the relation between kilos and pounds? There is a simple way out. Suppose you are trying to convert kilos into pounds. How would you go about it in the shortest manner possible? It’s simple.

1. Just double the kilos and add a decimal point to the answer.

2. Then, just add the two together.

3. Let’s take another example:
For instance, when you try to convert 86 kilos, just double it. That will give you 172. Now, add a decimal point to 172, (or just divide it by 10). That gives you 17.2. Now add the two numbers, i.e. 172 and 17.2. The result is 189.2. So, 86 kilos would give you 189.2 pounds. Ridiculously simple, isn’t it?

• Finally, a useful trick that will help you in temperature conversions! Here is an easy way to convert Fahrenheit to centigrade and vice versa. You’ve got to try this out!

1. When you are converting Fahrenheit to centigrade, just take away 30 from it and then divide by 2. You have your answer in a jiffy.

Want an example? Try this:

74 degree Fahrenheit -30=44. Divide this by 2 and you have 22. So, 74 degree is 22 degree centigrade.

2. If you want to convert a temperature from centigrade to Fahrenheit, just do the reverse. Just double the number and add 2 to it. Isn’t that fun?
There are a number of shortcuts you can use to make mathematics easier and fun. The easy steps we detailed above are just some of the tricks used by experts to get that edge over others. Try out these math shortcuts today and stun everyone with your speed!