## Numbers

In Decimal number system, there are ten symbols namely 0,1,2,3,4,5,6,7,8 and 9 called digits. A number is denoted by group of these digits called as numerals.

## Face Value

Face value of a digit in a numeral is value of the digit itself. For example in 321, face value of 1 is 1, face value of 2 is 2 and face value of 3 is 3.

## Place Value

Place value of a digit in a numeral is value of the digit multiplied by 10^{n} where n starts from 0. For example in 321:

- Place value of 1 = 1 x 10
^{0}= 1 x 1 = 1 - Place value of 2 = 2 x 10
^{1}= 2 x 10 = 20 - Place value of 3 = 3 x 10
^{2}= 3 x 100 = 300

## Types of Numbers

**Natural Numbers**– n > 0 where n is counting number; [1,2,3…]**Whole Numbers**– n ≥ 0 where n is counting number; [0,1,2,3…].**Integers**– n ≥ 0 or n ≤ 0 where n is counting number;…,-3,-2,-1,0,1,2,3… are integers.**Positive Integers**– n > 0; [1,2,3…]**Negative Integers**– n < 0; [-1,-2,-3…]**Non-Positive Integers**– n ≤ 0; [0,-1,-2,-3…]**Non-Negative Integers**– n ≥ 0; [0,1,2,3…]

**Even Numbers**– n / 2 = 0 where n is counting number; [0,2,4,…]**Odd Numbers**– n / 2 ≠ 0 where n is counting number; [1,3,5,…]**Prime Numbers**– Numbers which is divisible by themselves only apart from 1.**Composite Numbers**– Non-prime numbers > 1. For example, 4,6,8,9 etc.**Co-Primes Numbers**– Two natural numbers are co-primes if their H.C.F. is 1. For example, (2,3), (4,5) are co-primes.

## Divisibility

Following are tips to check divisibility of numbers.

**Divisibility by 2**– A number is divisible by 2 if its unit digit is 0,2,4,6 or 8.**Divisibility by 3**– A number is divisible by 3 if sum of its digits is completely divisible by 3.**Divisibility by 4**– A number is divisible by 4 if number formed using its last two digits is completely divisible by 4.**Divisibility by 5**– A number is divisible by 5 if its unit digit is 0 or 5.**Divisibility by 6**– A number is divisible by 6 if the number is divisible by both 2 and 3.**Divisibility by 8**– A number is divisible by 8 if number formed using its last three digits is completely divisible by 8.**Divisibility by 9**– A number is divisible by 9 if sum of its digits is completely divisible by 9.**Divisibility by 10**– A number is divisible by 10 if its unit digit is 0.**Divisibility by 11**– A number is divisible by 11 if difference between sum of digits at odd places and sum of digits at even places is either 0 or is divisible by 11.

## Tips on Division

- If a number n is divisible by two co-primes numbers a, b then n is divisible by ab.
- (a-b) always divides (a
^{n}– b^{n}) if n is a natural number. - (a+b) always divides (a
^{n}– b^{n}) if n is an even number. - (a+b) always divides (a
^{n}+ b^{n}) if n is an odd number.

## Division Algorithm

When a number is divided by another number then**Dividend = (Divisor x Quotient) + Reminder**

## Series

Following are formulaes for basic number series:

- (1+2+3+…+n) = (1/2)n(n+1)
- (1
^{2}+2^{2}+3^{2}+…+n^{2}) = (1/6)n(n+1)(2n+1) - (1
^{3}+2^{3}+3^{3}+…+n^{3}) = (1/4)n^{2}(n+1)^{2}

## Basic Formulaes

These are the basic formulae:

(a + b)^{2}= a^{2}+ b^{2}+ 2ab

(a - b)^{2}= a^{2}+ b^{2}- 2ab

(a + b)^{2}- (a - b)^{2}= 4ab

(a + b)^{2}+ (a - b)^{2}= 2(a^{2}+ b^{2})

(a^{2}- b^{2}) = (a + b)(a - b)

(a + b + c)^{2}= a^{2}+ b^{2}+ c^{2}+ 2(ab + bc + ca)

(a^{3}+ b^{3}) = (a + b)(a^{2}- ab + b^{2})

(a^{3}- b^{3}) = (a - b)(a^{2}+ ab + b^{2})

(a^{3}+ b^{3}+ c^{3}- 3abc) = (a + b + c)(a^{2}+ b^{2}+ c^{2}- ab - bc - ca)

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