__Quantitative Aptitude Notes on Simplification__

__BASIC FORMULAE__
1. (a+b)

^{2}=a^{2}+b^{2}+2ab
2. (a−b)

^{2}=a^{2}+b^{2}−2ab
3. (a +b)

^{2}− (a−b)^{2}=4ab
4. (a+b)

^{2}+ (a−b)^{2}=2(a^{2}+b^{2})
5. (a

^{2}–b^{2})= (a+b) (a−b)
6. (a+b+c)

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

^{3}+b^{3}) = (a+b) (a^{2}−ab+b^{2})
8. (a

^{3}–b^{3}) = (a−b) (a^{2}+ab+b^{2})
9. (a

^{3}+b^{3}+c^{3}−3abc)= (a+b+c) (a^{2}+b^{2}+c^{2}−ab−bc−ca)
10. If a+b+c=0, then a

^{3}+b^{3}+c^{3}=3abc.

__TYPES OF NUMBERS__

__1. Natural Numbers:__
Counting numbers 1, 2, 3, 4,
5 … are called natural
numbers

__2. Whole Numbers:__
All counting numbers together with
zero form the set of whole
numbers.

Thus,

(I) 0 is the only whole number
which is not a natural number.

(II) Every natural number is a
whole number.

__3. Integers:__
All natural
numbers, 0 and negatives of counting
numbers i.e.,…,−3,−2,−1,0,1,2,3,….. together form the set of integers.

**(i) Positive Integers:**1, 2, 3, 4….. is the set of all positive integers.

**(ii) Negative Integers:**−1, −2, −3… is the set of all negative integers.

**(iii) Non-Positive and Non-Negative Integers:**0 is neither positive nor negative.

So, 0,1,2,3,….
represents the set of non-negative
integers, while 0,−1,−2,−3,….. represents the set
of non-positive integers.

__4. Even Numbers:__
A number divisible by 2 is called
an even number, ex. 2, 4, 6, 8, etc.

__5. Odd Numbers:__
A number not divisible by 2 is
called an odd number. e.g. 1, 3, 5, 7, 9, 11 etc.

__6. Prime Numbers:__
A number greater than 1 is called
a prime
number, if it has exactly two factors, namely 1 and the number itself.

__7. Composite Numbers:__
Numbers greater
than 1 which are not prime, are
known as composite numbers, e.g., 4,6,8,9,10,12.

**Note:**
(i) 1 is neither prime nor
composite.

(ii) 2 is the only even number
which is prime.

(iii) There are 25 prime numbers
between 1 and 100.

__REMAINDER AND QUOTIENT:__"The remainder is r when p is divided by k" means p=kq+r the integer q is called the quotient.

__EVEN ,ODD NUMBERS__A number n is even if the remainder is zero when n is divided by 2: n=2z+ 0 or n=2z.

A number n is odd if the remainder
is one when n is divided by 2: n=2z+1.

**even X even = even**

**odd X odd = odd**

**even X odd = even**

**even + even = even**

**odd + odd = even**

**even + odd = odd**

__Some important tricks__- 1 + 2 + 3 + 4 + 5 + … + n = n(n + 1)/2
- (1
^{2}+ 2^{2}+ 3^{2}+ ..... + n^{2}) = n ( n + 1 ) (2n + 1) / 6 - (1
^{3}+ 2^{3}+ 3^{3}+ ..... + n^{3}) = (n(n + 1)/ 2)^{2} - Sum of first n odd
numbers = n
^{2} - Sum of first n even numbers = n (n + 1)