Quantitative Aptitude Notes on Simplification
BASIC FORMULAE
1. (a+b) 2=a2+b2+2ab
2. (a−b) 2=a2+b2−2ab
3. (a +b) 2− (a−b) 2=4ab
4. (a+b) 2+ (a−b) 2=2(a2+b2)
5. (a2–b2)= (a+b) (a−b)
6. (a+b+c) 2=a2+b2+c2+2(ab+bc+ca)
7. (a3+b3) = (a+b) (a2−ab+b2)
8. (a3–b3) = (a−b) (a2+ab+b2)
9. (a3+b3+c3−3abc)= (a+b+c) (a2+b2+c2−ab−bc−ca)
10. If a+b+c=0, then a3+b3+c3=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:
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
- (12 + 22 + 32 + ..... + n2) = n ( n + 1 ) (2n + 1) / 6
- (13 + 23 + 33 + ..... + n3) = (n(n + 1)/ 2)2
- Sum of first n odd numbers = n2
- Sum of first n even numbers = n (n + 1)