Number System Basics in English

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 10ⁿ where n starts from 0. For example in 321:

• Place value of 1 = 1 x 10⁰ = 1 x 1 = 1
• Place value of 2 = 2 x 10¹ = 2 x 10 = 20
• Place value of 3 = 3 x 10² = 3 x 100 = 300

Types of Numbers

1. Natural Numbers – n > 0 where n is counting number; [1,2,3…]
2. Whole Numbers – n ≥ 0 where n is counting number; [0,1,2,3…].
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…]
   0 is neither positive nor negative integer.
4. Even Numbers – n / 2 = 0 where n is counting number; [0,2,4,…]
5. Odd Numbers – n / 2 ≠ 0 where n is counting number; [1,3,5,…]
6. Prime Numbers – Numbers which is divisible by themselves only apart from 1.
7. Composite Numbers – Non-prime numbers > 1. For example, 4,6,8,9 etc.
8. 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.

1. Divisibility by 2 – A number is divisible by 2 if its unit digit is 0,2,4,6 or 8.
2. Divisibility by 3 – A number is divisible by 3 if sum of its digits is completely divisible by 3.
3. Divisibility by 4 – A number is divisible by 4 if number formed using its last two digits is completely divisible by 4.
4. Divisibility by 5 – A number is divisible by 5 if its unit digit is 0 or 5.
5. Divisibility by 6 – A number is divisible by 6 if the number is divisible by both 2 and 3.
6. Divisibility by 8 – A number is divisible by 8 if number formed using its last three digits is completely divisible by 8.
7. Divisibility by 9 – A number is divisible by 9 if sum of its digits is completely divisible by 9.
8. Divisibility by 10 – A number is divisible by 10 if its unit digit is 0.
9. 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

1. If a number n is divisible by two co-primes numbers a, b then n is divisible by ab.
2. (a-b) always divides (aⁿ – bⁿ) if n is a natural number.
3. (a+b) always divides (aⁿ – bⁿ) if n is an even number.
4. (a+b) always divides (aⁿ + bⁿ) if n is an odd number.

Division Algorithm

When a number is divided by another number thenDividend = (Divisor x Quotient) + Reminder

Series

Following are formulaes for basic number series:

1. (1+2+3+…+n) = (1/2)n(n+1)
2. (1²+2²+3²+…+n²) = (1/6)n(n+1)(2n+1)
3. (1³+2³+3³+…+n³) = (1/4)n²(n+1)²

Basic Formulaes

These are the basic formulae:

(a + b)2 = a2 + b2 + 2ab

(a - b)2 = a2 + b2 - 2ab

(a + b)2 - (a - b)2 = 4ab

(a + b)2 + (a - b)2 = 2(a2 + b2)

(a2 - b2) = (a + b)(a - b)

(a + b + c)2 = a2 + b2 + c2 + 2(ab + bc + ca)

(a3 + b3) = (a + b)(a2 -  ab + b2)

(a3 - b3) = (a - b)(a2 + ab + b2)

(a3 + b3 + c3 - 3abc) = (a + b + c)(a2 + b2 + c2 - ab - bc - ca)