Number System

N = {1, 2, 3, …}

Whole Numbers: In the natural numbers set if we include number 0 then the resulting set will be known as whole numbers.

W = {0, 1, 2, 3, ….}

Integers: The natural numbers along with zero and their negatives are known as integers.

I = {…., -3, -2, -, 0, 1, 2, 3, …}

Rational Numbers: The set of numbers which can be expressed in the form of p/q where q ≠ 0 and is known as rational numbers.

Q= { p/q : p and q are integers and q ≠ 0}

Irrational Numbers: Those numbers which cannot be expressed in the form of p/q, where q ≠ 0, are known as irrational numbers. For e.g. √2, √5, √7 etc. are irrational numbers.

Real Numbers: The set of those numbers which are either rational or irrational are known as real numbers.

Even Numbers: Those numbers which are divisible by 2 and are known as even numbers. For e.g. 4, 6, 8, 10, … are even numbers.

Odd Numbers: Those numbers which are not divisible by 2 and are known as odd numbers. For e.g. 3, 5, 9, 7, … are odd numbers.

Prime Numbers: A prime number is a number which has no factors besides it'self and unity, i.e. it is divisible only by it'self and 1 but not by any other number.

Composite Numbers: A composite number is one which has other factors including it'self and unity.

Test of Divisibility:

(i) Divisibility by 2 → Last digit divisible by 2
(ii) Divisibility by 4 → Last two digit's divisible by 4
(iii) Divisibility by 8 → Last three digit's divisible by 8
(iv) Divisibility by 16 → Last four-digit divisible by 16
(v) Divisibility by 3 → Sum of digit's divisible by 3
(vi) Divisibility by 9 → Sum of digit's divisible by 9
(vii) Divisibility by 27 → Sum of blocks of 3 (taken a right to left) divisible by 27
(viii) Divisibility by 7 → Remove the last digit, double it and subtract it from the truncated original number. Check if the number is divisible by 7
(ix) Divisibility by 11 → (sum of odd digit's) - (sum of even digit's) should be 0 or divisible by 11.

Basic summation properties:
(i) Sum of first n odd numbers is n²
(ii) Sum of first n even numbers is n(n + 1)
(iii) 1 + 2 + 3 + 4 + 5 + … + n = n(n + 1)/2
(iv) (1^² + 2^² + 3^² + ….. + n^²) = n ( n + 1 ) (2n + 1) / 6
(v) (1^³ + 2^³ + 3^³ + ….. + n^³) = (n(n + 1)/ 2)²