Numbers Introduction

Natural Number : Counting numbers 1,2,3,4,... are known as Natural Number.

The set of natural numbers can be represented by N = {1,2,3,4,...}

Whole Number : Natural Numbers including 0 is known as Whole number.

  1. The set of whole numbers can be represented by W = {0,1,2,3,4,...}
  2. Every natural number is a whole number.
  3. All whole number is a natural number excluding 0.

Integers : All Counting numbers and their negative including 0 are known as Integers.

The set of integers can be represented by I = {...,-3,-2,-1,0,1,2,3,4,...}

Positive Integers : The set {1,2,3,4,....} is a set of all positive integers.

Positive integers and natural numbers are synonyms.

Negative Integers : The set {-1,-2,-3,-4,....} is a set of all negative integers.

Non-positive and Non-negative Integers : 0 is neither positive and nor negative integers.

Even Number : An even number is a number that is exactly divisible by 2.

Examples : 2,4,6,8,144

Odd Number : An odd number is a number that is not exactly divisible by 2.

Examples : 1,3,5,7,9

Note : 0 is an even number.

Prime Number : A number is greater than 1 and which is divisible by 1 and itself is called Prime number.

Examples : 2,3,5,7,11,13

Note : 2 is only the prime number which is an even and other than 2 are odd prime numbers.

Composite Number : Natural numbers greater than 1 which are not prime are called composite numbers

Examples : 4,6,9,15

Co-Prime Number : Two numbers are Co-prime numbers, if their HCF is 1 is called Co-prime numbers.

Examples : (2,3),(4,5),(8,11)

Rational Number : The numbers of the for a/b, where a and b are integers and b ≠ 0

Examples : 2/5,4/3

Irrational Number : An Irrational Number is a real number that cannot be written as a simple fraction. In others words, Irrational numbers which when expressed in decimal form neither terminating nor repeating decimals.

Examples : √2 ,√5,Π

Algebraic Formulas
  1. $(a+b)^2=a^2+2ab+b^2$
  2. $(a-b)^2=a^2-2ab+b^2$
  3. $(a-b)(a+b)=a^2-b^2$
  4. $(a-b)^2+(a+b)^2=2(a^2+b^2)$
  5. $(a-b)^2-(a+b)^2=4ab$
  6. $(a+b)^3$ $=a^3+b^3+3a^2b+3ab^2$ $= a^3+b^3+3ab(a+b)$
  7. $(a-b)^3$ $= a^3-b^3-3a^2b+3ab^2$ $= a^3+b^3-3ab(a-b)$
  8. $a^3+b^3$ $= (a+b)(a^2-ab+b^2)$
  9. $a^3-b^3$ $= (a-b)(a^2+ab+b^2)$
  10. $a^3+b^3+c^3-3abc$ $=(a+b+c)$ $(a^2+b^2+c^2-ab-bc-ca)$