What are Integers?

Integers are the numbers that can be positive, negative or zero, but cannot be a fraction. These numbers are used to perform various arithmetic operations, like addition, subtraction, multiplication, and division. The symbol of integers is “Z“.

Z= {……-8,-7,-6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8,……}

The word integer originated from the Latin word “Integer” which means whole. It is a special set of whole numbers comprised of zero, positive numbers, and negative numbers and denoted by the letter Z.

Types of Integers

There are different types of numbers in Mathematics, They are:

  • Real Numbers
  • Natural Numbers
  • Whole Numbers
  • Rational numbers
  • Irrational numbers
  • Even and Odd Numbers, etc.

Integers in Maths

Rules of Integers

  • The Sum of two positive integrals is an integer
  • The Sum of two negative integers is an integer
  • The product of two positive integrals is an integer
  • The product of two negative integers is an integer
  • The Sum of an integer and its inverse is equal to zero
  • Product of an integer and its reciprocal is equal to 1

Properties

Closure Property:

According to the closure property of numbers, when two integers are added or multiplied together, it results in an integer only. If a and b are integers, then:

  • a + b = integer
  • a x b = integer

Examples:

2 + 5 = 7 (is an integer)

2 x 5 = 10 (is an integer)

Commutative Property:

According to the commutative property of integers, if a and b are two integers, then:

  • a + b = b + a
  • a x b = b x a

Examples:

3 + 8 = 8 + 3 = 11

3 x 8 = 8 x 3 = 24

But for subtraction and division, the commutative property does not obey.

Associative Property:

As per the associative property , if a, b and c are integers, then:

  • a+(b+c) = (a+b)+c
  • ax(bxc) = (axb)xc

Examples:

2+(3+4) = (2+3)+4 = 9

2x(3×4) = (2×3)x4 = 24

This property is applicable for addition and multiplication operations only.

Distributive property

According to the distributive property of integers, if a, b and c are integers, then:

a x (b + c) = a x b + a x c

Example: Prove that: 3 x (5 + 1) = 3 x 5 + 3 x 1

LHS = 3 x (5 + 1) = 3 x 6 = 18

RHS = 3 x 5 + 3 x 1 = 15 + 3 = 18

Since, LHS = RHS

Hence, proved.

Additive Inverse Property:

If a is an integer, then as per the additive inverse property of numbers,

a + (-a) = 0

Hence, -a is the additive inverse of integer a.

Multiplicative inverse Property:

If a is an integer, then as per the multiplicative inverse property of numbers,

a x (1/a) = 1

Hence, 1/a is the multiplicative inverse of integer a.

Identity Property:

a+0 = a

a x 1 = a

Example: -100,-12,-1, 0, 2, 1000, 989 etc…

As a set, it can be represented as follows:

Z= {……-8,-7,-6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8,……}

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