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Integers

Students can learn about Integers and working with them. Students can get help with Math problems from expert Math tutors available online.

Integers are the numbers in which the decimal part is zero. Integers include the set of natural numbers ( 1, 2, 3, ... ) , 0 and negative numbers ( natural numbers with negative sign ).

Representation of numbers

The above figure shows the representation on numbers on a number line. The set of integers can be represented as ( -?..., -3, -2, -1, 0, 1, 2, 3, ...? ).

Algebraic properties of integers:

Addition:

Sum of two integers in an integers

Example,

1) 6 + 3 = 9

2) 12 + 4 = 16

Irrespective of the number we take, the result will be an integer.

Subtraction:

Difference of two integers will be a an integers.

1) 6 - 3 = 3

2) 3 - 9 = -6

Multiplication:

Multiplication of two integers is an integer.

1) 3 x 4 = 12

2) 4 x 8 = 32

Division:

Division of two integers may or may not be an integer.

Example,

1) 4 ÷ 2 = 2 ; The result is an integer.

2) 100 ÷ 5 = 20 The result is an integer.

3) 5 ÷ 2 = 2.5 The number 2.5 is not an integer.

4) 6 ÷ 4 = 1.5 The result is not an integer.

Associativity over addition

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

Example,

2 + ( 3 + 5 ) = 2 + 8 = 10

( 2 + 3 ) + 5 = 5 + 5 = 10

Associativity over multiplication

a × (b × c) = (a × b) × c

Example,

2 x ( 3 x 4 ) = 2 x 12 = 24

( 2 x 3 ) x 4 = 6 x 4 = 24

Commutativity over addition

a + b = b + a

Example,

2 + 3 = 5

3 + 2 = 5

Commutativity over multiplication

a × b = b × a

Example,

2 x 3 = 6

3 x 2 = 6

Existence of identity

a + 0 = a ( additive )

a × 1 = a ( multiplicative )

Existence of additive inverse elements

a + (?a) = 0

Example,

3 + ( -3 ) = 3 - 3 = 0

Note:- Multiplicative inverse does not exists in integer range. For example,multiplicative inverse of 2 is 1/2 such that,

2 x 1/2 = 1

but, 1/2 is not an integer ( 1/2 belongs to rational numbers).

Distributive property

a × (b + c) = (a × b) + (a × c)

Example,

2 x ( 3 + 4 ) = 2 x 7 = 14

( 2 x 3 ) + ( 2 x 4 ) = 6 + 8 = 14

Note :If a × b = 0, then either a = 0 or b = 0 (or both)