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# Integers: Introduction & Properties With Examples

## Introduction to Integers

In Mathematics, integers are groups of whole valued- positive or negative numbers or zero. In other words, integers are numbers that do not include any fractional part. The set of integers are represented as shown below:

Z = { …-3, -2, -1, 0, 1, 2, 3}

The notation “Z” to represent the set of integers is derived from the German word “ Zahlen”. The word “Zahlen” means “numbers”. Integers larger than 0 are termed positive integers whereas integers lesser than 0 are termed negative integers.

Example:

5, 6, 7, 8, 0, -11, -13 are all integers.

Here,

( 5, 6, 7, 8 are posiitve integers)

(-11, and -13 are negative integers)

The values such as , 15,2, , are not integers.

## What are the 5 Different Properties of Integers?

The 5 different properties of integers are as follows:

1. Closure Property

The closure property says that when you add or subtract any 2 integers, you will always get the result as an integer. For example, if a and b are any 2 integers, then a + b, and a – b will become an integer.

Example 1:

• 5 – 7 = – 2
• 5 + (- 7) = – 2

In the above example, we can see the results are integers. The closure property under multiplication says that when any 2 integers will be multiplied,  their product will always be an integer. For example, If a and b are two integers then “ab” will also be an integer.

Example 2:

• 3 4 = 12
• – 4 3 = -12

In the above example, we can see, we get integers. The closure property does not follow the division of integers as the quotient of any 2 integers might be an integer or maybe not.

Example 3: – 4  – 8 = ½ is not an integer.

2. Commutative Properties of Addition And Multiplication

The commutative properties of addition and multiplication state that adding or multiplying integers will have the same result regardless of their arrangement. For example:

• a + b = b + a
• a b = b  a

Example 4:

1 + 3 = 3 + 1

4 = 4

LHS= RHS

Here, the result “4” is the same on both sides regardless of their arrangement.

Example 5:

3  4 = 4  3

12 = 12

LHS= RHS

Here, the result “12” is the same on both sides regardless of their arrangement.

3. Associative Properties of Addition and Multiplication

The associative property of addition and multiplication says that when 3 integers are added or multiplied in a group, the integers will have the same result regardless of the way they are grouped. For example:

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

Example 6:

2 + ( 4 + 5) = (2 + 4) + 5

2 + 9 = 6 + 5

11 = 11

Here, the result “11” is the same on both sides regardless of their grouping.

Example 7:

2  ( 4  5) = (2  4)  5

2  20 = 8  5

40 = 40

Here, the result “40” is the same on both sides regardless of their grouping.

4. Distributive Property of Multiplication Over Both Addition and Subtraction

The distributive property of multiplication over addition and subtraction is a very important property of an integer. This property says that when you multiply a sum or difference by the number, it obtains the same result as you multiply each addend or subtrahend by the number and then add or subtract the products together. Let us understand with an example:

Example 8:

8  ( 9 + 1) =  (8  9) + ( 8  1)

8  10 = 72 + 8

80 = 80

LHS = RHS

Example 9:

8  (9  – 1) =  (8  9) – ( 8  1)

8  8 = 72 –  8

64 = 64

LHS= RHS

5. Identity Property of Addition And Multiplication

The identity property of addition says that when addition of any integer is done to 0, the conclusion will be the integer itself. For example:

• a + 0 = 0
• -a + 0 = 0

Example 10:

• 2 + 0 = 2
• (-23) + 0 = -23

The identity property of multiplication says that when any integer is multiplied by 1, the result of this multiplication will be the integer itself. For example:

• a 1 = a

Example 11:

• 2 1 = 2

The identity property of multiplication also says that when any integer is multiplied by -1, the product will be opposite of that integer. For example:

• a -1 = – a

Example 12:

• 2 -1 = -2

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