A number that is written without a fractional component is referred to as an integer. Integers consist of the number zero, positive natural numbers, or counting numbers and their additive inverses or negative integers. Thus, 5, -8, 0 are all integers. However, numbers such as 9.69 and the square root of 3 (3) are not included in integers. In this article, we will learn about the different properties of integers and the associated concepts.

## Properties of integers

There are majorly five properties of integers as listed below.

## 1. Closure Property

Closure under addition states that the sum of any two integers will also be an integer.

m + n = p

where m, n, p are integers.

e.g. 8 + 2 = 10

Closure under subtraction states that the difference of any two integers will also be an integer.

m – n = q

where q is an integer

e.g. 8 – 2 = 6

Closure under multiplication states that if we multiply two integers their product will also be an integer.

mn = r

where r is an integer

e.g. 8*2 = 16

Division does not follow this rule.

## 2. Commutative Property

This property holds true for both addition and multiplication. It states that the order of adding or multiplying two integers does not matter, and the result will be the same. Thus, you can swap the terms, but the resultant sum or product will remain the same.

m + n = n + m

e.g. 8 + 2 = 2 + 8 = 10

mn = nm

e.g. 8*2 = 2*8 = 16

We cannot apply this property to subtraction and division.

## 3. Associative Property

The associative property states that the order of grouping for addition and multiplication of integers does not matter, and the resultant remains the same. In other words, you can group the integers in whichever way you like; however, that will not alter the sum or product.

(m + n) + y = m + (n + y)

Where m, n, y are integers

e.g. (8 + 2) + 4 = 8 + (2 + 4) = 14

(mn)y = m(ny)

e.g. (8*2)4 = 8(2*4) = 64

The associative property does not hold for subtraction and division

## 4. Distributive Property

The property can be distributive of multiplication over addition or distributive of multiplication over subtraction. It implies that you can distribute the ability of an operation over another operation. You can multiply the sum or difference of two or more integers by another integer will give the same answer as multiplying each number individually and then adding or subtracting products together.

m(n + y) = mn + my

e.g. 8(2 + 4) = 8*2 + 8*4 = 48

m(n – y) = mn – my

e.g. 8(2 – 4) = 8*2 – 8*4 = -16

## 5. Identity Property

This property can be applied to addition and multiplication but not to subtraction and division.

When an integer is added to zero, it results in the integer itself; hence, zero is called the additive identity.

m + 0 = m

e.g. 8 + 0 = 8

When an integer is multiplied with 1 it gives the same integer itself. One is the multiplicative inverse for any integer.

m * 1 = m

e.g. 8 * 1 = 8

These properties can be applied to all types of numbers falling under integers, such as whole numbers, prime numbers, composite numbers, etc.

### Conclusion

As integers form the basics of Mathematics hence, it is necessary for students to learn this topic well. A platform such as Cuemath is the best way to do so. They provide several resources that help in clearing student’s concepts while ensuring that they have fun during the studying process. Thus, a child will have a very strong foundation of the topic.

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