An **integer** is any positive or negative whole number, including zero without any fractional part. For example, -23, -4, 0, 6, 74, etc. are integers whereas, -2.2,-5/6, 4.7,3/7 are not integers. Integers are generally denoted by Z when represented in a set. The set of integers which is denoted by the alphabet Z is a denumerable set, which means that this set might contain an infinite number of integers and all these numbers can be denoted by a list that defines the identity of each element contained in the set.

For example, the set Z written as {…….-3, -2, -1, 0, 1, 2, 3, .…..} denotes that -678543 and 56884 are included in this set of integers but -2577.76, 45569.876 are not. There is a total of seven properties that are applicable in mathematics and as integers are an integral part of it, some of these even apply to integers. Here, we shall discuss the properties that apply to integers.

## What Properties Can Be Applied To Integers?

### Closure Property

The closure property of integers, based on addition and subtraction of the integers, states that if p and q are two integers then the sum or difference of these two will also always be an integer. For example, let p=7 and q=4 then p + q = 7 + 4= 11, which is an integer and also, p – q= 7- 4= 3, which is also an integer. In context to multiplication, it states that the product of two integers will always be an integer, which means if p and q are two integers then pq is also an integer. For example, taking the above values, p*q= 7*4= 28 is an integer.

### Associative Property

Associative property based on addition states that regardless of the order, numbers are grouped, the sum of numbers remains constant. For example, (1+n) + m= 1+ (n + m). Similarly, based on multiplication, irrespective of the order, the product of numbers grouped remains constant. For example, (1*n) * m= 1* (n* m). However, this property does not apply to subtraction and division.

### Commutative Property

The commutative law states that when any two numbers are added, say x and y and yield the result z then, these two numbers will give the same result even if their positions are swapped. For example, 7+2 yields 9, similarly 2+7 also yields 9. In addition to this, this property also holds for multiplication, however, it cannot be applied to subtraction and division as in these two cases, the results may differ.

### Distributive Property

Distributive property refers to the division of operations on a given set of numbers so that the equation becomes relatively easier to solve. Under this property, multiplication has a higher order of precedence. For example, in the equation x (y + z) the distributive property is applied as x*y + x* z= xy + xz which makes it quite easy to find the solution. However, the answer is the same in case we follow the order of the operation or the law distribution. The only difference lies in the complexity which is much reduced in the latter case.

### Identity Property

This property states that when zero is added to any number, it yields the number itself. Thus, zero is called additive identity. For example, p+0=p, in case p is an integer. In case we talk of multiplication, 1 is said to be the multiplicative identity as any number multiplied by 1 will amount to the number itself. For example, p*1= p, also any number multiplied by 0 yields 0.

Thus, these are the properties that apply to integers. In addition to integers, there are natural, whole, rational, real numbers also. So, if you want your child to grasp and understand these numbers properly so that mathematics does not become his fear, you should book a class for your child on **cuemath**. The coaches there are so well-versed with the concepts and they make the learning process so easy that children understand the concepts quickly.