Properties of Integers

In practically every discipline, integers aid in the computation of efficiency in positive or negative numbers. Integers provide information about one's current location. It also aids in determining how many more or fewer measures should be taken to achieve better results. Integers are defined as the set of all whole numbers but they also include negative numbers. Therefore, integers can be negative, i.e., -5, -4, -3, -2, -1, positive 1, 2, 3, 4, 5, and even include 0. An integer can never be a fraction, a decimal, or a percent. 

The integer set is denoted by the symbol “Z”. The set of integers are defined as follows:

Z = {-4, -3, -2, -1, 0, 1, 2, 3, 4}

Examples of integer are: -57, 0, -12, 19, -82, etc.

The integers can be represented as:

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

On the number, line integers are represented as follows:

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Here 0 is at the centre of the number line and is called the origin.

All numbers to the left of the (0) are negative integers with a prefix of a minus(-) sign, while all numbers to the right are positive integers with a prefix of a plus(+) sign, however, and they can be written without the + sign.

 In this article, we will study the different properties of integers.

Difference Between Integers and Whole Numbers

Properties of Integers

Integers have 5 main properties, they are mentioned below:

• Closure Property

• Associative Property

• Commutative Property

• Distributive Property

• Identity Property

Property 1: Closure Property

The closure property of integers under addition and subtraction states that the sum or difference of any two integers will always be an integer. if p and q are any two integers, p + q and p − q will also be an integer.

Example : 7 – 4 = 3;

7 + (−4) = 3; both are integers.

The closure property of integers under multiplication states that the product of any two integers will be an integer which means .if p and q are any two integers, p × q will also be an integer.

Example : 5 × 7 = 35;

(–4) × (7) = −28;both are integers.

Division of integers doesn’t hold for the closure property, i.e. the quotient of any two integers p and q, may or may not be an integer.

Example : (−3) ÷ (−12) = ¼, which is not an integer.

Property 2: Associative Property

Associative property refers to grouping. Associative property rules can be applied for addition and multiplication. If the associative property for addition and multiplication operation is carried out regardless of the order of how they are grouped, the result remains constant.

• Associative Property for Addition states that if 

(l + n) + m = l + ( n + m)

For example: (2 +  5) + 4 = 2 + (5 + 4) the answer for both the possibilities will be 11.

• Associative Property for Multiplication states that if

(l x n) x m = l x ( n x m)

For example: ( 2 x 3) x 5 = 2 x ( 3 x 5) the answer for both the possibilities will be 30.

Thus, we can apply the associative rule for addition and multiplication but it does not hold for subtraction and division.

Property 3: Commutative Property

Commutative law states that when any two numbers say x and y, in addition, gives the result as z, then if the position of these two numbers is interchanged we will get the same result z. Thus, we can say that commutative property states that when two numbers undergo swapping, the result remains unchanged. For example, 5 + 4 = 9 if it is written as 4 + 9, then also it will give the result 4. Similarly, the commutative property holds for multiplication. But it does not hold for subtraction and division.

Example: 7 + 2 = 2 + 7 = 9 

5 + 21 = 21 + 5 = 26

5 + 28 + 43 = 43 + 5 + 28 = 76

Property 4: Distributive Property

Distribute means, as the name itself implies, to divide something given equally. Distributive property means dividing the given operations on the numbers so that the equation becomes easier to solve. It states that “multiplication is distributed over addition.” For instance, take the equation a( b + c), when we apply the distributive property, we have to multiply a with both b and c and then add i.e. a x b + a x c = ab + ac.

Let us understand this concept with distributive property examples.

For example 3( 2 + 4) = 3 (6) = 18

or

By distributive law

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

  = 6 + 12

  = 18

Here we are distributing the process of multiplying 3 evenly between 2 and 4. We observe that whether we follow the order of the operation or distributive law the result is the same.

Property 5: Identity Property

Identity property states that when any zero is added to any number, it will give the same given number. Zero is called additive identity. 

For any integer p, p + 0 = p

The multiplicative identity property for integers says that whenever a number is multiplied by the number 1, it will give the integer itself as the result. Hence 1 is called the multiplicative identity for a number. 

For any integer p, p × 1 = p = 1 × p

If any integer multiplied by 0, the result will be zero: x × 0 = 0 =0 × x.

If any integer is multiplied by -1, the result will be opposite of the number: x × (−1) = −x = (−1) × x

Solved Examples

Example 1: Show that -37 and 25  follow commutative property under addition.

Solution :

Let a = -37 b = 25

Commutative property states that

a + b = b + a

L.H.S. = a + b 

= -37 + 25 

= -12

R.H.S. = b + a 

= 25 + (-37) 

= 25 – 37 

= -12

So, L.H.S. = R.H.S., i.e. a + b = b + a.

This means the two integers hold true commutative property under addition.

Example 2: Show that (-6), (-2), and (5) are associative under addition.

Solution :

Let a = - 6; b = - 2, and c = 5

Associative property for addition states that

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

L.H.S. = -6 + ( -2 + 5)

= – 6 + 3

= -3

R.H.S. = (- 6 + (-2)) + 5

= (- 6 - 2) + 5

= -8 + 5

= - 3

So, L.H.S. = R.H.S., i.e. a + (b + c) = (a + b) + c.

This proves that all three integers follow associative property under addition.

Types of Integers

Integers are divided into three groups:

• The quotient obtained when a positive integer is divided by a positive integer is a positive integer. For instance, (+6) (+3) = +2.

• The quotient obtained when a negative integer is divided by a negative integer is a positive integer. For instance, (-6) (-3) = +2.

• The quotient obtained when a positive integer is divided by a negative integer or a negative integer is divided by a positive integer is a negative integer. For instance, (-6) (+3) = -2 as well as (+6) (-3) = -2.

The Number Line

A number line is defined as a straight line with a "zero" point in the middle with positive and negative integers listed on both sides of zero, continuing indefinitely. A number line is an example of a tool that a student might use to solve addition and subtraction problems. 

Movement on the Number Line

• When we add positive integers on the number line, we move to the right.

• When we add negative integers on the number line, we move to the left.

Quiz Time

1. Whether -55 and 22 follow commutative property under subtraction.

2. State whether (-20) and (-4) follow commutative law under division?