What is an Integer?

What is an integer? An integer is any number including 0, positive numbers, and negative numbers. It should be noted that an integer can never be a fraction, a decimal or a per cent. Some examples of integers include 1, 3, 4, 8, 99, 108, -43, -556, etc.

All About Integers

Integers are a set of counting numbers (positive and negative), along with zero, that can be written without a fractional component. As mentioned above, an integer can be either positive, negative or zero.

All natural numbers are also integers that start from 1 and end at infinity.

All whole numbers are also integers, starting from 0 and ending at infinity.

Fact: All integers are real numbers but not all real numbers are integers. Since, real numbers also include rational and irrational numbers.

Symbol

The integers can be represented as:

Z = {……., -3, -2, -1, 0, 1, 2, 3, ……….}

Types of Integers

An integer can be of two types:

  • Positive Numbers
  • Negative Integer
  • 0

Some examples of a positive integer are 2, 3, 4, etc. while a few examples of negative integers are -2, -3, -5, etc.

In the number system, there are various types of numbers that come under the integer category. They are:

  • Whole numbers
  • Natural numbers
  • Odd and Even integers
  • Prime and composite numbers

Integer Rules

The rules, based on operations performed on integers are given below:

Addition rule

If the sign of both the integers is the same, then they are added such as:

  • (+) + (+) = +
  • (-) + (-) = –

Example:

  • 5 + 9 = 14
  • -5 + (-9) = -14

But if one of the numbers has a different sign, then it will lead to subtraction and output will contain a sign of the larger number. Let us understand with the help of examples.

  • (-10)+(2) = -10 + 2 = -8
  • (-2)+(10) = -2+10 = 8

Subtraction Rule

The sign of the first number stays the same, change subtraction to addition and change the sign of the second number. Once you have applied this rule, follow the rules for adding integers

  • (+) – (+) = (+) + (-); consider the sign of greater number
  • (-) – (-) = (-) + (+); consider the sign of greater number
  • (+) – (‐) = (+) + (+); answer will be positive
  • (‐) – (+) = (‐) + (‐); answer will be negative

Examples:

  • 9 – 6 = 3
  • -9 – (-6) = -9 + 6 = -3
  • 9 – (-6) = 9 + 6 = 15
  • -9 – (6) = -15

Multiplication and Division rules

If the signs are the same, multiply or divide and the answer is always positive.

  • (+) x (+) = + and (+) ÷ (+) = +
  • (‐) x (‐) = + and (‐) ÷ (‐) = +

If the signs are different, multiply or divide and the answer is always negative.

  • (+) x (‐) = – and (+) ÷ (‐) = ‐
  • (‐) x (+) = – and (‐) ÷ (+) = ‐

Solved Examples

  • 4 x 2 = 8 and 4 ÷ 2 = 2
  • (-4) x (-2) = 8 and (-4) ÷ (-2) = 2
  • (4) x (-2) = -8 and 4 ÷ (-2) = -2
  • (-4) x (2) = -8 and (-4) ÷ (2) = -2 

Integers Worksheet

Solve the following:

  1. 4+(-4) = __
  2. 13 – 2 + 9 = __
  3. 11 + 14 – 2 = __
  4. (+3) x (-7) = __
  5. (+4) x (+32) = __
  6. 0 x (+7) = __
  7. (+14) ÷ (-7) = __
  8. (-72) ÷ (-9) = __ 

Frequently Asked Questions on Integers

What are integers in the number system? Give examples.

An integer is a numerical value in a number system that is not fractional. They are positive and negative counting numbers including zero. For example, 33, 0, -33, are integers.

Is 0 an integer?

Yes, 0 is an integer.

What are the rules of integers?

The integers rules are:
Sum of two integers is an integer
Difference of two integers is an integer
Multiplication of two or more integers is an integer
Division of integers may or may not be an integer

How is an integer represented?

An integer is represented using the alphabet “Z”. For integers, Z = {…, -2, -1. 0. 1, 2, ……}

Are odd and even numbers integers?

Yes, odd and even numbers fall under the category of integers. Even whole numbers, prime numbers, and composite numbers are all integers.

What are the main properties of integers?

There are 5 main properties of integers which are:

  • Property 1: Closure property
  • Property 2: Commutative property
  • Property 3: Associative property
  • Property 4: Distributive property
  • Property 5: Identity Property

Learn more about the properties of integers here.

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