Integers | Definition, Examples & Types

The word integer originated from the Latin word “Integer” which means whole or intact. Integers are a special set of numbers comprising zero, positive numbers, and negative numbers. So, an integer is a whole number (not a fractional number) that can be positive, negative, or zero.

Examples of integers are -7, 1, 3, -78, 56, and 300. Examples of numbers that are not integers are -1.4, 5/2, 9.23, 0.9, 3/7.

In this article, we have covered everything about integers in maths, types of integers, examples, rules & arithmetic operations on integers.

Integers Definition

Integers are a fundamental concept in mathematics, representing a set of whole numbers that includes both positive and negative numbers, along with zero. Its symbol is "Z".

If a set is constructed using all-natural numbers, zero, and negative natural numbers, then that set is referred to as Integer. Integers range from negative infinity to positive infinity.

• Natural Numbers: Numbers greater than zero are called positive numbers. Example: 1, 2, 3, 4...
• Negative of Natural Numbers: Numbers less than zero are called negative numbers. Example: -1, -2, -3, -4...
• Zero (0) is neither positive nor negative.

Symbol of Integers

Set of integers is represented by the letter Z as shown below:

Z = {... -7, -6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7...}

Types of Integers

Integers are classified into three categories:

• Zero (0)
• Positive Integers (i.e. Natural numbers)
• Negative Integers (i.e. Additive inverses of Natural Numbers)

Zero

Zero is a unique number that does not belong to the category of positive or negative integers. It is considered a neutral number and is represented as "0" without any plus or minus sign.

Positive Integers

Positive integers, also known as natural numbers or counting numbers, are often represented as Z⁺. Positioned to the right of zero on the number line, these integers encompass the realm of numbers greater than zero.

Z⁺ → 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30,….

Negative Integers

Negative integers mirror the values of natural numbers but with opposing signs. They are symbolized as Z^–. Positioned to the left of zero on the number line, these integers form a collection of numbers less than zero.

Z^– → -1, -2, -3, -4, -5, -6, -7, -8, -9, -10, -11, -12, -13, -14, -15, -16, -17, -18, -19, -20, -21, -22, -23, -24, -25, -26, -27, -28, -29, -30,…..

Integers on a Number Line

As we have discussed previously, it is possible to visually represent the three categories of integers - positive, negative, and zero - on a number line.

Zero serves as the midpoint for integers on the number line. Positive integers occupy the right side of zero, while negative integers populate the left side. Refer to the diagram below for a visual representation.

Rules of Integers

Various rules of integers are,

• Addition of Positive Integers: When two positive integers are added together, the result is always an integer.
• Addition of Negative Integers: Sum of two negative integers results in an integer.
• Multiplication of Positive Integers: Product of two positive integers yields an integer.
• Multiplication of Negative Integers: When two negative integers are multiplied, the outcome is an integer.
• Sum of an Integer and Its Inverse: Sum of integer and its inverse is alays zero.
• Product of an Integer and Its Reciprocal: Product of an Integer and Its Reciprocal is always 1.

Arithmetic Operations on Integers

Four basic Maths operations performed on integers are:

• Addition of Integers
• Subtraction of Integers
• Multiplication of Integers
• Division of Integers

Addition of Integers

Addition of integers is similar to finding the sum of two integers. Read the rules discussed below to find the sum of integers.

Example: Add the given integers

• 3 + (-9)

• (-5) + (-11)

• 3 + (-9) = -6

• (-5) + (-11) = -16

Subtraction of Integers

Subtraction of integers is similar to finding the difference between two integers. Read the rules discussed below to find the difference between integers.

Example: Add the given integers

• 3 - (-9)

• (-5) - (-11)

• 3 - (-9) = 3 + 9 = 12

• (-5) - (-11) = -5 + 11 = 6

Multiplication of Integers

Multiplication of integers is achieved by following the rule:

• When both integers have same sign, the product is positive.

• When both integers have different signs, the product is negative.

Division of Integers

Division of integers is achieved by following the rule:

• When both integers have the same sign, the division is positive.
• When both integers have different signs, the division is negative.

Properties of Integers

Integers have various properties, the major properties of integers are:

• Closure Property
• Associative Property
• Commutative Property
• Distributive Property
• Identity Property
• Additive Inverse
• Multiplicative Inverse

Closure Property

Closure property of integers states that if two integers are added or multiplied together their result is always an integer. For integers p and q

• p + q = integer
• p × q = integer

Example:

(-8) + 11 = 3 (An integer)
(-8) × 11 = -88 (An integer)

Commutative Property

Commutative property of integers states that for two integers p and q

• p + q = q + p
• p × q = q × p

Example:

(-8) + 11 = 11 + (-8) = 3
(-8) × 11 = 11 × (-8) = -88

But the commutative property is not applicable to the subtraction and division of integers.

Associative Property

Associative property of integers states that for integers p, q, and r

• p + (q + r) = (p + q) + r
• p × (q × r) = (p × q) × r

Example:

5 + (4 + 3) = (5 + 4) + 3 = 12
5 × (4 × 3) = (5 × 4) × 3 = 60

Distributive Property

Distributive property of integers states that for integers p, q, and r

• p × (q + r) = p × q + p × r

For Example, Prove: 5 × (9 + 6) = 5 × 9 + 5 × 6

Solution:

LHS = 5 × (9 + 6) 
        = 5 × 15
        = 75

RHS = 5 × 9 + 5 × 6 
        = 45 + 30
        = 75

Thus, LHS = RHS Proved

Identity Property

Integers hold Identity elements both for addition and multiplication. Operation with the Identity element yields the same integers, such that

• p + 0 = p
• p × 1 = p

Here, 0 is Additive Identity, and 1 is Multiplicative Identity.

Additive Inverse

Every integer has its additive inverse. An additive inverse is a number that in addition to the integer gives the additive identity. For integers, Additive Identity is 0. For example, take an integer p then its additive inverse is (-p) such that

• p + (-p) = 0

Multiplicative Inverse

Every integer has its multiplicative inverse. A multiplicative inverse is a number that when multiplied to the integer gives the multiplicative identity. For integers, Multiplicative Identity is 1. For example, take an integer p then its multiplicative inverse is (1/p) such that

• p × (1/p) = 1

Applications of Integers

Integers extend beyond numbers, finding applications of integers in real life. Positive and negative values represent opposing situations. For instance, they indicate temperatures above and below zero. They facilitate comparisons, measurements, and quantification. Integers feature prominently in sports scores, ratings for movies and songs, and financial transactions like bank credits and debits.

Articles Related to Integers:

• Rational Number
• Irrational Number
• Real Numbers
• Properties of Integers

Integers Examples

Some examples on Integers are,

Example 1: Can we say that 7 is both a whole number and a natural number?

Solution:

Yes, 7 is both whole number and natural number.

Example 2: Is 5 a whole number and a natural number?

Solution:  

Yes, 5 is both a natural number and whole number.

Example 3: Is 0.7 a whole number?

Solution: 

No, it is a decimal.

Example 4: Is -17 a whole number or a natural number?

Solution: 

No, -17 is neither natural number nor whole number.

Example 5: Categorize the given numbers among Integers, whole numbers, and natural numbers,

• -3, 77, 34.99, 1, 100

Solution:

Numbers	Integers	Whole Numbers	Natural Numbers
-3	Yes	No	No
77	Yes	Yes	Yes
34.99	No	No	No
1	Yes	Yes	Yes
100	Yes	Yes	Yes

Integers Class 6 Worksheet

Integers are a fundamental concept in mathematics, especially introduced at the class 6 level, aiming to broaden the understanding of numbers beyond natural numbers and whole numbers. Worksheet on Integers for students to solve is added below:

Solve:

1. 23 + (-12)
2. 15 – 12
3. -14 + 14
4. (13) × (-17)
5. (4) × (12)
6. 0 × (-87)
7. (114) ÷ (-7)
8. (-7) ÷ (-3)

Answer Key:

1. 23 + (-12) = 11
2. 15 - 12 = 3
3. -14 + 14 = 0
4. 13 × (-17) = -221
5. 4 × 12 = 48
6. 0 × (-87) = 0
7. 114 ÷ (-7) = -16.29
8. -7 ÷ (-3) = 2.33

Read More: Practice Questions on Integers