Integers

Integers are groups of numbers that are defined as the union of positive numbers, and negative numbers, and zero is called an Integer. ‘Integer’ comes from the Latin language word that means ‘whole’ or ‘intact’. Integers do not include fractions and decimals.

What is an Integer?

If a set of all-natural numbers are constructed, whole numbers, and negative numbers then such a set is called an Integer set. It can be positive, negative, or zero. For Example 2, 5, -9, -17, 112, etc.

Definition of Integers

Integers include positive numbers, negative numbers, and zero. Integers are represented by the symbol Z such that,

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

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

Representation of Integers on a Number Line

Number Line is used in representing numbers visually with the help of a line. The position of any number can be easily identified with the help of a number line. In a number line, the center represents Zero, Positive numbers are included on the Right-hand Side, whereas Negative Numbers are on the Left-hand Side.

Operations on Integers

We can perform various operations on integers, basic operations on integers are:

• Addition
• Subtraction
• Multiplication
• Division

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.

Rules for Addition in Integers

Use the following rules for performing addition in integers.

If both Integers are of the same signs:

We add the absolute values of both integers and the sign of any integer is added to the answer.

If both Integers are of different signs:

We find the difference between the absolute values of both integers and the sign of the integer with a greater absolute value is added to the answer.

Example: Add the given integers: 

• 3 + (-9)
• (-5) + (-11)

Solution:

• 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.

Rules for Subtraction in Integers

Use the following rules for performing subtraction between integers.

• The sign of the subtrahend is changed and the operation is changed to addition.
• Use the rule of Addition to find the further result.

Example: Add the given integers: 

• 3 – (-9)
• (-5) – (-11)

Solution:

• 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.

Product of Sign	Resultant Sign	Example
(+) × (+)	+	9 × 3 = 27
(+) × (–)	–	9 × (-3) = -27
(–) × (+)	–	(-9) × 3 = -27
(–) × (–)	+	(-9) × (-3) = 27

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.

Division of Sign	Resultant Sign	Example
(+) ÷ (+)	+	9 ÷ 3 = 3
(+) ÷ (–)	–	9 ÷ (-3) = -3
(–) ÷ (+)	–	(-9) ÷ 3 = -3
(–) ÷ (–)	+	(-9) ÷ (-3) = 3

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 x 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 x q = q x 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

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

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

Thus, LHS = RHS Proved.

Identity Property of Integers

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.

Solved Example on Integers

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

FAQs on Integers

Question 1: What are integers?

Answer:

The union of zero, natural numbers, and their additive inverse is called Integers. It is mathematically denoted by the symbol Z.

Question 2: What are consecutive integers?

Answer:

Consecutive Integers are integers that are adjacent to each other on a number line. The difference between the two consecutive integers is “1”.

Question 3: Write the examples of integers.

Answer:

Examples of integers are -1, -9, 0, 1, 87, etc.

Question 4: What are the different types of integers?

Answer:

There are three different types of integers:

• Zero
• Positive Integers
• Negative integers

Question 5: Can integers be negative?

Answer:

Yes, integers can be negative. Negative Integers are -1, -4, and -55, etc.

Question 6: State whether the given statement is true or false: “all natural numbers are whole numbers”?

Answer:

True, all natural numbers are whole numbers but not vice versa.