By integer, we understand a number without any decimal or fractional part. It includes all whole numbers and negative numbers. Since whole numbers include zero and natural numbers, integers consist of:
They are represented by the symbol ‘Z’. Thus, integers are of 3 types: negative, zero, and positive. Together,
Z = {…… -4, -3, -2, -1, 0, 1, 2, 3, 4……}
In contrast, non-integers are decimals, fractions, or mixed numbers.
Integers on a Number Line
It helps to represent the numbers on a straight line visually and thus compares numbers placed at equal intervals on it. Similar to all other numbers, integers are also represented on a number line:
Integers on a Number Line
Thus, on a number line:
The number on the right is greater, corresponding to the number on the left
The positive integers are to the right side of 0
The negative integers are to the left side of 0
The zero is located in the middle of the positive and negative numbers
Operations on Integers
Like other numbers, there are 4 basic operations associated with integers: 1) Addition, 2) Subtraction, 3) Multiplication, and 4) Division. The specific rules while performing the 4 operations are discussed in their respective sections below.
However, there are some general rules we follow while performing these operations. They are:
When there is no sign before a number, it is considered positive
The absolute value of an integer is positive, i.e., |−5| = 5 and |5| = 5
Addition
While adding 2 integers with the same sign, we add the absolute values and then put the sign before the sum.
For example,
(+3) + (+7) = +10
(-5) + (-3) = -8
(+7) + (-4) = +3
(-8) + (+4) = -4
Practice:Find the value of 6 + (-12) using a number line.
Integers on a Number Line Example 2 – Answer 1
Here, the first number is 6, which is positive. So, from 0, we move 6 units to the right side. The following number is -12, carrying a negative. We move 12 units to the left side from 6.
Integers on a Number Line Example 2 – Answer 2
Thus, 6 + (-12) = -6
Subtraction
While subtracting two integers, we change the sign of the second number being subtracted and follow the addition rules.
For example,
(-8) – (+2) = (-8) + (-2) = -10
(+4) – (+1) = (+4) + (-1) = +3
(+4) – (+4) = (+4) + (-4) = 0
(+6) – (+8) = (+4) + (-8) = -2
Add the given integers: 6 + (-2)
Solution:
The absolute values of 6 and (-2) are 6 and 2, respectively. Their difference is 6 – 2 = 4 Now, among 6 and 2, 6 is the larger number, and its original sign ‘+’. Hence, the result will be positive Therefore, 6 + (-2) = 4
Subtract the given integers: 5 – 12
Solution:
Changing the sign of the second number and converting the expression into an addition problem (+5) + (-12) Now, the absolute values of 5 and -12 are 5 and 12, and the difference between the larger and the smaller number is 12 – 5 = 7 Among 5 and 12, 12 is the larger number and has a negative sign. Thus the result is -7
Multiplication
When multiplying two integers:
If both have the same sign, the result is positive. For example (+4) x (+5) = +20
If both have different signs, then the result is negative. For example (+4) x (-5) = – 20
The rules of multiplication are summarized below:
Product of Signs
Result
Examples
+ × +
+
5 × 4 = 20
+ × –
–
5 × -4 = -20
– × +
–
-5 × 4 = 20
– × –
+
-5 × -4 = 20
Multiply (-8) × 4
Solution:
According to the rules of multiplication, the product of a positive and a negative integer will have a negative sign followed by the result of multiplication. Here, (-8) × 4 = -32
Division
Here, the rules are similar to that of multiplication. When dividing two integers:
If both have the same sign, then the result is positive. For example (+8) ÷ (+2) = +4
If they have different signs, then the result is negative. For example (-12) ÷ (+4) = -3
The rules of division are summarized below:
Division of Signs
Result
Examples
+ ÷ +
+
8 ÷ 2 = 4
+ ÷ –
–
8 ÷ -2 = -4
– ÷ +
–
-8 ÷ 2 = -4
– ÷ –
+
-8 × -2 = 4
Divide (-18) ÷ 6
Solution:
According to the division rules, dividing a negative integer by a positive one will give a quotient with a negative sign. Here, (-18) ÷ 6 = -3
Properties
The main properties of integers are:
Closure Property
According to the closure property of integers, when two integers are added or multiplied, it results in an integer.
If ‘a’ and ‘b’ are integers, then:
a + b = integer, for example 3 + = 7 is an integer
a x b = integer, for example 3 × 4 = 12 is an integer
Commutative Property
According to the commutative property, interchanging the position of the operands in operation does not affect the result. It applies only to addition and multiplication.
For any 2 integers’ a’ and ‘b’:
a + b = b + a, for example 3 + 4 = 4 + 3 = 7
a × b = b × a, for example 3 × 4 = 4 × 3 = 12
Associative Property
According to the associative property, interchanging the grouping of two integers does not cause any change in the result of the operation. It applies to addition and multiplication.
For any integers’ a’, ‘b’, and ‘c’:
a + (b + c) = (a + b) + c. For example, 4 +(6 + 2) = (4 + 6)+ 2 = 12
a × (b × c) = (a × b) × c. For example, 4 x(6 × 2) = (4 × 6)x 2 = 48
Distributive Property
According to the distributive property of integers, an expression of the form a ×(b + c), where ‘a’, ‘b’ and ‘c’ are integers can be distributed and written as:
a ×(b + c) = (a × b) + (a × c), for example 3 x (5 + 2) = (3 × 5) + (3 × 2) = 21
Additive Inverse
According to the additive inverse property, adding an integer and its negative value always gives 0.
For any integer’ a’
a + (-a) = 0, for example 2 + (-2) = 0
Multiplicative Inverse
According to the property of multiplicative inverse, multiplication between any integer and its reciprocal always results in 1.
For any integer’ a’
a × ${\dfrac{1}{a}}$ = 1, for example 2 × ${\dfrac{1}{2}}$ = 1
Identity Property
Integers follow the identity property for addition and multiplication.
According to the identity property of addition, when 0 is added to an integer, it results in the integer itself.
For any integer’ a’
a + 0 = a, for example 7 + 0 = 7
According to the identity property of multiplication, when 1 is multiplied by any integer, the result is the integer itself.
For any integer’ a’a × 1 = a, for example 7 × 1 = 7
