What are Integers?
Integers are composed of whole numbers which can be positive, negative or zero. Integers do not include fractions. The numbers 0,1 ,2 and so on; -1, -2, -3, -4 and so on are examples of integers.
How to Subtract Integers?
The process of subtracting integers is similar to adding integers. Removing (or subtracting) a positive number is the same as adding a number whose value is the additive inverse (or opposite).
Let’s take an example: 4 – 2
The opposite or additive inverse of 2 is – 2,
so, 4 – 2 = 4 + ( – 2)
Similarly, removing (or subtracting) a negative number is the same way as adding a number whose value is the additive inverse (or opposite).
Let’s take an example: 4 – ( – 2).
The opposite or additive inverse of – 2 is 2,
So, 4 – ( – 2) = 4 + 2
There are two methods to subtract integers-
- Using a number line
- Using absolute values
Lets learn about these methods in detail.
Modeling Integer Subtraction using Number Line
We know that a number line can be used to find the sum of positive numbers, which involves movement to the right on the number line. Similarly, the difference between two positive numbers can be determined by a movement to the left on the number line.
- Subtracting a positive 𝒒-value is represented on the number line as moving to the left on a number line.
- Subtracting a negative 𝒒-value is represented on the number line as moving to the right on a number line.
Hence, to subtract an integer, simply add its opposite.
For example: 3 – 5 = ?
3 – 5 = 3 + ( – 5) = – 2
This example can be solved by either subtracting 5 from 3 or by adding – 5 to 3.
In both ways we get the same answer, that is, – 2.
Let’s find – 3 – 5 using the number line
Solution: First, draw an arrow from 0 to – 3 to represent – 3. Then draw an arrow 5 units to the left to represent subtraction of 5, or addition of – 5.
So, the answer will be – 3 – 5 = – 8.
Similarly let’s find – 3 – (- 8) using the number line.
Solution: Draw an arrow from 0 to – 3 to represent – 3. Then draw an arrow 8 units to the right to represent subtracting – 8, or adding 8.
So, the answer will be -3 – (-8) = 5.
Difference between Integers using Absolute Values
We know that subtracting an integer is the same as adding its additive inverse.
So when finding the difference between two integers, take the opposite of the subtrahend and add it to the minuend value.
Let’s understand this with the help of a few examples:
Find 4 – 11.
Solution: 4 – 11 = 4 + (- 11) (Adding the opposite of 11)
= – 7 (here, we added 4 and -11)
Therefore, the difference is -7.
Let’s take another example, find -7 – ( – 14)
Solution: -7 – ( – 14) = – 7 + 14 (Adding the opposite of – 14)
= + 7 (here, we added – 7 and 14)
Therefore, the difference is +7.
Find 8 – ( – 5).
Solution: 8 – ( – 5) = 8 + 5 (Adding the opposite of – 5)
= 13 (here, we added 8 and 5)
Therefore, the difference is + 13.
Solved Examples
Example 1: Find the difference using number line
A. – 5 – 2.
B. 6 – ( – 5)
C. 1 – 4.
Solution:
A. Firstly, draw an arrow from 0 to -5 to represent -5. Then draw an arrow 2 units to the left to represent subtracting 2, or adding -2.
So, the answer will be – 5 – 2 = – 7.
B. Draw an arrow from 0 to 6 to represent 6. Then draw an arrow 5 units to the left to represent subtracting – 5, or adding 5.
So, the answer will be 6 – ( – 5) = 11.
C. Firstly, draw an arrow from 0 to 1 to represent 1. Then draw an arrow 4 units to the left to represent subtracting 4, or adding – 4.
So, the answer will be 1 – 4 = – 3.
Example 2: Find the difference:
A. 8 – 3.
B. 9 – 17.
C. 10 – ( – 8).
D. – 12 – ( – 12).
Solution:
A. 8 – 3 = 8 + ( – 3) (Adding the opposite of 3)
= 5 (add 8 and – 3)
Therefore, the difference is 5.
B. 9 – 17 = 9 + ( – 17) (Adding the opposite of 17)
= –8 (add 9 and -17)
Therefore, the difference is – 8.
C. 10 – ( – 8) = 10 + 8 (Adding the opposite of -8)
= 18 (add 10 and 8)
Therefore, the difference is +18.
D. – 12 – ( – 12) = – 12 + 12 (Adding the opposite of – 12)
= 0 (add – 12 and 12)
Therefore, the difference is 0.
Example 3:
Joey purchased an electric toy car which costs $120. But Joey had only $112 dollars. How much extra money did Joey need to buy that electric car?
Solution:
The cost of the electric toy car = $120.
The money Joey had = $112.
To find the extra money Joey needs to buy that electric toy car, we need to subtract the cost of the car and the money Joey had.
The extra money Joey needed to buy the electric toy car = $120 – $112
= $120 + (- $112) (add opposite of 112)
= $8. (add 120 and – 112)
So, Joey needs $8 extra money to buy the electric toy car.
Example 4:
Joseph had $421 in his bank account. Joseph paid $210 for his house rent and spent $124 to buy a gift for his friend and he spent $19 for shopping. How much balance did Joseph have in his bank account ?
Solution:
Money joseph had in his account = $421.
The money he spent on paying house rent = $210.
The money he spent on shopping = $124.
The money he spent for his friend’s gift = $19
The money remaining in Joseph’s account = $421 – $210 – $124 – $19
= $421 + $( – 210 – 124 -19) (add opposites of 210, 124 and 19)
= $421 + $( – 353) (simplify)
= $68 (add)
Balance amount in Joseph’s bank account is $68.
The additive inverse of a number is defined as the value that when added to the original number gives zero as a result. It’s the amount we add to a number to make it equal to zero. The additive inverse of the number x is -x.
x + (-x) = x – x = 0
The identity property of subtraction states that subtracting zero from an integer yields the integer itself.
x – 0 = x for any integer x