# Subtracting Integers – Definition, Rules, Steps, Examples, Facts

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## What Is Meant by Subtracting Integers in Math?

Subtracting integers is the method of finding the difference between two integers. These two integers may have the same sign or different signs. The set of integers is represented by

Z={…,-3,-2,-1,0,1,2,3,…}.

If we subtract the integer b from the integer a, we write it as a – b.

Example: Subtract 7 from 9.

9 – 7 = 2

We can also write every subtraction problem as an addition problem. Replace the – sign by + sign, and replace the second integer by its additive inverse. This can be written symbolically as

a – b = a + (-b)

Finally, you can find the answer using the rules for adding integers.

Additive inverse of an integer is written by simply removing its sign, keeping only the numerical value.

Examples:

Additive inverse of 7 is -7.

Additive inverse of -9 is 9.

## Subtracting Integers: Definition

Subtracting integers is the method of finding the difference between two integers having the same or different signs.

Subtraction of integers can be written as the addition of the first number and the additive inverse of the second number.

## Subtrahend and Minuend

When we subtract one number from another, the number being subtracted is called subtrahend, the first number is called minuend.

Any addition fact can be written as a subtraction fact. Also, any subtraction fact can be written as an addition fact. So, the rules for adding integers and the rules for subtracting integers correspond to each other.

## How to Subtract Integers

The most simple rule to remember when subtracting integers is to convert the problem as the addition problem and use the rules for adding integers.

• Subtracting Integers with the Same Sign: Let’s understand how to subtract integers with the same signs.

i) Positive integer Positive integer

Here, the order of the numbers is important to decide the sign of the answer.

The answer of the result will be  positive when subtracting a smaller number from a larger number.

8-1=7

30 – 13 = 17

The answer of the result will be negative when subtracting a larger number from a smaller  number.

4 – 7 = -3

1-5=-4

ii) Negative integer Negative integer

(-5)-(-1) = (-5)+1 =-4

(-2) – (-5) = -2 + 5 = 3

• Subtracting Integers with Different Signs: Now, let’s understand how to subtract integers with different signs with examples.

i) Positive Integer Negative integer

It is clear that we are subtracting a smaller value from a larger value. So, the answer will always be positive in this case.

5-(-1) = 5 + 1 = 6

2 – (-5) = 2 + 5 = 3

ii) Negative Integer Positive Integer

It is clear that we are subtracting a larger value from a smaller value. So, the answer will always be negative in this case.

(-1)-5 = (-1) + (-5) = -6

(-2) – (5) = (-2) + (-5) = -7

## Properties of Subtraction of Integers

There are some properties related to the subtraction of integers. They are as follows:

• Closure property: The subtraction of any two integers is an integer. For any two integers a and b, the difference a -b is also an integer.
• Commutative property: Subtraction of integers is not commutative. For any two integers a and b, we have a – b b – a.

2 – 4 4 – 2  and -6 – (-3 ) -3 – (- 6)

• Associative property: The subtraction of integers is not associative. That is, for any three integers a, b and c, we have, a -(b – c) (a – b) – c

For example, 2 – ( 5 – 3) = 2 – 2 = 0

(2 – 5) – 3 = – 3 – 3 = – 6 ,

Here, 0 6

• Identity property: We have  a – 0 = a , for any integer a indicating that 0 is correct identity for subtraction.vIt’s important to note that 0 – a a is not the left identity

## Subtracting Integers on a Number Line

• Every subtraction fact can be written as an addition fact.
• Adding a positive integer is done by moving towards the right side (or the positive side) of the number line.
• Adding a negative integer is done by moving towards the left side (or the negative side) of the number line.
• Any one of the given integers can be taken as the base point from where we start moving on the number line.

Example: Subtract 4 from 2.

We have to find 2 – 4.

Express 2 – 4 as the addition fact.

2 – 4 = 2 + (-4).

Locate integer with greater absolute value, 2.

Start from 2, take 4 jumps to the left side as we are subtracting 4 to 2.

Therefore, -2 is the required answer.

• Integers are the negative numbers, zero, and positive numbers.
• Subtracting integers is the method of finding the difference between two integers with the same or different signs.
• To subtract two integers, add the opposite of the second integer to the first integer. This can be written symbolically as a
• – b = a + (- b)
• Adding a negative number is just like subtracting a positive number.
3 + (- 4) = 3 -(+4)
• Subtracting a negative number is just like adding a positive number.
• 3 -(- 4) = 3 + 4

## Conclusion

In this article, we learnt about subtraction of integers definition, rules, properties,steps and how to subtract integers on a number line. Let’s solve some examples and practice problems to understand the concept better.

## Solved Examples on Subtracting Integers

Example 1: Subtract: 46 from 80.

Solution:

We have to subtract two negative integers.

Here, we have to find (-80) – (-46).

(-80) – (-46)

=-80 + 46

= -(80-46)

=-34

Example 2: Subtract: 8 from 7.

Solution:

Here, we have to subtract two integers with different signs.

Let’s write it in the form of an expression,

7 – (- 8 )

= 7 + 8

= 15

Thus, 7 – (- 8 ) = 15

Example 3: Find 1 ( 5 ) using a number line.

Solution:

1 – (- 5 ) = 1 + 5

Start from 1 and take 5 jumps to the right.

Thus, 1 – (- 5 ) = 6

## Practice Problems on Subtracting Integers

1

### Subtraction of integers can be written as the ________ of the opposite number.

subtraction
multiplication
division
CorrectIncorrect
Subtraction of integers can be written as the addition of the opposite number.
2

### (-9)-(99)=

999
108
$-108$
$-90$
CorrectIncorrect
Correct answer is: $-108$
$(-9)-(99)=(-9)+(-99) = -108$
3

### $1000 - (- 1)=$

$-1001$
$1001$
999
$-999$
CorrectIncorrect
Correct answer is: $1001$
$1000- (- 1) = 1000 + 1 = 1001$
4

### When subtracting a positive integer from a negative integer, the answer will have

+ sign
- sign
the sign of the larger number
cannot be said
CorrectIncorrect
When subtracting a positive integer from a negative number, the answer will always have a negative sign since we subtract a larger number from a smaller number.
5

### Which subtraction fact does the given number line represent?

$3 + 5 = 8$
$8 \;-\; 3 = 5$
$8 \;-\; 5 = 3$
$8 + 3 = 11$
CorrectIncorrect
Correct answer is: $8 \;-\; 5 = 3$
Starting from 8, we took 5 steps back and landed on 3. So, the image shows the equation
$8 \;-\; 5 = 3$

## Frequently Asked Questions on Subtracting Integers

Integers include positive integers, negative integers, and zero. It is a number with no decimal or fraction part.

Set of integers = {…,−5,−4,−3,−2,−1, 0, 1, 2, 3, 4, 5,…}

In a subtraction equation, minuend is the number from which another number would be subtracted. A subtrahend is the term that denotes the number being subtracted from another.

The commutative property does not hold for subtraction. It means for any  two integers a and b, a – b ≠ b – a. For example, 2 – 4 ≠ 4 – 2

2 – 4 = – 2 and 4 – 2 = 2 and  -2 ≠ 2

Thus, commutative property does not hold true for subtraction.

The associative property does not hold for subtraction. It means for any three integers a, b, and c, a -(b – c) ≠ (a – b) – c

For example, 2 – ( 5 – 3) = 2 – 2 = 0

(2 – 5) – 3 = – 3 – 3 = – 6 ,

Here, 0 6

Thus, associative property does not hold true for subtraction.