# Properties of Subtraction – Closure, Identity, Commutative, Associative, Distributive

The basic four different arithmetic operations are addition, subtraction, multiplication, and division. To perform these arithmetic operations, you just need at least 2 numbers. Subtraction is one of the arithmetic operations that means removing objects from a group. It is represented using the minus sign “-“.

The other names of subtraction are minus, difference, deduct, less, decrease, and take away. For subtracting larger whole numbers, you can use subtraction with regrouping (borrowing) or quick subtraction or subtraction or addition methods. Interested candidates can read further sections to know five different properties of subtraction along with solved examples.

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## How to Subtract Two Whole Numbers?

Students can subtract two larger whole numbers by following these steps.

• Let us take two numbers one is minuend, second is subtrahend.
• Write the number that is to be subtracted from on the top and the number that is to be subtracted on the bottom.
• Begin the process from the rightmost digits of the numbers.
• If the minuend digit is lesser than the subtrahend digit, then barrow 10 from the next digit of minuend which is on the left side.
• Add 10 to the minuend digit and subtract the result from the subtrahend’s digit.
• Don’t forget to mention the borrowed value on the top of the digit.
• Then the next digit becomes (digit – 1)
• Repeat the steps till you are left with nothing on the left side.

### Properties of Subtraction

We are here to explain about five different properties of subtraction. Let us have a look at the following sections to get a clear idea about the topic.

Property 1: Closure Property

The difference between any two integers will also be an integer. If a, b are integers, then a – b is also an integer.

Examples:

15 – 5 = 10

259 – 8 = 251

45 – 48 = -3

16 – 0 = 16

Property 2: Identity Property

When a is an integer other than zero, then a – 0 = a but 0 – a is undefined.

Examples:

15 – 0 = 15 but 0 – 15 is not possibe

2 – 0 = 2 but 0 – 2 is undefined

84 – 0 = 84 but 0 – 84 is not defined

Property 3: Commutative Property

The commutative property of subtraction says that the swapping of terms will affect the difference value. If a and b are two integers, then (a – b) is never equal to (b – a).

Examples:

15 – 4 = 11 and 4 – 15 = -11

So, 15 – 4 ≠ 4 – 15

82 – 45 = 37 and 45 – 82 = -37

So, 82 – 45 ≠ 45 – 82

6 – 10 = -4 and 10 – 6 = 4

So, 6 – 10 ≠ 10 – 6

Property 4: Associative Property

The associative property of subtraction states that if you change the way of grouping numbers, then the result will be different. If a, b, c are three integers, then a – (b – c) is not equal to (a – b) – c.

Examples:

1. 15 – (10 – 2) = 15 – 8 = 7

(15 – 10) – 2 = 5 – 2 = 3

Therefore, 15 – (10 – 2) ≠ (15 – 10) – 2

2. 28 – (6 – 4) = 28 – 2 = 26

(28 – 6) – 4 = 22 – 4 = 18

Therefore, 28 – (6 – 4) ≠ (28 – 6) – 4

3. (156 – 120) – 10 = 36 – 10 = 26

156 – (120 – 10) = 156 – 110 = 46

Therefore, (156 – 120) – 10 ≠ 156 – (120 – 10)

Property 5: Distributive Property

The distributive property states that the integers are subtracted first and then multiplied by the result or multiply first and then subtraction later. If a, b, c are three integers, then a x (b – c) = a x b – a x c.

Examples:

1. 5 x (6 – 7) = 5 x (-1) = -5

5 x 6 – 5 x 7 = 30 – 35 = -5

So, 5 x (6 – 7) = 5 x 6 – 5 x 7

2. 16 x 25 – 16 x 20 = 400 – 320 = 80

16 x (25 – 20) = 16 x 5 = 80

So, 16 x 25 – 16 x 20 = 16 x (25 – 20)

Property 6:

If a, b, c are the whole numbers and a – b = c, then a = c + b.

Examples:

18 – 0 = 18 and 18 = 18 + 0

So, whenever zero is subtracted from any whole number, then we get the whole number.

18 – 18 = 0

So whenever a number is subtracted from the same number, then the difference is zero.

1850 – 1 = 1849 and 1849 + 1 = 1850.

### Solved Examples on Properties of Subtraction

Example 1:

Solve the following.

(i) 415 – 0

(ii) 710 – 2

(iii) 5645 – 455

Solution:

(i) The given integers are 415, 0

As per the identity property, if any number is subtracted from 0, then the difference is a whole number.

So, 415 – 0 = 415.

(ii) The given integers are 710, 2

710 – 2 = 708.

(iii)

10

5 6 4 5

– 4 5 5

5 1 9 0

5645 – 455 = 5190.

Example 2:

Find the missed numbers from the following.

(i) 258 – _____ = 0

(ii) ______ – 90 = 88

(iii) 1652 – ______ = 10

Solution:

(i) 258 – x = 0

We already know that a – b = c, then a = c + b.

258 – 0 = x

As per the identity property, if any number is subtracted from 0, then the difference is a whole number.

So, 258 – 0 = 0

(ii) x – 90 = 88

We already know that a – b = c, then a = c + b.

x = 88 + 90

x = 178

So, 178 – 90 = 88.

(iii) 1652 – x = 10

We already know that a – b = c, then a = c + b.

1652 – 10 = x

x = 1642

So, 1652 – 1642 = 10.

Example 3:

State whether the following statements are correct or not.

(i) 56 x (85 – 25) = 56 x 85 – 56 x 25

(ii) 182 – (72 – 38) = (182 – 72) – 38

(iii) 546 – 546 = 1

Solution:

(i) The given statement is 56 x (85 – 25) = 56 x 85 – 56 x 25

L.H.S = 56 x (85 – 25)

= 56 x 60

= 3360

R.H.S = 56 x 85 – 56 x 25

= 4760 – 1400

= 3360

So, L.H.S = R.H.S

Therefore, the stament is true.

(ii) The given statement is 182 – (72 – 38) = (182 – 72) – 38

L.H.S = 182 – (72 – 38)

= 182 – 34

= 148

R.H.S = (182 – 72) – 38

= (110) – 38

= 72

L.H.S ≠ R.H.S

Therefore, the statement is false.

(iii) The given statement is 546 – 546 = 1

We know that, if a number is subtracted from the same whole number, then the result is zero.

Therefore, the statement is false.

### FAQs on Properties of Subtraction

1. Write the different properties of subtraction?

The five various subtraction properties are closure property, identity property, distributive property, commutative property, and associative property.

2. Describe the terms minuend, subtrahend, and difference?

A minuend is a number that is to be subtracted from, a subtrahend is a number that is to be subtracted, and the difference is the result of subtracting one number from another number.

3. What is the difference between distributive property and associative property.

Distributive property for subtraction is true but associative property for subtraction is false. Distributive property is a x b – a x c = a x (b – c). Associative property is a – (b – c) ≠ (a – b) – c.