Informative line

# Subtraction on a Number Line

• Number line is quite an easy way of understanding operations on whole numbers.
• Suppose we want to perform subtraction of $$8$$ and $$15.$$
• To show subtraction of $$8$$ from $$15$$ on the number line, first draw a number line from $$0$$ to $$15$$.

• Start from $$15$$ and then move $$8$$ points backward (as we have to subtract $$8$$), by taking $$1$$ point at a time.

We reach the number $$7$$.

Thus, $$15-8=7$$

#### Which one of the following number lines represents $$8-4?$$

A

B

C

D

×

To show subtraction of $$4$$ from $$8$$ on the number line, start from $$8$$ and then move $$4$$ points backward (as we have to subtract $$4$$), by taking one point at a time.

We reach the number $$4$$.

Thus, $$8-4=4$$

Hence, option (D) is correct.

### Which one of the following number lines represents $$8-4?$$

A
B
C
D

Option D is Correct

# Closure Property (Subtraction)

• Closure property of subtraction states that the difference of two whole numbers is not always a whole number.

Case 1: Closure property is obeyed in subtraction of whole numbers

Whole number – whole number = whole number

• This case is only obeyed when the first term is greater than or equal to the second one.

$$a-b=c$$

$$(a\geq b)$$ and $$a,b,c$$ are whole numbers.

Example:

Consider $$a = 9$$ and $$b = 7$$

then $$a-b = 9 - 7 = 2$$

where $$9 ,7$$ and $$2$$ are all whole numbers and $$9$$ is greater than $$7$$.

Case 2: Closure property is not obeyed in subtraction of whole numbers

• This case is only obeyed when the first number is smaller than the second one. In this, the resultant is not a whole number.

$$a-b \neq c$$

$$b>a, \,a$$ and $$b$$ are whole numbers and $$c$$ is not a whole number.

Example:

Consider $$a = 7$$ and $$b = 9$$

$$a-b = 7 - 9 = -2$$

$$7, 9$$ are whole numbers and $$- 2$$ is not a whole number.

#### Given two numbers, $$4$$ and $$6$$. Which one of the following options represents that closure property is not obeyed in subtraction of whole numbers?

A $$4-6 = -2$$

B $$6-4 = 2$$

C $$4+6 =10$$

D $$4+6 = 6+4$$

×

According to the closure property of subtraction, the difference of two whole numbers is not always a whole number.

Option (A) represents that closure property is not obeyed in subtraction because the difference of two whole numbers is not a whole number.

Hence, option (A) is correct.

Option (B) represents that closure property is obeyed in subtraction because the difference of two whole numbers is also a whole number.

Hence, option (B) is incorrect.

Option (C) represents the closure property of addition.

Hence, option (C) is incorrect.

Option (D) represents the commutative property of addition.

Hence, option (D) is incorrect.

### Given two numbers, $$4$$ and $$6$$. Which one of the following options represents that closure property is not obeyed in subtraction of whole numbers?

A

$$4-6 = -2$$

.

B

$$6-4 = 2$$

C

$$4+6 =10$$

D

$$4+6 = 6+4$$

Option A is Correct

# Commutative Property

• Subtraction is not commutative for whole numbers, this means that when we change the order of numbers in subtraction expression, the result changes.

$$(a-b)\neq(b-a)$$

• For example: Consider the subtraction of two whole numbers $$5$$ and $$7$$.

$$\begin{array} {c c c} (7-5) & \neq & (5-7)\\ 2&\neq &-2 \end{array}$$

In both the expressions, answer is different.

• So, we can say that subtraction is not commutative for whole numbers.

#### Choose the correct option regarding the given numerical statement. $$3-4 \neq4-3$$

A It shows that closure property is not obeyed in subtraction.

B It shows that closure property is obeyed in subtraction.

C It shows that commutative property is not obeyed in subtraction.

D It shows that commutative property is obeyed in subtraction.

×

The given numerical statement is:

$$3-4 \neq 4-3$$

It shows that the difference of 3 and 4 changes, when we interchange their positions. Thus, it indicates that when we change the order of numbers in subtraction expression, the result changes.

So, it shows that commutative property is not obeyed in subtraction.

Hence, option (C) is correct.

### Choose the correct option regarding the given numerical statement. $$3-4 \neq4-3$$

A

It shows that closure property is not obeyed in subtraction.

.

B

It shows that closure property is obeyed in subtraction.

C

It shows that commutative property is not obeyed in subtraction.

D

It shows that commutative property is obeyed in subtraction.

Option C is Correct

# Associative Property

• Associative property does not work with subtraction, this means that when we change the order of numbers in subtraction, the result also changes.

$$a-(b-c)\neq(a-b)-c$$

For example:

$$1-(9-8)\neq(1-9)-8$$

$$1-1\neq(-8)-8$$

$$0\neq -16$$

Answers of both the expressions are not same.

So we can say that subtraction is not associative for whole numbers.

#### Which option represents the correct pair?                                                                                                                                                                                                Equation  Property (A) $$5 + (3+6) = (5+3) + 6$$ Commutative property of subtraction (B) $$5 - (6-3)\neq (5-6) - 3$$ Associative property of subtraction (C) $$5+6 \;= \;6+5$$ Closure property of subtraction (D) $$5-4 \;\neq\; 4-5$$ Commutative property of addition

A A

B B

C C

D D

×

Option (A) represents associative property of addition, which says that changing the grouping of the addends (terms) will not change the sum.

$$5+(3+6) = (5+ 3) + 6$$

Hence, option (A) is incorrect.

Option (B) represents associative property is not obeyed in subtraction. This means that when we change the order of numbers in subtraction, the result also changes.

$$5-(6-3) \neq (5-6)- 3$$

Hence, option (B) is correct.

Option (C) represents commutative property of addition, which says that the numbers can be added in any order and we still get the same answer.

$$5+6 = 6+5$$

Hence, option (C) is incorrect.

Option (D) represents commutative property is not obeyed in subtraction. This means that when we change the order of numbers in subtraction expression, the result also changes.

$$5-4 \; \neq\; 4-5$$

Hence, option (D) is incorrect.

### Which option represents the correct pair?                                                                                                                                                                                                Equation  Property (A) $$5 + (3+6) = (5+3) + 6$$ Commutative property of subtraction (B) $$5 - (6-3)\neq (5-6) - 3$$ Associative property of subtraction (C) $$5+6 \;= \;6+5$$ Closure property of subtraction (D) $$5-4 \;\neq\; 4-5$$ Commutative property of addition

A

A

.

B

B

C

C

D

D

Option B is Correct