Informative line

# Number Line

• A number line is a pictorial representation of real numbers.
• We can plot whole numbers on a number line. Number line can be drawn as long as we want but we can never find infinity on it.
• Zero lies at the middle of the number line.

• The positive numbers always lie on the right side of zero and negative numbers always lie on the left side of zero.

• A number which is farther to the right on the number line is greater than its previous numbers.

$$-4<-3<-2<-1<0<1<2<3<4$$

• The section of the number line between two numbers is called an interval.

• If the numbers are placed in their correct order, then the interval of numbers is always equal.

#### Which point represents whole number $$6$$ on the number line? [Each interval is equal]

A Point $$A$$

B Point $$B$$

C Point $$C$$

D Point $$D$$

×

Since whole numbers start from zero $$(0)$$, so starting from zero and moving $$6$$ points forward, the point which comes on the number line represents $$6.$$

So, point $$C$$ represents the whole number $$6$$ on the number line.

Hence, option (C) is correct.

### Which point represents whole number $$6$$ on the number line? [Each interval is equal]

A

Point $$A$$

.

B

Point $$B$$

C

Point $$C$$

D

Point $$D$$

Option C is Correct

# Closure Property

• Closure property is obeyed for division only if the division of two whole numbers is a whole number.

Case 1: Closure property is obeyed in division for whole numbers

Division of any two whole numbers can be a whole number.

Whole number $$\div$$ whole number = whole number

This case is only obeyed when divisor is a factor of dividend.

For example:

$$8 \div 4 = 2$$

$$4$$ is a factor of $$8$$ and both are whole numbers. So, the result is also a whole number i.e. 2.

Case 2: Closure property is not obeyed in division for whole numbers

Division of any two whole numbers is not always a whole number.

Whole number $$\div$$ whole number $$\neq$$ whole number

In this case, divisor is not a factor of dividend. so the quotient is not a whole number.

For example: $$2 \div 4 = \dfrac{1}{2}$$

$$2$$ and $$4$$, both are whole numbers but $$\dfrac{1}{2}$$ is not a whole number.

#### Which one of the following options represents that the closure property is obeyed in division for whole numbers?

A $$12 \div2 = 6$$

B $$2 \div 12 = \dfrac{1}{6}$$

C $$4 \div 12 = \dfrac{1}{3}$$

D $$9 \div 18 = 0.5$$

×

Closure property is obeyed in division only if the division of two whole numbers is a whole number.

Option (A) represents that closure property is obeyed in division for whole numbers as on division of two whole numbers i.e. ($$12$$ and $$2$$), the result is also a whole number $$(6)$$.

Hence, option (A) is correct.

Option (B) represents that closure property is not obeyed in division for whole numbers because $$\dfrac{1}{6}$$ is not a whole number.

Hence, option (B) is incorrect.

Option (C) represents that closure property is not obeyed in division for whole numbers because $$\dfrac{1}{3}$$ is not a whole number.

Hence, option (C) is incorrect.

Option (D) represents that closure property is not obeyed in division for whole numbers because $$0.5$$ is not a whole number.

Hence, option (D) is incorrect.

### Which one of the following options represents that the closure property is obeyed in division for whole numbers?

A

$$12 \div2 = 6$$

.

B

$$2 \div 12 = \dfrac{1}{6}$$

C

$$4 \div 12 = \dfrac{1}{3}$$

D

$$9 \div 18 = 0.5$$

Option A is Correct

# Zero Property of Division

• The zero property of division have two cases.

Case (i): If we divide zero by a whole number, the answer will be zero.

$$0 \;\div 12 = 0$$

Case (ii): If a whole number is divided by zero, then the problem cannot be solved.

$$12 \div 0$$

#### Solve: $$100 \div 0$$

A 0

B Undefined

C 1

D 100

×

Whole numbers divided by zero is undefined.

So, $$100 \div 0$$ is undefined.

Hence, option (B) is correct.

### Solve: $$100 \div 0$$

A

0

.

B

Undefined

C

1

D

100

Option B is Correct

# Division on Number line

• Number line is quite easy way of understanding operation of whole numbers.
• To show division of 15 by 3 on the number line, we start from 15 and move 3 points backward at a time.

• Now, keep on making such moves till we reach zero.

• We made 5 moves of 3 points each.
• Thus, the number of moves of 3 steps represent the result.

So, $$15 \div 3 = 5$$ (moves)

#### Which one of the following number lines represents $$20 \div 4 = ?$$

A

B

C

D

×

To show division of 20 by 4 on the number line, start from 20 and move 4 points backward at a time.

Now, keep on making such moves till we reach zero.

We made 5 moves of 4 units each.

Thus, $$20 \div 4 = 5$$

Hence, option (D) is correct.

### Which one of the following number lines represents $$20 \div 4 = ?$$

A
B
C
D

Option D is Correct

# Commutative Property

• Division is not commutative for whole numbers. This means that if we change the order of numbers in the division expression the result also changes.

$$(a \div b)\neq(b\div a)$$

• For example: Consider division of two whole numbers $$8$$ and $$4$$.

$$(8 \div 4)\neq(4 \div 8)\\ \;\;\;\;2\;\;\;\;\;\;\;\;\;\;\;\;\dfrac{1}{2}$$

In both the expressions, answer is not same.

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

#### Choose the correct statement regarding the expression: $$9 \div 3 \neq 3 \div 9$$

A It shows closure property is not obeyed in division.

B It shows commutative property is not obeyed in division.

C It shows commutative property is obeyed in division.

D It shows closure property is obeyed in division.

×

Given expression: $$9 \div 3 \neq 3 \div 9$$

The given expression represents commutative property but this property is not obeyed in division because on interchanging the numbers the result also changes.

So, the expression shows commutative property is not obeyed in division.

Hence, option (B) is correct.

### Choose the correct statement regarding the expression: $$9 \div 3 \neq 3 \div 9$$

A

It shows closure property is not obeyed in division.

.

B

It shows commutative property is not obeyed in division.

C

It shows commutative property is obeyed in division.

D

It shows closure property is obeyed in division.

Option B is Correct

# Associative Property

• Associative property does not work with division. This means that when we change the grouping of numbers, the result also changes.

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

• For example:

$$1\div (4\div2)\neq(1\div4)\div2$$

$$1\div2\neq\dfrac{1}{4}\div2$$

$$\dfrac{1}{2}\neq\dfrac{1}{8}$$

Answer of both the expressions are not same.

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

#### Choose the correct statement regarding the expression: $$9\div(10\div2)\neq(9\div10)\div2$$

A It shows closure property is obeyed in division.

B It shows associative property is obeyed in division.

C It shows commutative property is not obeyed in division.

D It shows associative property is not obeyed in division.

×

Given expression: $$9\div(10\div2)\neq(9\div10)\div2$$

The given expression represents associative property but this property is not obeyed in division because on changing the grouping of numbers, the result also changes.

So, the expression shows associative property is not obeyed in division.

Hence, option (D) is correct.

### Choose the correct statement regarding the expression: $$9\div(10\div2)\neq(9\div10)\div2$$

A

It shows closure property is obeyed in division.

.

B

It shows associative property is obeyed in division.

C

It shows commutative property is not obeyed in division.

D

It shows associative property is not obeyed in division.

Option D is Correct