- 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.

A Point \(A\)

B Point \(B\)

C Point \(C\)

D Point \(D\)

- 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.

A \(12 \div2 = 6\)

B \(2 \div 12 = \dfrac{1}{6}\)

C \(4 \div 12 = \dfrac{1}{3}\)

D \(9 \div 18 = 0.5\)

- 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.

\(12 \div 0\)

The answer is undefined.

- 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)

- 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.

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.

- 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.

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.