Natural Numbers & Whole Numbers — Exercise 3(D)

Show that:

(i) division of whole numbers is not closed.

(ii) any whole number divided by 1, always gives the number itself.

(iii) every non-zero whole number divided by itself gives 1 (one).

(iv) zero divided by any non-zero number is zero only.

(v) a whole number divided by 0 is not defined.

For each part, given above, give two suitable examples.

Answer

(i) Division of whole numbers is not closed because the result of dividing one whole number by another is not always a whole number.

Examples: 5 ÷ 8 and 12 ÷ 24 are not whole numbers, etc.

(ii) Any whole number divided by 1, always gives the number itself.

Examples: 5 ÷ 1 = 5, and 16 ÷ 1 = 16, etc.

(iii) Every non-zero whole number divided by itself gives 1.

Examples: 8 ÷ 8 = 1, and 12 ÷ 12 = 1, etc.

(iv) Zero divided by any non-zero whole number gives zero only.

Examples: 0 ÷ 6 = 0, and 0 ÷ a = 0, if a ≠ 0, etc.

(v) A whole number divided by 0 is not defined.

Examples: 7 ÷ 0 is not defined, and 16 ÷ 0 is not defined, etc.

If x is a whole number such that x ÷ x = x; state the value of x.

Answer

For every non-zero whole number, dividing it by itself gives 1, i.e. x ÷ x = 1.

So, x ÷ x = x means x = 1.

Hence, the value of x is 1.

Fill in the blanks:

(i) 987 ÷ 1 = ...............

(ii) 0 ÷ 987 = ...............

(iii) 336 - (888 ÷ 888) = ...............

(iv) (23 ÷ 23) - (437 ÷ 437) = ...............

Answer

(i) Any whole number divided by 1 gives the number itself.

987 ÷ 1 = 987.

(ii) Zero divided by any non-zero whole number gives 0.

0 ÷ 987 = 0.

(iii) Any non-zero whole number divided by itself gives 1.

⇒ 888 ÷ 888 = 1

⇒ 336 - (888 ÷ 888) = 336 - 1 = 335.

(iv) Any non-zero whole number divided by itself gives 1.

⇒ 23 ÷ 23 = 1 and 437 ÷ 437 = 1

⇒ (23 ÷ 23) - (437 ÷ 437) = 1 - 1 = 0.

Which of the following statements are true?

(i) 12 ÷ (6 × 2) = (12 ÷ 6) × (12 ÷ 2)

(ii) a ÷ (b - c) = $\dfrac{a}{b} - \dfrac{a}{c}$

(iii) (a - b) ÷ c = $\dfrac{a}{c} - \dfrac{b}{c}$

(iv) (15 - 13) ÷ 8 = (15 ÷ 8) - (13 ÷ 8)

(v) 8 ÷ (15 - 13) = $\dfrac{8}{15} - \dfrac{8}{13}$

Answer

(i) L.H.S. = 12 ÷ (6 × 2) = 12 ÷ 12 = 1.

R.H.S. = (12 ÷ 6) × (12 ÷ 2) = 2 × 6 = 12.

Since L.H.S. ≠ R.H.S., this statement is false.

(ii) Division is not distributive over subtraction in this form, so a ÷ (b - c) ≠ $\dfrac{a}{b} - \dfrac{a}{c}$.

This statement is false.

(iii) (a - b) ÷ c = $\dfrac{a - b}{c} = \dfrac{a}{c} - \dfrac{b}{c}$.

This statement is true.

(iv) L.H.S. = (15 - 13) ÷ 8 = 2 ÷ 8 = $\dfrac{2}{8} = \dfrac{1}{4}$.

R.H.S. = (15 ÷ 8) - (13 ÷ 8) = $\dfrac{15}{8} - \dfrac{13}{8} = \dfrac{2}{8} = \dfrac{1}{4}$.

Since L.H.S. = R.H.S., this statement is true.

(v) L.H.S. = 8 ÷ (15 - 13) = 8 ÷ 2 = 4.

R.H.S. = $\dfrac{8}{15} - \dfrac{8}{13}$, which is a negative number.

Since L.H.S. ≠ R.H.S., this statement is false.

Hence, statements (iii) and (iv) are true.