Natural Numbers and Whole Numbers

Q1: Show that:
For each part, given above, give two suitable examples.

i. division of whole numbers is not closed.

Step 1: Division is closed if dividing any two whole numbers always gives a whole number.
Step 2: Check with examples: \[ 6 \div 3 = 2 \quad \text{(whole number)} \] \[ 7 \div 3 = \frac{7}{3} = 2.333\ldots \quad \text{(not a whole number)} \]Step 3: Since division of two whole numbers can give a non-whole number, division is not closed in whole numbers.
Answer: Division of whole numbers is not closed.

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

Step 1: Check examples: \[ 12 \div 1 = 12 \] \[ 0 \div 1 = 0 \]Step 2: Dividing any whole number by 1 returns the number itself.
Answer: Any whole number ÷ 1 = the number itself.

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

Step 1: Check examples: \[ 7 \div 7 = 1 \] \[ 25 \div 25 = 1 \]Step 2: Dividing a non-zero whole number by itself always gives 1.
Answer: Non-zero whole number ÷ itself = 1.

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

Step 1: Check examples: \[ 0 \div 5 = 0 \] \[ 0 \div 123 = 0 \]Step 2: Zero divided by any non-zero number is zero.
Answer: 0 ÷ any non-zero number = 0.

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

Step 1: Division by zero is undefined in mathematics.
Step 2: Examples: \[ 5 \div 0 = \text{undefined} \] \[ 0 \div 0 = \text{undefined} \]Answer: Division by zero is not defined.

Q2: If x is a whole number such that \(x\ \div\ x\ =\ x\); state the value of x.

Step 1: Given: \[ \frac{x}{x} = x \]Step 2: For division \(\frac{x}{x}\) to be defined, \(x \neq 0\). Also, \(\frac{x}{x} = 1\) for all \(x \neq 0\).So, the equation becomes: \[ 1 = x \]Step 3: Therefore, \[ x = 1 \]Answer: x = 1

Q3: Fill in the blanks

i. \(987\div1=\)________

Step 1: Divide 987 by 1: \[ 987 \div 1 = 987 \] Answer: 987

ii. \(0\div987=\)________

Step 1: Divide 0 by 987: \[ 0 \div 987 = 0 \] Answer: 0

iii. \(336-(888\div888)=\)_________

Step 1: Calculate \(888 \div 888\): \[ 888 \div 888 = 1 \] Step 2: Substitute and solve: \[ 336 – 1 = 335 \] Answer: 335

iv. \((23\div23)-(437\div437)=\)_________

Step 1: Calculate each division: \[ 23 \div 23 = 1, \quad 437 \div 437 = 1 \] Step 2: Subtract: \[ 1 – 1 = 0 \] Answer: 0

Q4: Which of the following statements are true?

i. \(12\div(6\times2)=(12\div6)\times(12\div2)\)

Step 1: Evaluate left side: \[ 12 \div (6 \times 2) = 12 \div 12 = 1 \] Step 2: Evaluate right side: \[ (12 \div 6) \times (12 \div 2) = 2 \times 6 = 12 \] Step 3: Check equality: \[ 1 \neq 12 \] Answer: False

ii. \(a\div(b-c)=\frac{a}{c}-\frac{b}{c}\)

Step 1: Given: \[ a \div (b – c) = \frac{a}{c} – \frac{b}{c} \] This is not generally true by division or fraction rules.
Answer: False

iii. \((15-13)\div8=(15\div8)-(13\div8)\)

Step 1: Calculate left side: \[ (15 – 13) \div 8 = 2 \div 8 = \frac{1}{4} \] Step 2: Calculate right side: \[ (15 \div 8) – (13 \div 8) = \frac{15}{8} – \frac{13}{8} = \frac{2}{8} = \frac{1}{4} \] Step 3: Since both sides equal \(\frac{1}{4}\), the statement is true.
Answer: True

iv. \(8\div(15-13)=\frac{8}{15}-\frac{8}{13}\)

Step 1: Calculate left side: \[ 8 \div (15 – 13) = 8 \div 2 = 4 \] Step 2: Calculate right side: \[ \frac{8}{15} – \frac{8}{13} = \text{not equal to } 4 \] Since \(\frac{8}{15} – \frac{8}{13}\) is approximately: \[ 0.5333 – 0.6154 = -0.0821 \neq 4 \]Answer: False