Natural Numbers and Whole Numbers

Q1: Consider two whole numbers a and b such that a is greater than b.

i. Is \(a – b\) a whole number? Is this result always true?

Step 1: Whole numbers are closed under subtraction only when the minuend is greater than or equal to the subtrahend.
Step 2: Given: \(a > b\), and both are whole numbers.
Step 3: Let \(a = 9\), \(b = 4\) \[ a – b = 9 – 4 = 5 \] 5 is a whole number.
Conclusion: Yes, \(a – b\) is a whole number if \(a \geq b\).
Answer: Yes, it is a whole number. This result is true when \(a \geq b\).

ii. Is \(b – a\) a whole number? Is this result always true?

Step 1: Given: \(a > b\)
Step 2: Let \(a = 7\), \(b = 3\) \[ b – a = 3 – 7 = -4 \] Step 3: \(-4\) is not a whole number.
Conclusion: No, \(b – a\) is not a whole number when \(b < a\).
Answer: No, it is not a whole number. This result is always true.

Q2: Fill in the blanks

i. \(8 – 0 =\) ____, and \(0 – 8 =\) ________
8 – 0 ≠ 0 – 8, this shows subtraction of whole numbers is not ______

Step 1: \[ 8 – 0 = 8, \quad 0 – 8 = -8 \] Step 2: Since \(8 ≠ -8\), subtraction is not commutative.
Answer: 8; -8; commutative

ii. \(5 – 10 =\) _______, which is not a ______
Subtraction of ______ is not closed.

Step 1: \[ 5 – 10 = -5 \] Step 2: \(-5\) is not a whole number.
Answer: -5; whole number; whole numbers

iii. \(7 – 18 =\) _____ and \((7 – 18) – 5 =\) ________
\(18 – 5 =\) ______ and \(7 – (18 – 5) =\) ________
Is \((7 – 18) – 5 = 7 – (18 – 5)\)?

Step 1: \[ 7 – 18 = -11 \Rightarrow (-11) – 5 = -16 \] Step 2: \[ 18 – 5 = 13 \Rightarrow 7 – 13 = -6 \] Step 3: \[ -16 \ne -6 \] This shows subtraction is not associative.
Answer: -11; -16; 13; -6; No

Q3: Write the identity number, if possible for subtraction of whole numbers.

Step 1: In addition, the identity number is 0, because: \[ a + 0 = a = 0 + a \]Step 2: Let’s check if any number behaves like an identity in subtraction.
Try 0: \[ a – 0 = a \quad \text{(this holds true)} \] But, \[ 0 – a \ne a \quad \text{(this is not true for } a \ne 0) \]Step 3: Subtraction is not commutative or associative, and there’s no number that satisfies: \[ a – x = a \quad \text{and} \quad x – a = a \]Conclusion: There is no identity element for subtraction of whole numbers because no number satisfies the identity condition in all cases.
Answer: No identity number exists for subtraction of whole numbers.

Q4: Fill in the blanks

i. \(12 \times (9 – 6) =\) ________ = ________

Step 1: First solve inside the bracket: \[ 9 – 6 = 3 \] Now multiply: \[ 12 \times 3 = 36 \]Answer: \(12 \times 3 = 36\)

ii. \(12 \times 9 – 12 \times 6 =\) ______ = ________

Step 1: \[ 12 \times 9 = 108, \quad 12 \times 6 = 72 \] Now subtract: \[ 108 – 72 = 36 \]Answer: 108 – 72 = 36

iii. Is \(12 \times (9 – 6) = 12 \times 9 – 12 \times 6\)? ________.

Step 1: LHS: \(12 \times (9 – 6) = 12 \times 3 = 36\)
RHS: \(12 \times 9 – 12 \times 6 = 108 – 72 = 36\)
Answer: Yes

iv. Is this type of result always true?

Step 1: Yes, this is an example of the distributive property of multiplication over subtraction: \[ a \times (b – c) = a \times b – a \times c \]Answer: Yes, always true by distributive property.

Q5: Fill in the blanks

i. \((16 – 8) \times 24 =\) ________ = _______

Step 1: Calculate inside the bracket: \[ 16 – 8 = 8 \] Multiply: \[ 8 \times 24 = 192 \]Answer: \(8 \times 24 = 192\)

ii. \(16 \times 24 – 8 \times 24 =\) ________ – _______ = _______

Step 1: Calculate each multiplication: \[ 16 \times 24 = 384, \quad 8 \times 24 = 192 \] Subtract: \[ 384 – 192 = 192 \]Answer: \(384 – 192 = 192\)

iii. Is \((16 – 8) \times 24 = 16 \times 24 – 8 \times 24\)? __________

Step 1: LHS = \(192\)
RHS = \(192\)
Since LHS = RHS,
Answer: Yes

iv. Is this type of result always true?

Step 1: Yes, this follows the distributive property of multiplication over subtraction: \[ a \times (b – c) = a \times b – a \times c \]Answer: Yes, it is always true.

Q6: Find the difference between the largest number of four digits and the smallest number of six digits.

Step 1: Identify the largest four-digit number: \[ \text{Largest four-digit number} = 9999 \]Step 2: Identify the smallest six-digit number: \[ \text{Smallest six-digit number} = 100000 \]Step 3: Find the difference: \[ \text{Difference} = \text{Smallest six-digit number} – \text{Largest four-digit number} \] \[ = 100000 – 9999 \]Step 4: Calculate the subtraction: \[ 100000 – 9999 = 90001 \]Answer: 90001

Q7: Find the difference between the smallest number of eight digits and the largest number of five digits.

Step 1: Identify the smallest eight-digit number: \[ \text{Smallest eight-digit number} = 10,000,000 \]Step 2: Identify the largest five-digit number: \[ \text{Largest five-digit number} = 99,999 \]Step 3: Find the difference: \[ \text{Difference} = \text{Smallest eight-digit number} – \text{Largest five-digit number} \\ = 10,000,000 – 99,999 \]Step 4: Calculate the subtraction: \[ 10,000,000 – 99,999 = 9,900,001 \]Answer: 9,900,001