Chapter Test Paper — Set 1

Answer: (b) 0

Natural numbers are the counting numbers: 1, 2, 3, 4, ... They start from 1. Zero (0) is not a natural number; it belongs to the set of whole numbers. Options (a), (c), and (d) are all natural numbers.

Answer: (b) Brahmagupta

The Indian mathematician Brahmagupta (7th century CE) formally defined zero and its arithmetic properties in his work Brahmasphutasiddhanta. He was the first to treat zero as a number and establish rules for arithmetic operations involving it.

Answer: (c) Integer

−7 is a negative integer. Integers include all positive whole numbers, zero, and all negative whole numbers: ..., −3, −2, −1, 0, 1, 2, 3, ...
−7 is NOT a natural number (which start from 1), NOT a whole number (which start from 0), and NOT irrational (it can be written as −7/1, a ratio of two integers).

Answer: 0 (Zero)

The set of whole numbers is W = {0, 1, 2, 3, ...}. The smallest element of this set is 0.

Answer: True

Every natural number IS a whole number. The set of natural numbers ℕ = {1, 2, 3, ...} is a subset of the whole numbers W = {0, 1, 2, 3, ...}. So ℕ ⊂ W.

Answer: Undefined

Division by zero is undefined. There is no number that when multiplied by zero gives a non-zero result. For example, a ÷ 0 is not defined for any value of a.

(i) (−8) + 5:
When adding a positive and a negative integer, subtract the smaller absolute value from the larger and keep the sign of the larger.
|−8| = 8 > |5| = 5, so the answer is negative.
(−8) + 5 = −(8 − 5) = −3 [1 mark]

(ii) (−3) − (−7):
Subtracting a negative number is the same as adding its positive:
(−3) − (−7) = (−3) + 7 = +(7 − 3) = 4 [1 mark]

Natural Numbers (ℕ): {1, 2, 3, 4, 5, ...}   Examples: 1, 7, 25 [½ mark]

Whole Numbers (W): {0, 1, 2, 3, 4, ...}   Examples: 0, 3, 12 [½ mark]

Integers (ℤ): {..., −3, −2, −1, 0, 1, 2, 3, ...}   Examples: −5, 0, 8 [1 mark]

Relationship: ℕ ⊂ W ⊂ ℤ

(i) 0.75 as p/q:
0.75 = 75/100 = 3/4   (dividing numerator and denominator by 25)
So 0.75 = 3/4 [1 mark]

(ii) −3 as p/q:
Any integer n can be written as n/1.
So −3 = −3/1 [1 mark]

√9 is a RATIONAL number. [1 mark]

Reason: √9 = 3, because 3 × 3 = 9.
Since 3 is a whole number, it can be written as 3/1, which is in the form p/q where p = 3, q = 1, and q ≠ 0.
Therefore √9 = 3 is a rational number. [1 mark]

There is exactly ONE whole number that is not a natural number: 0 (Zero). [1 mark]

Explanation:
Natural Numbers: ℕ = {1, 2, 3, 4, ...}
Whole Numbers: W = {0, 1, 2, 3, 4, ...}
The only element in W that is not in ℕ is 0. All other whole numbers (1, 2, 3, ...) are also natural numbers. [1 mark]

A set is closed under an operation if performing that operation on any two members of the set always gives a result that is also in the set.

(i) ℕ is NOT closed under Subtraction: [1.5 marks]
• 3 − 5 = −2 ∉ ℕ (negative number, not natural)
• 2 − 7 = −5 ∉ ℕ
Since subtraction of two natural numbers can give a negative result (not a natural number), ℕ is NOT closed under subtraction.

(ii) ℕ is NOT closed under Division: [1.5 marks]
• 3 ÷ 2 = 1.5 ∉ ℕ (not a whole number)
• 5 ÷ 7 = 5/7 ∉ ℕ (a fraction)
Since division of two natural numbers can give a non-natural result, ℕ is NOT closed under division.

Method: Convert to equivalent fractions with a common denominator, then find numbers between them.

1/4 = 3/12    and    1/3 = 4/12

To find 3 rationals between 3/12 and 4/12, multiply numerator and denominator by 10:
1/4 = 30/120    and    1/3 = 40/120

Three rational numbers between 30/120 and 40/120 are:
31/120, 35/120 (= 7/24), 39/120 (= 13/40) [1 mark each]

Any three valid rationals between 1/4 and 1/3 are acceptable.

Number Line:
← ··· −3 ·· −1 ·· 0 ···· 2 ······ 5 ··· →
(Draw equally spaced points from −3 to 5, marking each value) [1 mark]

Classification: [2 marks]
• Integers (ℤ): −3, −1, 0, 2, 5 (all of them are integers)
• Whole Numbers (W): 0, 2, 5 (non-negative integers)
• Natural Numbers (ℕ): 2, 5 (positive integers only)

Note: −3 and −1 are integers only (not whole or natural numbers).

Distributive Property: a × (b + c) = a × b + a × c

Here: a = 1/2, b = 1/3, c = 1/4

LHS:
1/2 × (1/3 + 1/4)
= 1/2 × (4/12 + 3/12)   [finding common denominator] [1 mark]
= 1/2 × 7/12
= 7/24 [1 mark]

RHS:
(1/2 × 1/3) + (1/2 × 1/4)
= 1/6 + 1/8
= 4/24 + 3/24
= 7/24 [1 mark]

Since LHS = RHS = 7/24, the distributive property is verified. ✓

Theorem: √2 is irrational.

Proof by Contradiction:

Step 1 — Assumption: [1 mark]
Assume, for the sake of contradiction, that √2 is rational.
Then √2 = p/q, where p and q are integers, q ≠ 0, and p/q is in its lowest terms (i.e., HCF(p, q) = 1).

Step 2 — Square both sides: [1 mark]
(√2)² = (p/q)²
2 = p²/q²
p² = 2q²
Therefore p² is even (divisible by 2).

Step 3 — Show p is even: [1 mark]
If p² is even, then p must also be even.
(Because: if p were odd, say p = 2m+1, then p² = 4m²+4m+1, which is odd. Contradiction.)
So p = 2k for some integer k.

Step 4 — Show q is even: [1 mark]
Substituting p = 2k into p² = 2q²:
(2k)² = 2q²
4k² = 2q²
q² = 2k²
Therefore q² is even, which means q is also even.

Step 5 — Contradiction and Conclusion: [1 mark]
We have shown that both p and q are even, meaning they have 2 as a common factor.
But this contradicts our assumption that HCF(p, q) = 1 (i.e., p/q is in lowest terms).
Therefore, our assumption was wrong.
Hence, √2 is irrational. ■

(a) Definitions: [2.5 marks — 0.5 each]

• Natural Numbers (ℕ): The counting numbers starting from 1. ℕ = {1, 2, 3, 4, ...}

• Whole Numbers (W): Natural numbers together with zero. W = {0, 1, 2, 3, ...}

• Integers (ℤ): All positive and negative whole numbers including zero. ℤ = {..., −2, −1, 0, 1, 2, ...}

• Rational Numbers (ℚ): Numbers that can be expressed in the form p/q, where p and q are integers and q ≠ 0. Their decimal expansion is either terminating or non-terminating recurring.

• Irrational Numbers: Numbers that cannot be expressed in p/q form. Their decimal expansion is non-terminating and non-recurring.

(b) Examples: [2.5 marks — 0.5 each]

• Natural Number: 7
• Whole Number: 0
• Integer: −4
• Rational Number: 3/5 (= 0.6)
• Irrational Number: √2 (= 1.41421...)

Why 1 is neither prime nor composite:

Definition of a Prime Number: A natural number greater than 1 that has exactly two distinct factors — 1 and itself. [½ mark]

For the number 1: [1 mark]
The factors of 1 are: 1 (only one factor).
• It does not satisfy the prime definition because it does not have two distinct factors.
• It is not composite because composite numbers have more than 2 factors.

Conclusion: The number 1 is a unit in the number system. It is the multiplicative identity (1 × n = n for any n). Mathematicians specially exclude 1 from both prime and composite categories to maintain the uniqueness of prime factorization (Fundamental Theorem of Arithmetic). [½ mark]