Number Systems

What you'll learn

• Plot rational numbers accurately on a number line
• Perform all four operations on rational numbers with confidence
• Find square roots using prime factorisation and long division
• Find cube roots using prime factorisation

Key concepts

Rational Numbers on the Number Line

A rational number p/q can be placed on the number line by dividing unit segments into q equal parts and counting p parts from 0.

Plotting key examples:

Density property: Between any two rational numbers, infinitely many rational numbers exist.

Mean method to find a rational number between a and b: (a + b) / 2

Worked Example: Find 3 rational numbers between 1/4 and 1/2. Step 1: 1/4 = 2/8 and 1/2 = 4/8 → 3/8 lies between them. Step 2: Mean of 1/4 and 3/8 = 5/16; mean of 3/8 and 1/2 = 7/16 Answer: 5/16, 3/8, 7/16

Operations on Rational Numbers

Addition / Subtraction — find LCM of denominators:

(5/6) + (−7/9): LCM = 18 → 15/18 − 14/18 = 1/18

(−3/4) − (−5/6): LCM = 12 → −9/12 + 10/12 = 1/12

Multiplication: (p/q) × (r/s) = pr/qs

(−7/8) × (−12/21) = 84/168 = 1/2

Division: (p/q) ÷ (r/s) = (p/q) × (s/r)

(−5/6) ÷ (10/9) = (−5/6) × (9/10) = −45/60 = −3/4

Properties summary:

Squares and Square Roots

Square of a number n = n × n = n²

Properties of perfect squares:

• A perfect square never ends in 2, 3, 7, or 8
• A perfect square ending in 1 has square root ending in 1 or 9
• A perfect square ending in 6 has square root ending in 4 or 6

Method 1 — Prime Factorisation: Group prime factors in pairs. Each pair contributes one factor to the square root.

Worked Example: √1764 1764 = 2 × 2 × 3 × 3 × 7 × 7 = (2)² × (3)² × (7)² √1764 = 2 × 3 × 7 = 42

Worked Example (check if perfect square): Is 2028 a perfect square? 2028 = 2² × 3 × 169 = 2² × 3 × 13² 3 is unpaired → NOT a perfect square. Multiply by 3 to make it one: 2028 × 3 = 6084 = 78².

Method 2 — Long Division Method:

Steps for √529:

1. Pair digits from right: 5 | 29
2. Largest square ≤ 5 is 4 (2²). Quotient starts with 2. Remainder = 5 − 4 = 1.
3. Bring down 29 → 129. Double quotient: 2×2 = 4. Find digit d so (40+d)×d ≤ 129. Try d=3: 43×3=129. ✓
4. Quotient = 23, remainder = 0. So √529 = 23

Long division for decimals:

√2 ≈ 1.414 (pair as 2.00 00 00, work 2 decimal places at a time)

Cubes and Cube Roots

Cube of a number n = n × n × n = n³

Properties of cubes:

• Cube of an even number is even; cube of an odd number is odd
• Cube of a negative number is negative: (−3)³ = −27

Units digit of cube vs units digit of number:

Cube Root — Prime Factorisation: Group prime factors in triples. Each triple contributes one factor.

Worked Example: ∛3375 3375 = 3 × 3 × 3 × 5 × 5 × 5 = 3³ × 5³ ∛3375 = 3 × 5 = 15

Worked Example: ∛13824 13824 = 2⁷ × 108 → Let's factorise properly: 13824 = 2 × 6912 = 2 × 2 × 3456 = ... = 2⁶ × 216 = 64 × 216 Wait: 13824 = 2³ × 2³ × 3³ × ... Let's use: 13824 = 24³ ÷? → 24³ = 13824. ∛13824 = 24

Is a number a perfect cube? Check: all prime factors must appear in multiples of 3.

Is 1536 a perfect cube? 1536 = 2⁹ × 3 → 3 appears once (not a multiple of 3) → Not a perfect cube

Quick check

1. Find 4 rational numbers between −1/2 and −1/3.
2. Evaluate: (−7/15) × (5/14) ÷ (−2/3)
3. Find √7056 using prime factorisation.
4. Find the smallest number by which 675 must be multiplied to make it a perfect square.
5. Find ∛17576 using prime factorisation.

Open the Practice tab for graded questions on Number Systems.