Number Systems

What you'll learn

• Identify irrational numbers and distinguish them from rationals
• Place irrational numbers on the number line geometrically
• Apply laws of exponents and radicals
• Rationalise the denominator of expressions

Key concepts

Irrational Numbers

Rational number — can be written as p/q (p, q integers, q ≠ 0). Decimal form is either terminating or repeating.

Irrational number — CANNOT be written as p/q. Decimal form is non-terminating and non-repeating.

Proof that √2 is irrational (by contradiction): Assume √2 = p/q in lowest terms (HCF(p,q)=1). Then 2 = p²/q², so p² = 2q². This means p² is even → p is even → p = 2m. Substituting: 4m² = 2q² → q² = 2m² → q is even. But then both p and q are even, contradicting HCF(p,q)=1. Therefore √2 is irrational.

Between any two rationals, irrationals exist; between any two irrationals, rationals exist.

Real numbers R = Rationals ∪ Irrationals

Real Numbers (R)
├── Rational Numbers (Q)
│   ├── Integers (Z)
│   │   ├── Natural Numbers (N): 1, 2, 3, …
│   │   ├── Whole Numbers (W): 0, 1, 2, …
│   │   └── Negative integers: −1, −2, …
│   └── Non-integer fractions: 1/2, −3/7, …
└── Irrational Numbers: √2, √3, π, e, …

The Real Number Line

Every real number corresponds to exactly one point on the number line, and every point corresponds to exactly one real number.

Key positions:

← … −√3 −√2 −1  0  1  √2  √3  π  … →
         ↑      ↑
      ≈−1.732  ≈1.732

Representing Surds on the Number Line

√n can be located geometrically using the Pythagorean theorem.

Method — Representing √2:

1. Draw a number line. Mark O (0) and A (1).
2. Draw OB ⊥ OA at A with AB = 1 unit.
3. OB = √(OA² + AB²) = √(1 + 1) = √2
4. With O as centre and radius OB, mark √2 on the number line.

Method — Representing √3:

1. Use the point √2 already marked as P.
2. Erect a perpendicular of length 1 at P.
3. Distance from O to the top = √(√2)² + 1² = √(2+1) = √3
4. Arc back to number line.

General pattern: To represent √n, if √(n−1) is already marked, erect a unit perpendicular and swing the hypotenuse back.

Spiral of Theodorus — successive right triangles each with legs √(n−1) and 1 produce √n.

Representing √x for any x > 0 (graphical method):

1. On a line, mark AB = x and BC = 1 (so AC = x + 1).
2. Find the midpoint M of AC.
3. Draw semicircle with diameter AC.
4. Erect perpendicular at B to meet the semicircle at D.
5. BD = √x

Worked Example: Represent √5 on number line. Mark AB = 5, BC = 1 on a line. AC = 6. Midpoint M of AC at 3 from A. Semicircle radius = 3. B is 2 from M. BD = √(3²−2²) = √(9−4) = √5. ✓

Laws of Exponents and Radicals

Laws of Exponents (for real base a > 0, rational exponents m, n):

Laws of Radicals:

Worked Example: Simplify √72 72 = 36 × 2 = 6² × 2 √72 = 6√2

Worked Example: Simplify (√5)⁴ = (5^(1/2))⁴ = 5^(4/2) = 5² = 25

Worked Example: Simplify ∛(2⁶) = (2⁶)^(1/3) = 2² = 4

Rationalising the Denominator

Goal: Remove surds from the denominator.

Case 1 — Single surd in denominator:

Multiply numerator and denominator by the surd.

Example: 5/√3 = (5 × √3) / (√3 × √3) = 5√3 / 3

Example: 3/(2√7) = 3√7 / (2 × 7) = 3√7 / 14

Case 2 — Binomial surd denominator (a + √b or √a + √b):

Multiply by the conjugate: (a − √b) or (√a − √b). Uses the identity (x+y)(x−y) = x² − y².

Worked Example: Rationalise 1/(√5 + 2) Conjugate = (√5 − 2) = (√5 − 2) / [(√5 + 2)(√5 − 2)] = (√5 − 2) / (5 − 4) = (√5 − 2) / 1 = √5 − 2

Worked Example: Rationalise (√3 + 1) / (√3 − 1) Multiply by (√3 + 1): = (√3 + 1)² / [(√3)² − 1²] = (3 + 2√3 + 1) / (3 − 1) = (4 + 2√3) / 2 = 2 + √3

Why rationalise? It produces a simpler standard form and makes comparisons and further calculations easier.

Quick check

1. Prove that √3 is irrational.
2. Represent √7 on a number line using a geometric construction (describe the steps).
3. Simplify: (2^(3/2) × 3^(1/2)) / (√6)
4. Rationalise the denominator: 7 / (3 + √2)
5. Simplify: (√5 − √3)(√5 + √3) + (√2)⁶

Open the Practice tab for graded questions on Number Systems.