Irrational Numbers

• What makes a number irrational and how it differs from rational numbers.
• Familiar irrational numbers: √2, √3, π, and non-perfect square/cube roots.
• How to decide whether a number under a root is rational or irrational.
• That the decimal expansion of an irrational number is non-terminating and non-repeating.
• Real-life contexts where irrationals appear: diagonals of squares, circles, and certain geometric constructions.

1. Definition — An irrational number cannot be written as p/q for any integers p, q (q ≠ 0). Its decimal goes on forever without a repeating block.

2. Square roots — √n is rational only when n is a perfect square (e.g. √4 = 2, √25 = 5). Otherwise it is irrational (√2, √3, √5, √7).

3. Famous constants — π ≈ 3.14159… (ratio of circumference to diameter of a circle) and e are irrational. π cannot be expressed as a fraction.

4. Sum and product rules — √2 + √3 is irrational. √2 × √2 = 2 is rational. A rational × irrational (non-zero) is irrational.

5. Locating on number line — Irrationals like √2 can be placed using geometry: construct a right triangle with legs 1 and 1; the hypotenuse has length √2.

6. Where it shows up — Diagonal of a 1 m × 1 m tile (√2 m), circumference of a circular garden (2πr), and Pythagorean distances in navigation.

Show that √50 is irrational and simplify it.

Step 1 — Factor: √50 = √(25 × 2) = √25 × √2 = 5√2
Step 2 — 5 is rational, √2 is irrational
Step 3 — Product of non-zero rational and irrational is irrational
Answer: √50 = 5√2, which is irrational

Application: A square room has side 5 m. The diagonal is 5√2 ≈ 7.07 m — useful for estimating the longest straight plank that fits corner to corner.

• Assuming every root is irrational (√16 = 4 is rational).
• Thinking π = 22/7 exactly (22/7 is only an approximation).
• Believing the sum of two irrationals is always irrational (√2 + (−√2) = 0, which is rational).
• Rounding irrationals too early in multi-step calculations.

• Is √18 rational or irrational? Simplify it.
• Which is irrational: √49, √10, or 0.5?
• Between which two consecutive integers does √20 lie?
• Why is π not a rational number?

Open the Practice tab for graded questions on Irrational Numbers.