Rational & Irrational Numbers
Year 10 (IGCSE) 🔢 Number Identify, classify and work with rational and irrational numbers.
🔢 Rational Numbers
A rational number can be written exactly as a fraction $\frac{p}{q}$ where $p$ and $q$ are integers and $q \neq 0$.
Rational examples: $\frac{3}{4}$, $-5$, $0.7$, $0.\overline{3} = \frac{1}{3}$, $\sqrt{9} = 3$
All terminating decimals and all recurring decimals are rational numbers.
√ Irrational Numbers
An irrational number cannot be written as a fraction. Its decimal expansion never terminates and never repeats.
Irrational examples: $\sqrt{2} = 1.41421\ldots$, $\pi = 3.14159\ldots$, $e = 2.71828\ldots$
$\sqrt{n}$ is irrational whenever $n$ is a positive integer that is not a perfect square.
🔄 Converting Recurring Decimals to Fractions
Any recurring decimal can be converted to a fraction using a short algebraic method.
- Let $x$ = the recurring decimal.
- Multiply by a power of 10 to shift the repeating block one full cycle.
- Subtract to eliminate the recurring part.
- Solve for $x$ and simplify the fraction.
Example: Convert $0.\overline{27}$. Let $x = 0.272727\ldots$. Then $100x = 27.272727\ldots$. Subtract: $99x = 27$, so $x = \frac{27}{99} = \frac{3}{11}$.
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Interactive Demonstration — Rational & Irrational Numbers
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Interactive demonstration available in the Calculator tab!
🔢 Number Type Classifier
Check if √n is rational or irrational, and simplify the surd.