What is the Decimal Expansion of an Irrational Number? The decimal expansion of an irrational number is non-terminating (never ends) and non-repeating (no pattern repeats). You can keep calculating digits forever, and they will never settle into a repeating cycle.
Examples:
- \(\sqrt{2} = 1.4142135623730950488...\) (never ends, no repeating pattern)
- \(\pi = 3.14159265358979323846...\) (never ends, no repeating pattern)
- \(\sqrt{3} = 1.7320508075688772935...\)
Contrast with Rational Numbers:
- Terminating decimals: \(\frac{1}{4} = 0.25\) (ends)
- Repeating decimals: \(\frac{1}{3} = 0.333333...\) (repeats 3)
- Irrational: \(1.414213...\) (never ends and never repeats)
How do we know an irrational number's decimal never repeats? If a decimal expansion eventually repeats, the number can be written as a fraction (rational number). Since irrational numbers cannot be written as fractions, their decimal expansions cannot repeat.
Approximating Irrational Numbers: We often use approximations (rounded values) for irrational numbers in calculations:
- \(\pi \approx 3.14\) or \(3.1416\)
- \(\sqrt{2} \approx 1.414\)
- \(\sqrt{3} \approx 1.732\)