Irrational Numbers – Numbers That Never Repeat
An irrational number is a number that cannot be written as a fraction of two integers. Its decimal expansion goes on forever without ever terminating and without ever settling into a repeating pattern – a genuinely different kind of infinity from the repeating decimals of rational numbers.
The discovery of irrational numbers is one of the most dramatic stories in the history of mathematics. The ancient Greek Pythagorean school believed that “all is number,” meaning that everything in the universe could be described using whole numbers and their ratios. According to legend, one of their own members, Hippasus of Metapontum (around the 5th century BCE), proved that √2 – the length of the diagonal of a simple unit square – could not be written as any such ratio, shattering the Pythagorean worldview. The legend claims Hippasus was thrown overboard and drowned by his fellow Pythagoreans for revealing this unsettling truth, though historians debate how much of the story is embellishment – what's certain is that the discovery of irrationality was genuinely disruptive to Greek mathematics.
Is √2 Really Irrational?
The classic proof (by contradiction): assume √2 = p/q in lowest terms. Squaring gives 2 = p²/q², so p² = 2q², meaning p² (and therefore p) is even. Writing p = 2k gives q² = 2k², meaning q is also even – but then p and q share a factor of 2, contradicting “lowest terms.” So no such fraction can exist.
√9 = 3, a whole number (9 is a perfect square), so √9 is rational, not irrational.
Famous Irrational Numbers
| Number | Approximate Value | Where It Comes From |
|---|---|---|
| π (pi) | 3.14159… | Ratio of a circle's circumference to its diameter |
| e (Euler's number) | 2.71828… | Base of natural logarithms; continuous growth |
| √2, √3, √5… | 1.41421…, 1.73205…, 2.23607… | Square roots of any non-perfect-square whole number |
Real-Life Application
- Circles: any calculation involving π (circumference, area, volume).
- Diagonals: the diagonal of a square with side 1 is √2.
- Growth and finance: continuous compound growth uses e.
Key Takeaways
- An irrational number cannot be written as a fraction of two integers.
- Its decimal expansion never terminates and never repeats.
- Hippasus's discovery of √2's irrationality shook the ancient Pythagorean belief that all numbers were ratios.
- π, e, and the square roots of non-perfect squares are all famous irrational numbers.
Practice: Irrational Numbers
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