What is an irrational number?
GeneralClass 10CBSEAnswered 25 May 2026
Answer
An irrational number cannot be expressed as a fraction of two whole numbers. As a decimal, it goes on forever without repeating any pattern.
Contrast with rational numbers: 1/3 = 0.333... goes on forever but repeats (the 3 repeats) making it rational. An irrational number neither terminates nor repeats.
Famous examples:
- Pi (π) = 3.14159265358979... the ratio of a circle's circumference to its diameter
- √2 = 1.41421356... the length of the diagonal of a square with side 1. Proven irrational by ancient Greek mathematicians.
- e = 2.71828182... the base of natural logarithms
- The square root of any prime number
Simple proof that √2 is irrational: assume √2 = a/b in lowest terms. Then a² = 2b², so a is even (a = 2k). Then (2k)² = 2b², so b² = 2k², meaning b is also even. But if both a and b are even, a/b was not in lowest terms — a contradiction. Therefore √2 cannot be a fraction.
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