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Irrational numbers
Decimals that go on forever with no pattern. Can't be written as a fraction of two whole numbers.
π — the digits never settle
Pi's decimal expansion is irrational. No pattern, no repeat, no end.
3.1415926535897
Famous irrational numbers
- √2 ≈ 1.41421… — the diagonal of a unit square.
- π ≈ 3.14159… — circle ratio.
- e ≈ 2.71828… — natural growth.
- φ ≈ 1.61803… — golden ratio.
Why √2 is irrational
Suppose √2 = a/b in lowest terms. Then 2b² = a², so a is even. Write a = 2k, then b² = 2k² so b is even too. Both even contradicts "lowest terms." So √2 can't be written as a fraction.