The Irrationality of √2
Use proof by contradiction to show that the square root of 2 cannot be written as a fraction of integers.
Use proof by contradiction to show that the square root of 2 cannot be written as a fraction of integers.
See how assuming that √2 is rational forces both the numerator and denominator of a fraction to be even.
A fraction in lowest terms cannot have both numerator and denominator even, so the assumption that √2 is rational leads to a contradiction.
After completing this lesson you should be able to:
Assume, for contradiction, that √2 = a/b, where a and b are integers with no common factor.
Squaring gives 2 = a²/b², so a² = 2b².
Therefore a² is even, so a is even. Let a = 2k.
Substituting gives 4k² = 2b², so b² = 2k².
Thus b is also even.
So a and b share a factor of 2, contradicting the assumption that a/b was in lowest terms.
Hence, √2 is irrational.
What contradiction appears in the proof?
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