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Number theory · Lesson 4

12 min

Why root 2 is not a fraction

1
Suppose root 2 equals p over q, a fraction in lowest terms.
Skipped the primer on proof by contradiction — your skill check shows you have it.
2
Squaring, p squared equals 2 q squared. So p squared is even, and so is p. Write p equals 2 k.
3
Then 4 k squared equals 2 q squared, so q squared equals 2 k squared and q is even too.
Next, you explain why p² even forces p even. In your own words.
4
Both are even, so the fraction was not in lowest terms. The assumption fails. ∎

Integers
Primes
Rationals
Irrationals(this lesson)
The real line