Example. Prove that 13 is irrational.
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Answer:
13 is rational number
how can we prove that it is irrrational
Answer:
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Puzzling's avatar
Puzzling
Lv 7
5 years ago
Let's suppose √13 is a *rational* number. Then we can write √13 as the ratio of two integers in lowest terms. So we could write:
√13 = a/b, where a and b are integers, b is not zero and a/b is in lowest terms (have no factors in common).
Square both sides:
13 = a²/b²
a² = 13b²
That means a² is evenly divisible by 13 resulting in an integer (b²).
But since 13 is prime, also means that a is divisible by 13.
We could therefore write:
a = 13k, for some positive integer k
Substituting back we have:
(13k)² = 13b²
13 * 13k² = 13b²
Cancel a 13 from both sides:
13k² = b²
b² = 13k²
Using the same logic, we can show that b² is divisible by 13 and therefore b is divisible by 13.
But this is a contradiction. We assumed that a and b were integers that had no common factors such that a/b was in lowest terms. But we just showed that if we started with this assumption, we would have 13 as a factor of both.
Answer:
√13 cannot be rational, proof by contradiction