Practice question · Put in order
Order the steps of the classic proof by contradiction that the square root of 2 is irrational.
- Deduce that a is even, so a = 2k, giving b squared = 2 k squared
- Conclude the square root of 2 is irrational
- Deduce that b is also even, contradicting 'lowest terms'
- Write it as a fraction a/b in lowest terms
- Square and rearrange to get a squared = 2 b squared
- Assume the square root of 2 is rational
Hints
- A proof by contradiction opens by assuming the OPPOSITE of the goal.
- The contradiction must strike the assumption itself, here, the 'lowest terms' condition.
Show the answer
- Assume the square root of 2 is rational
- Write it as a fraction a/b in lowest terms
- Square and rearrange to get a squared = 2 b squared
- Deduce that a is even, so a = 2k, giving b squared = 2 k squared
- Deduce that b is also even, contradicting 'lowest terms'
- Conclude the square root of 2 is irrational
Why
Assume the negation, derive consequences until something impossible appears, then conclude the assumption was false. The contradiction here is that a and b are both even despite being in lowest terms, so no such fraction exists.
Practise Proof by Contradiction and Contrapositive
The app has 5 more questions on this lesson, and keeps your place in the course. Mathematics I is free to start.
More questions on Proof by Contradiction and Contrapositive
- Sort each opening line by the proof technique it belongs to.
- Select every statement that is a CORRECT negation of the one shown beside it.
- Euclid proved there are infinitely many primes by assuming a finite list and building N = p₁p₂…pₖ + 1. Why…
- Proof by contrapositive and proof by contradiction both begin by assuming a negation. What distinguishes…
- A conditional and its contrapositive always have the same truth value.