Why can the square root of 2 never be a fraction? Learn to prove it, step by step, with a proof by contradiction. Then prove the same for other roots, and find out when a sum or a product of such numbers is irrational.
3 lessons, about 1 h 15 min3 simulations11 problems3 practice questionsPython in every lesson
Created by StemPy LabA proof shows that a statement is true in every case, and many examples are not enough. In a proof by contradiction we suppose that the statement is false, reason correctly, and reach something impossible. Then the statement must be true. With the Fundamental Theorem of Arithmetic we prove a key fact: if a prime divides , then it divides .
A calculator shows only a few digits of , so it cannot tell us whether the digits ever repeat. We prove that is irrational by contradiction: if it were a fraction in lowest terms, both its top and its bottom would be even. The same proof works for the square root of any prime. It fails for a perfect square such as , because .
We prove two rules: a rational plus an irrational number is irrational, and a non-zero rational times an irrational number is irrational. With them, and the irrationality of square roots of primes, we prove that numbers such as are irrational. Two irrational numbers can give a rational or an irrational sum or product.
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