Irrational Square Roots

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Proof by Contradiction and Square Roots

Based on “Rational and Irrational Numbers” and “The Square Root of 2 Is an Irrational Number” in section 3.3 of the textbook, and this discussion of irrational square roots.

The Square Root of 6

Can you adapt the proof that √2 is irrational to show that √6 is irrational?

To start, how did the proof that √2 is irrational work? It was a proof by contradiction, with 4 main parts:

1. Assume the root is rational, i.e., its reduced form looks like m/n, where m and n are integers, n isn’t 0, and m and n have no common factors other than 1.
2. Show that m must be even
3. Show that n must  be even
4. Now you have a contradiction: 2 is common factor for two even numbers, but m and n have no common factors.

Try applying these ideas to proving that the square root of 6 is irrational. We developed the proof in LaTeX, mostly by figuring out step by step how to adapt the outline above to the new claim. Most steps in the outline correspond directly to parts of the proof for √6. The only exception is showing that n is even, which is a little trickier than for √2. The finished proof is the second one in a set of examples of proof by contradiction, whose LaTeX source and PDF output files are available in Canvas.

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Proof by cases.

Please read section 3.4 in the textbook.

Please also contribute to this discussion of proof by cases.

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