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Proof the Square Root of 2 is Irrational

This week we will prove that the square root of 2 is irrational. However, before we get to this, I present the weekly logic puzzle.

Logic Puzzle

An Arab sheikh tells his two sons to race their camels to a distant city to see who will inherit his fortune. The one whose camel is slower wins. After wandering aimlessly for days, the brothers ask a wise man for guidance. Upon receiving the advice, they jump on the camels and race to the city as fast as they can. What did the wise man say to them?

The answer is at the bottom of the newsletter.

Proof the Square Root of 2 is Irrational

We will use the method of contradiction for our proof. This means I’ll disprove that that square root of 2 is rational, leaving the alternative that it is irrational.

The definition of a rational number is that it can be expressed as a ratio of two integers. Let’s call them p and q. That means an irrational number may not be expressed this way. For purposes of our proof by contradiction, let’s say for now that the square root of can be expressed as p q , where the fraction is reduced to its lowest terms. So we have:

√2 = p q

2 = p2 q2   (Squaring both sides)

2q2 = p2

At this point, p must be even because if the square of a number is even, then the number itself is even. Likewise, the square of an odd number is also odd. Thus, we can say p=2k, where k is some integer.

2q2 = (2k)2

2q2 = 4k2

q2 = 2k2

By the same logic, q must be even as well. So, both p and q are both even. However, we assumed at the beginning that p and q were reduced to their lowest form. Yet, if they are both even, then they can both be divided by 2.

Thus, the original assumption that √2 = p q has been proven false. Thus the alternative must be true that √2 is irrational.

Logic Puzzle Solution

The wise man said, “Switch camels and race to the distant city.”