Why no fraction works

The idea

Assuming √2 = p/q in simplest form gives 2q² = p², so p is even (a number whose square is even is even); writing p = 2k gives q² = 2k², so q is even too, a contradiction.

Suppose it were a fraction

Suppose √2 were a fraction p/q, written in simplest form, so p and q have no common factor except 1. What equation do we get if we square both sides and clear the fraction?

Squaring √2 = p/q and multiplying by q² gives:

• 2q² = p²
• q² = 2p²
• p = 2q

The eight steps

Put the eight steps of Hippasus's proof in order. Tap the step that comes next.

p and q both even

Assuming √2 = p/q in simplest form gives 2q² = p², so p is even (a number whose square is even is even). Writing p = 2k gives q² = 2k², so q is even too. This is a contradiction, because p/q was in simplest form.

So √2 cannot be written as p/q: it is irrational.

Notes

If √2 = p/q in simplest form, then 2q² = p², so p and then q are even: a contradiction. So √2 is irrational.

Check yourself

In 2q² = p², why is p² an even number?

Put p = 2k in 2q² = p². Then 2q² = (2k)² = 4k². Divide both sides by 2: q² = ?k². What is ?

Answer: 2

4k² ÷ 2 = 2k², so q² = 2k².

The square of a number is even. What can we say about the number?

What is the contradiction in the proof?

• Because p is a prime number. p is not known to be prime. The reason is the factor 2 in 2q².
• Because q is odd. We do not know yet whether q is odd or even. p² = 2q² is even because of the factor 2.
• It equals 2 times a whole number — correct. Yes. 2q² is 2 multiplied by the whole number q².
• The number itself is even — correct. Yes. A number whose square is even is even.
• The number could be odd. An odd number times an odd number is odd, so an odd number has an odd square.
• The number must be 2. Many even numbers work: 4² = 16, 6² = 36, and so on.
• 2q² = p². That is just an equation we got by squaring.
• p and q are both even, but p/q was in simplest form — correct. Yes. Both even means they share the factor 2.
• p = 2k. That is how an even number is written. It is not a contradiction.