Look-and-see proofs that the square root of 2 is irrational

SqCarpets3

The names tell us that the square root of −1 is “imaginary”, but the numbers we’re more used to are “real”.

In fact, they are equally real and equally imaginary. Certainly, √2 is just as “imaginary” as √−1.

Concede for the sake of argument that whole numbers and fractions are more “real”. They are more familiar, and used more everyday. (No-one says they’ll be “back in √2 minutes”).

But √2 is not equal to any fraction. It can never be worked out as an exact decimal. If all you know is whole numbers and fractions, then you have to “imagine” an extension of your idea of number in order for 2 to have a square root.

Proof? Look at these diagrams. If the left-hand diagram shows two copies of a green square with whole-number sides which is exactly half the area of a bigger square, then the right-hand diagram shows us that the squares formed by the “left-out” areas (blue) are exactly half the area of the square formed by the overlap of the two green squares (red).

SqCarpets3

So, suppose √2 is equal to a fraction p/q. Reduce p/q to lowest terms. Then p2 is the smallest whole-number-sided square which is exactly twice the area of another whole-number-sided square.

But it can’t be; because if it is, then, by the diagram, there is a smaller whole-number-sided square which is exactly twice the area of another whole-number-sided square.

That’s a contradiction, so the assumption that √2 is equal to a fraction p/q must be wrong. ∎

Alternatively:

If √2 is equal to a fraction p/q, then reduce p/q to lowest terms, and the smallest whole-number-sided right-angled isosceles (45-degree) triangle has hypotenuse p and smaller sides q.

Imagine that triangle cut out on a bit of paper. Fold the upright smaller side over onto the hypotenuse, as shown in the diagram. Then in the bottom left-hand corner you create a smaller whole-number-sided right-angled isosceles (45-degree) triangle.

Kiselev

Contradiction again, so again the assumption that √2 is equal to a fraction p/q must be wrong. ∎

We tend not to be aware that we’re committing ourselves to a whole new concept of number when we agree that there is such a number as √2, but that’s what we’re doing.

√−1 is a further extension of the concept of number, but better described as a further stretch of imagination in order to comprehend reality than as “imaginary” as opposed to “real”.