We normally use Base 10 to count, 0-9. Sometimes we use Base 2, 0-1, or Base 16, 0-F, when working with computers.

But any number can be used as the base for a number system. Those of us old enough to remember log books will be familiar with that. And the fact that you can readily convert between bases, “10” = “A”.

So take a circle, the circumference of a circle is given by 2πR  where R is the radius. Now, if we go for Base π then the circumference of any circle would be expressed as 2R. By definition, 2R = the diameter of the circle so in Base π the diameter and the circumference of a circle are the same.

But the diameter is a straight line and as such can be expressed as a square, put 4 matches in a straight line and you will see what I mean.

So the circumference of the circle can be expressed as a square.

Hence it is shown that you can square a circle.

QED. 🙂