Question

What is the square root of 2?

Answer 1

`sqrt(2) ~~ 99/70` is an irrational number.

Explanation:

The square root of `2` is a number which when multiplied by itself gives `2`.

Note that any positive number actually has two square roots - a positive and a negative one. That having been said, "the square root" is usually taken to mean the positive square root, also known as the principal square root.

The square root of `2` is an irrational number, so cannot be represented as a fraction in the form `p/q` where `p, q` are integers.

It can be represented as a continued fraction, written:

`sqrt(2) = [1;bar(2)] = 1+1/(2+1/(2+1/(2+1/(2+1/(2+...)))))`

We can truncate this continued fraction early in order to get rational approximations to `sqrt(2)`

For example:

`sqrt(2) ~~ [1; 2, 2, 2] = 1+1/(2+1/(2+1/2)) = 17/12 = 1.41bar(6)`

One fun way to calculate rational approximations is using an integer sequence defined recursively by:

`{ (a_0 = 0), (a_1 = 1), (a_(n+2) = 2a_(n+1)+1) :}`

The first few terms of this sequence are:

`0, 1, 2, 5, 12, 29, 70, 169`

The ratio between successive terms tends towards `sqrt(2)+1`

So:

`sqrt(2) ~~ 169/70-1 = 99/70 = 1.4bar(142857)`

The approximation `sqrt(2) = 99/70 = 297/210` is the ratio of sides of a sheet of A4 paper in mm.

If we want more accuracy, just calculate a few more terms of the sequence first.