What is the square root of 2?

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.