What is the square root of 2?

1 Answer
Aug 14, 2018

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.