Question

What is the square root of 5?

Answer 1

The square root of `5` can't be simplified father than it already is, so here is `sqrt5` to ten decimal places:

`sqrt5~~2.2360679775...`

Answer 2

`sqrt(5) = 2+1/(4+1/(4+1/(4+1/(4+1/(4+...))))) ~~ 2889/1292 ~~ 2.236068` is an irrational number.

Explanation:

All positive numbers normally have two square roots, a positive one and a negative of the same size. We denote the positive (a.k.a. principal) square root of `n` by `sqrt(n)`.

A square root of a number `n` is a number `x` such that `x^2 = n`. So if `x^2 = n` then also `(-x)^2 = n`.

However, popular usage is that "the square root" refers to the positive one.

Suppose we have a positive number `x` which satisfies:

`x = 2+1/(2+x)`

Then multiplying both sides by `(2+x)` we get:

`x^2+2x = 2x+5`

Then subtracting `2x` from both sides we get:

`x^2=5`

So we have found:

`sqrt(5) = 2+1/(2+sqrt(5))`

`color(white)(sqrt(5)) = 2+1/(4+1/(4+1/(4+1/(4+1/(4+...)))))`

SInce this continued fraction does not terminate, we can tell that `sqrt(5)` cannot be represented as a terminating fraction - i.e. a rational number. So `sqrt(5)` is an irrational number a little smaller than `2 1/4 = 9/4`. For better rational approximations you can terminate the continued fraction after more terms.

For example:

`sqrt(5) ~~ 2+1/(4+1/4) = 2+4/17 = 38/17 ~~ 2.235`

Unpacking these continued fractions can be a little tedious, so I generally prefer to use a different method, namely the limiting ratio of an integer sequence defined recursively.

Define a sequence by:

`{ (a_0 = 0), (a_1 = 1), (a_(n+2) = 4a_(n+1)+a_n) :}`

The first few terms are:

`0, 1, 4, 17, 72, 305, 1292, 5473`

The ratio between terms will tend to `2+sqrt(5)`.

So we find:

`sqrt(5) ~~ 5473/1292 - 2 = 2889/1292 ~~ 2.236068`