Question

What is the square root of `31` ?

Answer 1

`5.56`

Explanation:

All you need to do is take a calculator then press this sign `sqrt` then put the number under it.

Answer 2

`+-5.56776436283`

Explanation:

`because sqrt31 = +-5.56776436283`
However if you are using this in algebra i would leave it as `sqrt31` as it is easier than writing `+-5.56776436283`.

Answer 3

`sqrt(31)` is an irrational number approximately equal to:

`120041/21560 ~~ 5.5677644`

Explanation:

Since `31` is a prime number, its square root is an irrational number.

Actually, every non-zero number has two square roots.

In the case of `31` they are written `sqrt(31)` and `-sqrt(31)`.

`sqrt(31)` is the called the principal square root and is the positive one.

`-sqrt(31)` is the negative square root.

When people say "the square root", they generally mean "the principal square root", i.e. the positive one.

Note that:

`5^2 = 25 < 31 < 36 = 6^2`

So the (positive) square root of `31` is somewhere between `5` and `6`.

Using a calculator:

`sqrt(31) ~~ 5.56776436283`

but we don't need a calculator to find rational approximations to `sqrt(31)`.

There are at least `25` methods we could use, but one effective one is the Babylonian method:

If `a` is an approximation to `sqrt(n)`, then a better one is:

`(a^2+n)/(2a)`

Since `31` is about half way between `25=5^2` and `36=6^2`, a good first approximation to `sqrt(31)` is `11/2`.

So a better approximation is:

`((11/2)^2+31)/(2*11/2) = (121/4+124/4)/11 = 245/44`

Repeat to get a better approximation:

`((245/44)^2+31)/(2*245/44) = 120041/21560 ~~ 5.56776438`