# What is the square root of 82?

##### 2 Answers

#### Explanation:

S is the number of which you're aproxximating its sqaure root. In this case

Heres what this means and how it is used:

First, take a guess, what might the square root of 82 be?

the square root of 81 is 9, so it must be sligthly higher than 9 right?

Our guess will be

Inserting 9.2 as "x" in the formula will give us

This will be the next number we put into the equation. This is because we started with a guess of 9.2 =

Let's say we did the same calculation 100 times! Then we would have

Enough talking, let's do some actual calculations!

We start with our guess

Now do the same with the new number:

Let's do it one last time:

That means

And there you have it!

Sorry if all my talking was annoying. I tried to explain it in-depth and in a simple way, which is always nice if you're not very familiar with a certain field in mathematics. I don't see why some people has to be so posh when explaining mathematics :)

#### Explanation:

The prime factorisation of

#82 = 2*41#

Since there are no square factors,

However, note that

Since this is of the form

#sqrt(82) = [9;bar(18)] = 9+1/(18+1/(18+1/(18+1/(18+...))))#

More generally:

#sqrt(n^2+1) = [n;bar(2n)] = n+1/(2n+1/(2n+1/(2n+1/(2n+...))))#

More generally still:

#sqrt(n^2+m) = n+m/(2n+m/(2n+m/(2n+m/(2n+...))))#

In any case, we can use the continued fraction to get rational approximations to

For example:

#sqrt(82) ~~ [9;18] = 9+1/18 = 163/18 = 9.0bar(5)#

#sqrt(82) ~~ [9;18,18] = 9+1/(18+1/18) = 2943/325 = 9.05bar(538461)#

#sqrt(82) ~~ [9;18,18,18] = 9+1/(18+1/(18+1/18)) = 53137/5868 ~~ 9.05538513974#

A calculator tells me that:

#sqrt(82) ~~ 9.0553851381374#

So you can see that our approximations are accurate to just about as many significant digits as the total number of digits in the quotient.