# What is the square root of 89?

##### 1 Answer

The square root of

#### Explanation:

Since

You can approximate it using a Newton Raphson method.

I like to reformulate it a little as follows:

Let

Choose

Iterate using the formulas:

#p_(i+1) = p_i^2 + n q_i^2#

#q_(i+1) = 2 p_i q_i#

This will give a better rational approximation.

So:

#p_1 = p_0^2 + n q_0^2 = 19^2 + 89 * 2^2 = 361+356 = 717#

#q_1 = 2 p_0 q_0 = 2 * 19 * 2 = 76#

So if we stopped here, we would get an approximation:

#sqrt(89) ~~ 717/76 ~~ 9.434#

Let's go one more step:

#p_2 = p_1^2 + n q_1^2 = 717^2 + 89 * 76^2 = 514089 + 514064 = 1028153#

#q_2 = 2 p_1 q_1 = 2 * 717 * 76 = 108984#

So we get an approximation:

#sqrt(89) ~~ 1028153/108984 ~~ 9.43398113#

This Newton Raphson method converges fast.

Actually, a rather good simple approximation for

#sqrt(89) ~~ 500/53 ~~ 9.43396#

If we apply one iteration step to this, we get a better approximation:

#sqrt(89) ~~ 500001 / 53000 ~~ 9.4339811321#

**Footnote**

All square roots of positive integers have repeating continued fraction expansions, which you can also use to give rational approximations.

However, in the case of

#sqrt(89) = [9; bar(2, 3, 3, 2, 18)] = 9+1/(2+1/(3+1/(3+1/(2+1/(18+1/(2+1/(3+...)))))))#

The approximation