What is the square root of 89?

1 Answer
Oct 1, 2015

Answer:

The square root of #89# is a number which when squared gives #89#.

#sqrt(89) ~~ 9.434#

Explanation:

Since #89# is prime, #sqrt(89)# cannot be simplified.

You can approximate it using a Newton Raphson method.

I like to reformulate it a little as follows:

Let #n = 89# be the number you want the square root of.

Choose #p_0 = 19#, #q_0 = 2# so that #p_0/q_0# is a reasonable rational approximation. I chose these particular values since #89# is about halfway between #9^2 = 81# and #10^2 = 100#.

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.

#color(white)()#
Actually, a rather good simple approximation for #sqrt(89)# is #500/53#, since #500^2 = 250000# and #89 * 53^2 = 250001#

#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#

#color(white)()#
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)# the continued fraction expansion is a little messy so not so nice to work with:

#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 #500/53# above is #[9; 2, 3, 3, 2]#