19 Jun The square root of ##89## is a number which when squared giv
The square root of ##89## is a number which when squared gives ##89##. ##sqrt(89) ~~ 9.434## 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]##
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