19 Jun ##sqrt(42) ~~ 8479/1350 = 6.48bar(074) ~~ 6.4807407## ##42=2
##sqrt(42) ~~ 8479/1350 = 6.48bar(074) ~~ 6.4807407## ##42=2*3*7## has no square factors so ##sqrt(42)## cannot be simplified. it is an irrational number between ##6## and ##7## Note that ##42 = 6*7 = 6(6+1)## is in the form ##n(n+1)## Numbers of this form have square roots with a simple continued fraction expansion: ##sqrt(n(n+1)) = [n;bar(22n)] = n + 1/(2+1/(2n+1/(2+1/(2n+1/(2+…)))))## So in our example we have: ##sqrt(42) = [6;bar(2 12)] = 6+1/(2+1/(12+1/(2+1/(12+1/(2+…)))))## We can truncate the continued fraction early (preferably just before one of the ##12##’s) to get good rational approximations for ##sqrt(42)##. For example: ##sqrt(42) ~~ [6;2122] = 6+1/(2+1/(12+1/2)) = 337/52 = 6.48bar(076923)## ##sqrt(42) ~~ [6;2122122] = 6+1/(2+1/(12+1/(2+1/(12+1/2)))) = 8479/1350 = 6.48bar(074) ~~ 6.4807407## This approximation will have approximately as many significant digits as the sum of the significant digits of the numerator and denominator hence stop after ##7## decimal places.
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