Without direct evaluation, determine which of:
√5 + √7 - √12 and √6+ √8 - √14 is greater.

I suppose it depends what you mean by 'direct evaluation'.

Jer proposed a puzzle some time ago which led to an approximation method for square roots, without working them out exactly: sqrty=x-n/(2x-1), where x^2 is the nearest square greater than y, and n=(x-y) The larger the number, the more accurate the result, so we multiply everything by say 10: (10√5 + 10√7 - 10√12)- (10√6+10√8 - 10√14), so the numbers are the square roots of 500, 600, etc.. 

Then

 sqrt500  x-n/(2x-1), x=23, n=29; 22.3556
 sqrt600  x-n/(2x-1), x=25, n=25; 24.4898
 sqrt700  x-n/(2x-1), x=27, n=29; 26.4528
 sqrt800  x-n/(2x-1), x=29, n=41; 28.2807
 sqrt1200 x-n/(2x-1), x=35, n=25; 34.6377
 sqrt1400 x-n/(2x-1), x=38, n=44; 37.4133

(22.3556+24.4898-26.4528)-(28.2807+34.6377-37.4133)=-5.1125, so the second expression is greater.

If I was on a desert island with a hundred cannibal mathematicians and no pocket calculator, I'd use a bigger multiplier.

Edited on February 23, 2018, 11:56 pm

Posted by broll on 2018-02-23 23:39:30