Can you explain why the fourth root of 9.1 is practically the same as 33/19? You shouldn't use even a calculator!

So I asked myself was there something special about 91/10 and 33/19 or are there lots of examples that could be used to create this problem?

First, what do I require? I want two fractions a/b and c/d such that (a*c^4)/(b*d^4) is almost 1. More specifically, we will require that the numerator and denominator differ by exactly one:  |b*c^4-a*d^4|=1 . With thought you can see that if we have a pair of fractions that meet this condition, then the pair of inverses also work. So off the bat, we see 10/91 and 19/33 would have worked, but using 9.1 was a nice bonus.

Furthermore, I want a,b,c and d to all be at most 2-digit numbers.

Guess what, e.g.'s pair has the fraction closest to 1 of all 106 pairs (and their inverses) that meet these conditions! And there are only 8 (and inverses) that have reasonably small errors (my opinion :-) ).

My second favorite of these: fourth root of 39 is very close to 2.5. My favorite: the fourth root of 76/15 is very closer to 3/2 but the fourth root of 86/17 is even closer!

Yes, I used a calculator :-)

I edited with more care towards my statements about error.

Edited on June 7, 2005, 4:54 pm

Posted by owl on 2005-06-07 05:46:16