By the way, did you know many roots can be approximated VERY closely by averaging the closest fractions above and below their values? It's something I noticed with the "22/7 squared is ALMOST 10" thing; 10/7 squared is 100/49, a little more than 2, and 7/5 squared is 49/25, a little LESS than 2. The average is easy to find (fractions where the denominator of one is the numerator of the other work best for this: )

(10/7+7/5)/2=

50+49/2X35=

99/70=

1 29/70.

There, the square root of two approximated as a rational number IN MY HEAD. How accurate is it?

Well, I actually used Excel for 29/70, 'cos I'm lazy (it's really just 0.428571... -0.0.142857... after all) and it gave 1.414285714. For the square root of 2 it gives 1.414213562. So treating 99/70 as the square root of 2 is accurate to four significant figures. If you're curious, squaring the former value gives 2.000204082, accurate to three significant figures. It's more precise than the tables of squares printed in my HS textbook for Honors Trig; they only went to three decimal places, but I can get an extra magnitude of precision by averaging simple fractions. The only real challenge is finding fractions whose squares are close to integers (e.g. 9/4 squared is 81/16, a little more than 5, and 20/9 squared is 400/81, a little less, yielding 161/72 as an average; 2 17/72=2.236111111 rather than 2.236067977, and the square is 5.000192901. )