I just found an easy way you could do sqrt that I didn't even know about tbh.
wiki wrote: |
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To calculate \sqrt{S}, where S = 125348, to 6 significant figures, use the rough estimation method above to get
x_0 = 6 \cdot 10^2 = 600.000. \,
x_1 = \frac{1}{2} \left(x_0 + \frac{S}{x_0}\right) = \frac{1}{2} \left(600.000 + \frac{125348}{600.000}\right) = 404.457.
x_2 = \frac{1}{2} \left(x_1 + \frac{S}{x_1}\right) = \frac{1}{2} \left(404.457 + \frac{125348}{404.457}\right) = 357.187.
x_3 = \frac{1}{2} \left(x_2 + \frac{S}{x_2}\right) = \frac{1}{2} \left(357.187 + \frac{125348}{357.187}\right) = 354.059.
x_4 = \frac{1}{2} \left(x_3 + \frac{S}{x_3}\right) = \frac{1}{2} \left(354.059 + \frac{125348}{354.059}\right) = 354.045.
x_5 = \frac{1}{2} \left(x_4 + \frac{S}{x_4}\right) = \frac{1}{2} \left(354.045 + \frac{125348}{354.045}\right) = 354.045.
Therefore, \sqrt{125348} \approx 354.045 \,. |
http://en.wikipedia.org/wiki/Methods_of_computing_square_roots#Babylonian_method
*edit it looks weird when I paste it..lol
Basically you do something like
sqrt of 15 to 6 signifigant values ( can be anything the more you have the more accurate ) would be
s = 15
x0 = 6 * 100 = 600
x1 = 1/2( x0 + s/x0 )= 1/2( 600 + 15/600 ) = 300.0125
x2 = 1/2( x1 + s/x1 ) = 1/2( 300.0125 + 15/300.0125 ) = 150.0312
x3 = 1/2( x2 + s/x2 ) = 1/2( 150.0312 + 15/150.0312 ) = 75.0656
x4 = 1/2( x3 + s/x3 ) = 1/2( 75.0656 + 15/75.0656 ) = 37.6327
x5 = 1/2( x4 + s/x4 ) = 1/2( 37.6327 + 15/37.6327 ) = 19.0156
x6 = 1/2( x5 + s/x5 ) = 1/2( 19.0156 + 15/19.0156 ) = 9.9022
As you can see that isn't the correct value you want to go till xn is almost the same if not the same as xn - 1 so I'll keep going The reason you need more for 15 vs that large number is because its such a small value.
x7 = 1/2( x6 + s/x6 ) = 1/2( 9.9022 + 15/9.9022 ) = 5.7085
x8 = 1/2( x7 + s/x7 ) = 1/2( 5.7085 + 15/5.7085 ) = 4.1679
x9 = 1/2( x8 + s/x8 ) = 1/2( 4.1679 + 15/4.1679 ) = 3.8834
x10 = 1/2( x9 + s/x9 ) = 1/2( 3.8834 + 15/3.8834 ) = 3.8730
x11 = 1/2( x10 + s/x10 ) = 1/2( 3.8730 + 15/3.8730 ) = 3.8730
sqrt( 15 ) is approx 3.8730
let me check the calculator that my computer comes with...
3.8730