How do you find the square root of 2?
1 Answer
Use a continued fraction to find rational approximations.
Explanation:
We can find rational approximations in several ways. Here I show a method called continued fractions...
Consider the number
Then:
#t = sqrt(2)+1#
#= 2 + (sqrt(2)-1)#
#= 2 + ((sqrt(2)-1)(sqrt(2)+1))/(sqrt(2)+1)#
#= 2 + (2-1)/(sqrt(2)+1)#
#= 2 + 1/(sqrt(2)+1)#
#= 2 + 1/t#
Given that:
#t = 2 + 1/t#
notice that we can substitute this expression for
#t = 2 + 1/(2+1/t)#
and again:
#t = 2 + 1/(2+1/(2+1/t))#
In fact:
#t = 2 + 1/(2+1/(2+1/(2+1/(2+1/(2+...)))))#
Now remember
#sqrt(2) = 1 + 1/(2+1/(2+1/(2+1/(2+1/(2+...)))))#
This is called a continued fraction.
There is a shorter notation for a continued fraction using square brackets. Using this notation we can write:
#sqrt(2) = [1;2,2,2,2,2,...] = [1;bar(2)]#
To find a rational approximation for
For example:
#sqrt(2) ~~ [1;2,2,2] = 1+1/(2+1/(2+1/2)) = 1+1/(2+2/5) = 1+5/12 = 17/12 ~~ 1.41bar(6)#
For more accuracy, truncate a little later:
#sqrt(2) ~~ [1;2,2,2,2,2] = 1+1/(2+1/(2+1/(2+1/(2+1/2)))) = 99/70 = 1.4bar(142857)#
This is actually the same accuracy as an approximation to
In fact