How do you find the square root of 2?

Question

Isaiah Anthony · Accepted Answer

Use a continued fraction to find rational approximations.

#sqrt(2)# is an irrational number, not expressible in the form #p/q# for integers #p, q#. There are multiple ways to find rational approximations, and in this example I'll focus on one called continued fractions. Consider the number #t = sqrt(2)+1# Next: #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# Considering that #t = 2 + 1/t# notice that we can substitute this expression for #t# on the right hand side to find: #t = 2 + 1/(2+1/t)# and once more: #t = 2 + 1/(2+1/(2+1/t))# Actually: #t = 2 + 1/(2+1/(2+1/(2+1/(2+1/(2+...)))))# Now remember #t = sqrt(2) + 1#, so we have: #sqrt(2) = 1 + 1/(2+1/(2+1/(2+1/(2+1/(2+...)))))# We refer to this as a continuing fraction. Square brackets are used in a shorter notation for continued fractions, which allows us to write: #sqrt(2) = [1;2,2,2,2,2,...] = [1;bar(2)]# To find a rational approximation for #sqrt(2)# we can truncate this continued fraction early. As an illustration: #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)# You can truncate a little later for greater accuracy: #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 #sqrt(2)# as the ratio of the sides of a sheet of A4 (#297"mm" xx 210"mm"#). In fact #sqrt(2)# is closer to #1.41421356237#

Jameson Allen · Answer

undefined The square root of 2 can be found using a calculator or by using methods such as the Babylonian method or Newton's method for approximation. As an exact value, the square root of 2 is an irrational number, which means it cannot be expressed as a finite decimal or fraction. Therefore, the most common representation of the square root of 2 is the symbol $ \sqrt{2} $.