Calculators and computers use Newton's Method to compute square
roots. On this slide, we'll see how to compute √2.
Finding the square root of 2 is the same thing as solving x2−2=0. So we set f(x)=x2−2, f′(x)=2x, and apply the
recursive formula xn+1=xn−f(xn)f′(xn)=xn−x2n−22xn.
It only takes a few steps to get 10 or 20
decimal places.
xn
x
f(x)=x2−2
f′(x)=2x
x−f(x)f′(x)
1.4142135623731
x1
1
-1
2
1−−12=3/2
1_.5000000000000
x2
32
14
3
32−1/43=172
1.41_66666666667
x3
1712
1144
176
1712−1/14417/6=577408
1.41421_56862745
x4
577408
1166464
577204
665857470832
1.41421356237_47
(The computed value is correct up to the final underlined
digit.)