# Thread: Finding square root. Newton's method

1. Hi The Science Forum members!

Could somebody tell me how Newton came to an idea for:

New (x) = 0.5 * (x + b/x)

where b is the number that we are looking square root for?

I understand (x^2+b)/2 but (x^2+b)/2x do not understand.

Please help.  2.

3. Originally Posted by scientist91
Hi The Science Forum members!

Could somebody tell me how Newton came to an idea for:

New (x) = 0.5 * (x + b/x)

where b is the number that we are looking square root for?

I understand (x^2+b)/2 but (x^2+b)/2x do not understand.

Please help.
Using use notation (replace new(x) by y for notation simplification)

y=x-(b-x^2)/2x = (b+x^2)/2x = (b/x +x)/2.

The first "equality" is obtained from the first two terms of the Taylor series for
f(u) around u0, where u corresponds to b and u0 corresponds to x^2, with f(u) being the square root.  4. Newton often thought geometrically - geometric reasoning, involving pictures and triangles and straightedge constructions was a much more important part of a mathematical education then than now.

So if you want to follow Newton's probable approach, draw a picture of the function and make lines and intersections with a ruler as you proceed with the iteration. Then you can see how the tangent line via the derivative hands you the b/x, how x and b/x bracket the real root, , and how averaging them tightens the bracket at each step (with a suitable choice of x and function).

With a little thought, you don't need the tangent (or the derivative, or any calculus): if x is less than the square root, b/x will be greater, and the average brings you in. If x is greater, b/x will be smaller, once again a bracket and tightening.  Bookmarks
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