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.

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)Originally Posted by scientist91
y=x(bx^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.
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.
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