Re: Not Just the US With Education Problems

Walter Bushell <proto@xxxxxxx> wrote:
In article <g2hqdq$in7$4@xxxxxxxxxxxxxxxxx>,
Paul J Gans <gans@xxxxxxxxx> wrote:

What happens is that many kids, not understanding how
to compute a square root decide that they will never
master math.

The algorithm for the computation isn't important. The
concept of the square root is.

A caculator allows students to play with square roots
without having to know the algorithm -- which is complex
and very difficult to prove to grade or high school
students.

x[n+1]=(x[n]+c/x[n])/2 where c is the number we are taking the square
root of.

is hard to prove, given a sloppy guess it converges fast as a little
experience will show. Not only that errors do not accumulate.

Yup. That's Newton's method. Easy enough to prove once
one has a year of calculus under one's belt.

The method I was taught in school was a manual method rather
like long division. One bunched the number whose root you
wanted into pairs of digits, and went on from there. It
actually is a longhand method of completing the square.

I don't know why they never taught Newton's method. It
is fairly quick even longhanded. And it converges as you
know, quadratically, so that the number of correct digits
doubles each time through. So one can get the first digit
almost in one's head. One more pass gets two, which involves
only a two digit division. The next pass gets four, which
is about what precision we worked to by hand.

And it converges no matter what guess you start with. That's
not always true for Newton's method, but it is for taking
square roots.

--
Norton.

.

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