### Mr. Matsumoto's
finding square roots by iteration.

I thought of a "new method" of calculating square root of n.

x^2=n

x^2+mx =n+mx (m is a guess number-Don)

(factor out x on the left side, then divide by m+x )

x=(n+mx)/(m+x)

(to top, add m^2 and subtract m^2, then
do some algebra to get)

x=m+(n-m^2)/(m+x)

If n is nearly m^2, so this value converge rapidly.
I iterate f(x)= m+(n-m^2)/(m+x).

For example: to find the square root of 5, n=5, pick m=2, initial
value
for x=1

2+(5-2^2)/(2+1)=2.3333...

2+(5-2^2)/(2+2.3333...)=2.230769...

2+(5-2^2)/(2+2.230769...)=2.236363...

2+(5-2^2)/(2+2.236363...)=2.236051...

2+(5-2^2)/(2+2.236051...)=2.236068...

These were calculated by MS Excel.

Mr. Matsumoto has read Don's book. He contacted Don by email with this
idea as well a method of using iteration in solving quadratic equations in
a
way that is different from the 10 ways Don shows in Ch. 8. Don shows
another way
to do square roots by iteration in Ch. 10. It's
exciting when people are willing to share their ideas. And it's exciting
to learn new things...There is never an end to learning!!

To other discoveries

To order Don's
materials

Mathman
home