[Roots and Powers from Division (! ?)] - A New Kind of Science: The NKS ForumA New Kind of Science: The NKS Forum
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Roots and Powers from Division (! ?)
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Posted by: Val Smith
Sorry if this has been noticed before and maybe well known,
but to me it's very profound and I should have heard of it before.
It must be the simplest way of calculating
y^x and roots I ever saw. It just hit me last night.
And it's just too easy,simple,elegant.
To find all the powers of N,
subtract N from a "gazillion"
and calculate the inverse
(simply make a fraction).
A Gazillion is thus defined any power of 10
and the higher it is the greater the precision.
The quotient will periodically produce
a new power of N.
Example:
1/(1,000,000-16)=0. 00001 00016 00256 65536...
or something like that. A neat digital matrix
of the powers of 16 for example.
Eventually overflows will simply carry over
on top of the previous power if you chose
an insufficiently big oodle to work with.
Such errors are not useless IMHO
but inspire revelations of number theory.
Posted by: Todd Rowland
Very nice.
It is known in the form of the power series expansion of
1/1-x == 1+x + x^2+...
where x== N/10^n
Though it should be pointed out that when N^k < 10^n the power shows up, but for larger k, the digits overlap.
E.g.,
N[1/(1000000 - 16), 40]==
1.000016000256004096065537
compared to the powers of 16, i.e., 1, 16, 256, 4096, 65536, 1048576.
The power series has motivated new ideas in number theory, and surely will continue doing so.
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