Electron wrote:
> analogously to [division] computing the quotient by counting how many tim=
es
> you can subtract the divisor, you can compute the square root of a number
> counting how many times you can subtract sequential odd numbers
> (1,3,5,7,9,...).
>=20
> [ try it -- it works!]. e.g. 25 -1-3-5-7-9 =3D 0
>=20
> OK, but can someone explain me *why* it works? :-)

Of course. Just look at the differences between successive squares:

1^2 - 0^2 =3D 1
2^2 - 1^2 =3D 3
3^2 - 2^2 =3D 5
4^2 - 3^2 =3D 7

This is simply the discrete form of the general concept of differentiation.
The derivative of a 2nd-order function (a quadratic) is a 1st-order (linear=
)
function, and the derivative of a linear function is a constant (0-order
function).

Similarly, the discrete differences between squares is an arithmetic sequen=
ce,
and the differences between the numbers in that sequence is a constant (2).

Can you guess how you might compute an integer cube root?

-- Dave Tweed
--=20
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