Q: Use Newton's method to approximate .
I know that "you are supposed to" let
and from there on proceed with Newton's method. I was wondering why we one cannot instead let
I realized that it is to no avail once you plug it all in into
but I am not understanding why it would be wrong to let . Is it merely because the sought value is not an intersection with the x-axis for ?
the point is, √10 isn't rational. so to define it in terms of rational numbers, we need a sequence of rational numbers that converges to √10. the sequence:
isn't such a sequence, even though it clearly converges to √10.
any number "is what it is". that's not so useful as asking "how big is it?", and doing THAT entails approximation.
once could be perverse and actually obtain an approximation of √10 in terms of rational expressions in pi (use f(x) = x2- 10 and x0 = pi) although i am unsure as to WHY one might do so.
when we say "approximate" we usually mean: "approximate by rational numbers", or sometimes even: "approximate by finite decimals" (a much more restrictive class of rational numbers).