Editing Multiplicative inverse (section)

== Algorithms ==
The reciprocal may be computed by hand with the use of [[long division]].

Computing the reciprocal is important in many [[division algorithm]]s, since the quotient ''a''/''b'' can be computed by first computing 1/''b'' and then multiplying it by ''a''. Noting that <math>f(x) = 1/x - b</math> has a [[Zero of a function|zero]] at ''x'' = 1/''b'', [[Newton's method]] can find that zero, starting with a guess <math>x_0</math> and iterating using the rule:

:<math>x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)} = x_n - \frac{1/x_n - b}{-1/x_n^2} = 2x_n - bx_n^2 = x_n(2 - bx_n).</math>

This continues until the desired precision is reached. For example, suppose we wish to compute 1/17 ≈ 0.0588 with 3 digits of precision. Taking ''x''<sub>0</sub> = 0.1, the following sequence is produced:
:''x''<sub>1</sub> = 0.1(2 − 17 × 0.1) = 0.03
:''x''<sub>2</sub> = 0.03(2 − 17 × 0.03) = 0.0447
:''x''<sub>3</sub> = 0.0447(2 − 17 × 0.0447) ≈ 0.0554
:''x''<sub>4</sub> = 0.0554(2 − 17 × 0.0554) ≈ 0.0586
:''x''<sub>5</sub> = 0.0586(2 − 17 × 0.0586) ≈ 0.0588
A typical initial guess can be found by rounding ''b'' to a nearby power of 2, then using [[bit shift]]s to compute its reciprocal.

In [[constructive mathematics]], for a real number ''x'' to have a reciprocal, it is not sufficient that ''x'' ≠ 0. There must instead be given a ''rational'' number ''r'' such that 0&nbsp;&lt;&nbsp;''r''&nbsp;&lt;&nbsp;|''x''|. In terms of the approximation [[algorithm]] described above, this is needed to prove that the change in ''y'' will eventually become arbitrarily small.

[[File:X to x power showing minimum.svg|thumb|Graph of f(''x'') = ''x''<sup>''x''</sup> showing the minimum at (1/''e'', ''e''<sup>−1/''e''</sup>).]]

This iteration can also be generalized to a wider sort of inverses; for example, [[Invertible matrix#Newton's method|matrix inverses]].