Editing Multiplicative inverse (section)

== Complex numbers ==
As mentioned above, the reciprocal of every nonzero complex number <math>z=a+bi</math> is complex. It can be found by multiplying both top and bottom of 1/''z'' by its [[complex conjugate]] <math>\bar z = a - bi</math> and using the property that <math>z\bar z = \|z\|^2</math>, the [[absolute value]] of ''z'' squared, which is the real number {{math|''a''<sup>2</sup> + ''b''<sup>2</sup>}}:

:<math>\frac{1}{z} = \frac{\bar z}{z \bar z} = \frac{\bar z}{\|z\|^2} = \frac{a - bi}{a^2 + b^2} = \frac{a}{a^2 + b^2} - \frac{b}{a^2+b^2}i.</math>

The intuition is that
:<math>\frac{\bar z}{\|z\|}</math>
gives us the [[complex conjugate]] with a [[Magnitude (mathematics)|magnitude]] reduced to a value of <math>1</math>, so dividing again by <math>\|z\|</math> ensures that the magnitude is now equal to the reciprocal of the original magnitude as well, hence:
:<math>\frac{1}{z} = \frac{\bar z}{\|z\|^2}</math>
In particular, if ||''z''||=1 (''z'' has unit magnitude), then <math>1/z = \bar z</math>. Consequently, the [[imaginary unit]]s, {{math|±''i''}}, have [[additive inverse]] equal to multiplicative inverse, and are the only complex numbers with this property. For example, additive and multiplicative inverses of {{math|''i''}} are {{math|1=−(''i'') = −''i''}} and {{math|1=1/''i'' = −''i''}}, respectively.

For a complex number in polar form {{math|1=''z'' = ''r''(cos φ + ''i'' sin φ)}}, the reciprocal simply takes the reciprocal of the magnitude and the negative of the angle:

:<math>\frac{1}{z} = \frac{1}{r}\left(\cos(-\varphi) + i \sin(-\varphi)\right).</math>

[[File:Reciprocal integral.svg|thumb|Geometric intuition for the integral of 1/''x''. The three integrals from 1 to 2, from 2 to 4, and from 4 to 8 are all equal. Each region is the previous region halved vertically and doubled horizontally. Extending this, the integral from 1 to 2<sup>''k''</sup> is ''k'' times the integral from 1 to 2, just as ln 2<sup>''k''</sup> = ''k'' ln 2.]]