Multiplicative inverse
Complex numbers

The property that x x = 1 {\displaystyle {\frac {x}{x}}=1} which holds for real numbers also holds for complex numbers, a + b i a + b i = z z = 1 {\displaystyle {\frac {a+bi}{a+bi}}={\frac {z}{z}}=1}. In order to find the multiplicative inverse of a complex number z = a + b i {\displaystyle z=a+bi}, we must find this 1 z {\displaystyle {\frac {1}{z}}}. As mentioned above, the reciprocal of every nonzero complex number z = a + b i {\displaystyle z=a+bi} is complex. It can be found by multiplying both top and bottom of 1/ z by its complex conjugate z ¯ = a − b i {\displaystyle {\bar {z}}=a-bi} and using the property that z z ¯ = ‖ z ‖ 2 {\displaystyle z{\bar {z}}=\|z\|^{2}}, the absolute value of z squared, which is the real number a + b:

The intuition is that

gives us the complex conjugate with a magnitude reduced to a value of 1 {\displaystyle 1}, so dividing again by ‖ z ‖ {\displaystyle \|z\|} ensures that the magnitude is now equal to the reciprocal of the original magnitude as well, hence:

In particular, if || z ||=1 (z has unit magnitude), then 1 / z = z ¯ {\displaystyle 1/z={\bar {z}}}. Consequently, the imaginary units, ± i, have additive inverse equal to multiplicative inverse, and are the only complex numbers with this property. For example, additive and multiplicative inverses of i are −(i) = − i and 1/ i = − i, respectively.

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