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# Multiplicitive inverse of a Complex Number

### Examples of Multiplicitive Inverses

Before we go over how multiplicative inverses work for complex numbers, let's quickly review how multiplicative inverses work for real rational numbers like the ones below.

$$2 \text{ and } \frac 1 2$$ because $$2 \cdot \frac 1 2 = \red 1$$
$$7 \text{ and } \frac 1 7$$ because $$2 \cdot \frac 1 7 = \red 1$$

Remember that the multiplictive inverse of a given number is what you multiply that number by in order to have a product of $$\red 1$$.

#### Ok, so how do we do this for complex numbers?

$$a + bi$$ = ?

The multiplicitive inverse of any complex number $$a + bi$$ is $$\frac{1}{ a + bi}$$.

However, since $$i$$ is a radical and in the denominator of a fraction, many teachers will ask you to rationalize the denominator. To rationalize the denominator just multiply by the complex conjugate of the original complex number (which is now in the denominator). Remember that multiplying two complex conjugates produces a real product, hence a rationalized denominator! We walk through this entire process in the example below.right-click