Complex Numbers: Conjugate Arithmetic

We know that for a complex number $z=a+bi$, the complex conjuagte of $z$, denoted $\overline{z}$, is $\overline{z}=a-bi$.

The complex conjugate has some pleasing arithmetic properties. All of these properties can be proved very easily by simply decomposing the complex numbers into their real and imaginary components, or by expressing the complex numbers in polar form, so I will not prove them here.

$$\overline{x}+\overline{y}=\overline{x+y}$$ $$\overline{x}-\overline{y}=\overline{x-y}$$ $$\overline{x}\cdot\overline{y}=\overline{xy}$$ $$\frac{\overline{x}}{\overline{y}}=\overline{x/y}$$

The four simplest computational operations all allow us to separate or combine two complex numbers under the conjugate. This is a very useful fact whenever one is dealing with complex numbers.

Here are some other useful computational facts to keep in mind:

$$z\cdot\overline{z}=|z|^2$$ $$|z|=|\overline{z}|$$ $$\arg(\overline{z})=-\arg(z)$$

(If you're unfamiliar with the magnitude $|z|$, see here. Don't worry too much about the argument ($\arg(z)$) for now; you will encounter it soon here.)

Notice the fact that $z\cdot\overline{z}$ and $z+\overline{z}$ are both real numbers. In fact, unless $z$ is real, $\overline{z}$ is the only number, real or complex, that satisfies this property.

For this reason, to simplify a fraction with a complex number in the denominator, one may multiply both the numerator and denominator by the conjugate to make the denominator a real number. This is similar to rationalizing the denominator by using an irrational number's radical conjugate.

Also, if a nonreal number is a root of a polynomial with real coefficients, then that number's conjugate is also a root of that polynomial. This also leads to the fact that every polynomial of odd degree has at least one real root, without any vague continuity argument.