14.1 Introduction

The complex conjugate of $x+yi$ is just $x-yi$ - that is, when we take the complex conjugate of a number, the real part remains unchanged, but the sign of the complex part is flipped (i.e. positive numbers become negative and vice versa).

The complex conjugate can be denoted in a number of ways. For a complex number $z$, we can write the conjugate as $\overline{z}$ (it is also sometimes written as $z^{*}$).

This is a useful theorem (well it shows up sometimes), and the proof uses the standard method for proving identities (Section 6.2).

Another operation we can perform is division. To divide two complex numbers (which we can call $w$ and $z$), we do something similar to rationalising the denominator (except for complex numbers). Note that $\sqrt{-1}$ behaves just like a surd (except that it’s not in the set of real numbers), so it shouldn’t come as any great surprise that the same techniques are useful here as for surds.

We want to make the denominator real, so we just rationalise it,

If we plot a complex number, then the modulus is the \saylength of the complex number. This is exactly the same as how the magnitude of the vector in $\mathbb{R}^{2}$ (all real-number valued coordinates. e.g. $(1,2)$, $(\pi,6)$, etc.) is defined as

Proof: we can prove this by direct computation. If $z$ is a complex number then there exist $x$ and $y$ such that $z=x+iy$, and therefore

Note that we used the difference of two squares identity (see Section 6.2) to show this.

Proof: let $z=a+bi$ and $w=c+di$, then we proceed with the standard method of proving identities (Section 6.2),

What remains is to prove that

We can work backwards here, and obtain that

Which is what we needed to prove (note that it is of paramount importance that our proof - which it does - flows in both directions; that is, that every step is reversible, see Section 6.2 for more on this).

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  $(zw)^{*}=z^{*}w^{*}$

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  $\left(\frac{z}{w}\right)^{*}$

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  $\left(z^{*}\right)^{*}$

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  $\left\lvert z\right\rvert=\left\lvert z^{*}\right\rvert$

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  $\left\lvert zw\right\rvert=\left\lvert z\right\rvert\left\lvert w\right\rvert$

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  $\left\lvert\frac{z}{w}\right\rvert=\frac{\left\lvert z\right\rvert}{\left\lvert w\right\rvert}$

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  $\left\lvert z+w\right\rvert\leqq\left\lvert z\right\rvert+\left\lvert w\right\rvert$

We can prove this using the standard method for proving identities (see Section 6.2 for more details of the method) - pick one side and work to the other side using reversible steps.