# Complex plane

## Contents

# Complex plane#

From the definition of complex numbers it is clear that there is a natural correspondence between a set of complex numbers \(\mathbb{C}\) and points in a plane: to every complex number \(z = x +iy\) we can *uniquely* assign a point \(P(x, y)\) in the \(\mathbb{R}^2\) plane. We call a plane in which every point has a corresponding complex number assigned to it a **complex plane** (or Argand plane). However, for simplicity we would normally simply say “in a point \(z\)” when we mean “in a point (of a complex plane) assigned a complex number \(z\).”

```
z = complex(3, 2)
fig = plt.figure()
ax = fig.add_subplot(111)
plt.plot(z.real, z.imag, 'o', zorder=10)
plt.plot([3, 3], [0, 2], '--k', alpha=0.8)
plt.plot([0, 3], [2, 2], '--k', alpha=0.8)
ax.text(2.7, -0.25, "$Re(z)$", fontsize=12)
ax.text(-0.7, 1.95, "$Im(z)$", fontsize=12)
ax.arrow(-1., 0, 4.5, 0, shape='full', head_width=0.1, head_length=0.2)
ax.arrow(0, -1.5, 0, 4, shape='full', head_width=0.1, head_length=0.2)
ax.axis('equal')
ax.set_xlim(-2, 4)
ax.set_ylim(-2, 4)
ax.axis('off')
plt.show()
```

Having represented complex numbers geometrically, let us revisit the terms we introduced in the first notebook.

Re \(z\) is equal to the abscissa and Im \(z\) to the ordinate of a point \(P\). We therefore call x-axis the

*real axis*and y-axis the*imaginary axis*.The modulus of a complex number \(|z|\) is equal to the distance between the corresponding point \(P\) and the origin of the complex plane. In general, the distance between points \(P_1\) and \(P_2\) assigned to numbers \(z_1\) and \(z_2\) is equal to the modulus of the difference of those two points:

If a number \(z\) is assigned a point \(P(x, y)\), then the complex conjugate \(z^*\) is assigned a point \(P^*(x, -y)\), which is just a reflection of \(P\) with respect to the real axis.

Every point \(P(x, y)\) has a corresponding vector \(\vec{OP}\) from the origin to \(P\). The sum \(z_1 + z_2\) therefore represents vector addition \( \overrightarrow{OP_1} + \overrightarrow{OP_2} \).

Because of the vector analogy, we can also say that complex numbers satisfy the *triangle inequalities*:

# Trigonometric form#

We can also observe the complex plane in polar coordinates \((r, \phi)\) which are related to the Cartesian coordinates \((x, y)\) through:

This leads to the *trigonometric form* of a complex number \(z\):

where \(r\) is the **modulus** (or magnitude), equal to the absolute value of the complex number, a direct consequence of Pythagora’s theorem:

and the polar angle \(\varphi\) is defined (to a multiply \(2\pi\)) by:

The angle \(\varphi\) is called the **argument** (or phase) of a complex number \(z\) and we write \(\text{Arg}(z) = \varphi\). A special case is the complex number \(z=0\) for which \(r = 0\) and its argument is not defined.

Finding \(\varphi\) is delicate because \(\tan\) is a multivalued function. To avoid ambiguity, the simplest choice is \(n = 0\) so that the interval is of length \(2\pi\) and \( - \pi < \text{arg}(z) \leq \pi \). The value of \(\text{Arg}(z)\) with \(n=0\) is called the **principal value** of the argument. With this:

THe relationship between \(\text{Arg}(z)\) and \(\text{arg}(z)\) is therefore:

Multiplying two complex numbers results in their absolute values being multiplied and the arguments being added:

For the multiplicative inverse we have

Properties of \(|z|\) and arg(\(z\))

Let us summarise our findings in the form of the following equalities:

where \(z_1, z_2 \neq 0\). By induction we get:

# Polar form#

The property \( \text{arg}(z_1 \cdot z_2) = \text{arg}(z_1) + \text{arg}(z_2) \) might remind us of logarithms, where \(\log (a \cdot b) = \log a + \log b\). This is not a coincidence!

The exponential function, which we will look at in the next chapter, allows us to write complex numbers in a *polar form*:

where we have used the **Euler’s formula**:

In this representation, certain operations become much easier. For example,

for all powers \(n \in \mathbb{Z}\). If we write this using a complex number with unit modulus we recover **de Moivre’s formula**:

Tip: Trigonometric identities

Trigonometric identities are also much easier to be recovered this way. For example, let us think about angle addition \(\theta + \varphi\).

We Apply Euler’s formula to each complex number.

Equate real and imaginary parts on both sides:

The reader is encouraged to try to recover other trigonometric identities.