Complex plane

Mathematics Methods 2

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:

\[ d(T_1, T_2) = \sqrt{(x_1 - x_2)^2 + (y_1 - y_2)^2} = | z_1 - z_2 |\]

• 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:

\[ | z_1| - |z_2| \leq |z_1 + z_2| \leq |z_1| + |z_2| \]

Trigonometric form

Complex plane - Argand diagram

Source: Wikipedia

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

\[ x = r \cos \varphi, \quad y = r \sin \varphi \]

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

\[ z = r(\cos \varphi + i \sin \varphi), \]

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

\[ r = \sqrt{x^2 + y^2} = |z| \]

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

\[ \tan (\varphi + 2n \pi) = \frac{y}{x}, \quad n \in \mathbb{Z}, x \neq 0. \]

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:

\[ \text{arg}(1) = 0, \quad \text{arg}(i) = \frac{\pi}{2}, \quad \text{arg}(-1) = \pi, \quad \text{arg}(-i) = -\frac{\pi}{2}, \quad \text{etc.} \]

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

\[ \text{Arg}(z) = \text{arg}(z) + 2n \pi, \quad n \in \mathbb{Z}.\]

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

Angle addition formulae

\[\begin{split} \sin(\alpha + \beta) = \sin \alpha \cos \beta + \cos \alpha \sin \beta \\ \cos (\alpha + \beta) = \cos \alpha \cos \beta - \sin \alpha \sin \beta\end{split}\]

\[\begin{split} \begin{align} z_1 \cdot z_2 & = r_1(\cos \varphi_1 + i \sin \varphi_1) \cdot r_2(\cos \varphi_2 + i \sin \varphi_2) \\ & = r_1 r_2 (\cos (\varphi_1 + \varphi_2) + i \sin(\varphi_1 + \varphi_2)) \end{align} \end{split}\]

For the multiplicative inverse we have

\[\begin{split} \begin{align} z^{-1} & = \frac{1}{r(\cos \varphi + i \sin \varphi)} \cdot \frac{cos \varphi - i \sin \varphi}{cos \varphi - i \sin \varphi} = \frac{\cos \varphi - i \sin \varphi}{r( \cos^2 \varphi + \sin^2 \varphi)} \\ & = \frac{1}{r} (\cos \varphi - i \sin \varphi) \end{align} \end{split}\]

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

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

\[ | z_1 \cdot z_2 | = |z_1| \cdot |z_2| = r_1 r_2, \quad \text{arg}(z_1 \cdot z_2) = \text{arg}(z_1) + \text{arg}(z_2), \]

\[|z^{-1}| = |z|^{-1} = r^{-1}, \quad \text{arg}(z^{-1}) = -\text{arg}(z) \]

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

\[ |z^n| = |z|^n, \quad \text{arg}(z^n) = n\text{arg}(z), \quad \forall n \in \mathbb{Z}. \]

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:

\[ z = r e^{i \varphi} \]

Euler’s identity

For a special angle \(\varphi = \pi\) the Euler formula gives the Euler’s identity:

\[ e^{i \pi} + 1 = 0 \]

It is considered to be one of the most beautiful equations in mathematics.

where we have used the Euler’s formula:

\[ e^{i \varphi} = \cos(\varphi) + i \sin (\varphi). \]

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

\[ z_1 \cdot z_2 = r_1 r_2 e^{i(\varphi_1 + \varphi_2)}, \quad z^n = r^n e^{in \varphi} \]

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

\[ (\cos \varphi + i \sin \varphi)^n = (e^{i \varphi})^n = e^{in \varphi} = \cos (n \varphi) + i \sin (n \varphi). \]

Tip: Trigonometric identities

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

\[ e^{i(\theta + \varphi)} = e^{i \theta} e^{i \varphi} \]

We Apply Euler’s formula to each complex number.

\[\begin{split}\begin{align} \cos (\theta + \varphi) + i \sin(\theta + \varphi) & = (\cos \theta + i \sin \theta)(\cos \varphi + i \sin \varphi) \\ & = \cos \theta \cos \varphi + i \cos \theta \sin \varphi + i \sin \theta \cos \varphi + i^2 \sin \theta \sin \varphi \\ & = (\cos \theta \cos \varphi - \sin \theta \sin \varphi) + i(\cos \theta \sin \varphi + \sin \theta \cos \varphi) \end{align} \end{split}\]

Equate real and imaginary parts on both sides:

\[\begin{split}\cos (\theta + \varphi) = \cos \theta \cos \varphi - \sin \theta \sin \varphi \\ \sin (\theta + \varphi ) = \cos \theta \sin \varphi + \sin \theta \cos \varphi \end{split}\]

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