2.3 Complex Numbers

Definition of Complex Numbers¶

Imaginary Components¶

Complex numbers expand the real numbers by adding the imaginary square root of -1

thus $\sqrt{-9}=3i, \sqrt{-2}=\sqrt{2}i$, etc.

Complex: Real plus Imaginary¶

Speaking more broadly, a complex number has both real and imaginary components. A complex number $z$ can be expressed as

with $x$ being the real components and $y$ the imaginary. Of course, we can treat any purely real or imaginary number as a complex number by setting $y$ or $x$ to zero, respectively.

Complex Conjugate¶

A key property of complex number is the complex conjugate $z^{*}$ defined as

i.e. the complex conjugate leaves the real components unchanged and flips the sign of the imaginary components. From this it immediately follows that any purely real number is its own complex conjugate.

Magnitude of Complex Numbers¶

The magnitude of a complex number, denoted as $|z|$, is given as

where we have used the fact that $-b^2i^2=b^2$.

Eq. (4) suggests that complex numbers may be viewed as vectors in the real plane with length $|z|$.

You may recognize Figure 1 as an example of an isomorphism between $\mathbb{C}$ and $\mathbb{R^2}$. If you’re not familiar with the term isomorphism, you can consider it a fancy way of saying “mapping.”

Polar Form¶

Figure 1 suggests the polar form of complex with magnitude $r=|z|$ and angle $\theta=\arctan{(\frac{y}{x})}$, allowing the substitution

While Eq. (5) may be suggestive as a heuristic, we will see in the next section that it actually touches upon a deep concept in mathematics.

Euler’s Formula: The “Jewel” of Mathematics¶

Euler’s formula, called “our jewel” by Richard Feynman, establishes a profound connection between trigonometry and complex numbers.

Euler’s Formula:

Or, multiplying through by $r$ for magnitudes different than 1:

Euler’s Identity¶

When $\theta=\pi$, we have:

or

This is known as Euler’s identity, which relates five fundamental mathematical constants: $e$, $i$, $\pi$, 1, and 0.

Proof of Euler’s Formula¶

Let $f(\theta) = \frac{\cos(\theta) + i \sin(\theta)}{e^{i\theta}}$

Since $\frac{df}{d\theta} = 0$, $f(\theta)$ is a constant, and we need only establish its value at a single point.

At $\theta = 0$:

Therefore $f(\theta) = 1$ for all $\theta$, or:

Applications of Euler’s Formula¶

Solving Problems Involving $i$¶

Euler’s formula can help solve otherwise extremely difficult problems involving $i$. To see this, note that given that $e^{i\theta} = \cos(\theta) + i \sin(\theta)$, at $\theta=\frac{\pi}{2}$ we have

Solving $i^i$¶

Eq. (16) allows us to solve problems that at first may seem intractable. For example, let us calculate $i^i$:

Somewhat surprisingly, it turns out that $i^i$ is a real number slightly greater than $\frac{1}{5}$.

Sine and Cosine Addition Formulae¶

Use of Euler’s formula can simplify trigonometric derivations by transforming geometric problems into algebraic ones. A pair of trigonometric identities that we will make use of later in the book are the sine and cosine addition formulae, given as

and

While these can be proven purely geometrically, Euler’s formula gives us an arguably far simpler algebraic method to prove them instead. Using $\Re(z)$ to denote the real component of $z$ and $\Im(z)$ to denote the imaginary component, we have

and

Note that

Combining Eq. (22) with Eq. (23) we arrive at the sine addition formula

Similarly, combining Eq. (21) with Eq. (23) gives us the cosine addition formula