WhizMath

Introduction: What are Complex Numbers?

Welcome to a fascinating realm of mathematics where numbers take on a new dimension! In our journey through mathematics, we've primarily dealt with real numbers—numbers that can be plotted on a single number line. However, what happens when we encounter equations like $x^2 + 1 = 0$? There are no real solutions for $x$, as the square of any real number is non-negative.

This is where complex numbers come into play. They extend the concept of real numbers by introducing an "imaginary" unit, allowing us to solve equations that are otherwise unsolvable within the real number system. Complex numbers are not just theoretical constructs; they are indispensable tools in fields like electrical engineering, physics, signal processing, and fluid dynamics.

In this lesson, we will demystify complex numbers, starting from their fundamental definition, exploring their arithmetic, and visualizing them in the complex plane. Prepare to expand your mathematical horizons and discover the elegance and utility of these extraordinary numbers!

Chapter 1: The Imaginary Unit and Basic Form

1.1 Defining the Imaginary Unit ($i$)

The cornerstone of complex numbers is the imaginary unit, denoted by $i$ (or sometimes $j$ in engineering contexts). It is defined as:

$i = \sqrt{-1}$

From this definition, it immediately follows that:

$i^2 = -1$

This property is fundamental and allows us to work with square roots of negative numbers.

Powers of $i$:

• $i^1 = i$
• $i^2 = -1$
• $i^3 = i^2 \cdot i = -1 \cdot i = -i$
• $i^4 = i^2 \cdot i^2 = (-1) \cdot (-1) = 1$

The powers of $i$ cycle with a period of 4. This pattern is very useful for simplifying higher powers of $i$. For example, $i^{25} = i^{4 \cdot 6 + 1} = (i^4)^6 \cdot i^1 = 1^6 \cdot i = i$.

1.2 Standard Form of a Complex Number

A complex number $z$ is generally expressed in the standard form (or rectangular form):

$z = a + bi$

Where:

• $a$ is the real part of $z$, denoted as $\text{Re}(z)$.
• $b$ is the imaginary part of $z$, denoted as $\text{Im}(z)$.
• Both $a$ and $b$ are real numbers.

Example 1.2.1:

• $z = 3 + 4i$: Real part is 3, imaginary part is 4.
• $z = -2 - 7i$: Real part is -2, imaginary part is -7.
• $z = 5$: This is a real number, but can be written as $5 + 0i$. So, real numbers are a subset of complex numbers.
• $z = -6i$: This is a pure imaginary number, can be written as $0 - 6i$. Real part is 0, imaginary part is -6.

Chapter 2: Operations with Complex Numbers

2.1 Addition and Subtraction

To add or subtract complex numbers, you simply add or subtract their corresponding real parts and imaginary parts separately.

Let $z_1 = a + bi$ and $z_2 = c + di$.

Addition:

$z_1 + z_2 = (a + c) + (b + d)i$

Subtraction:

$z_1 - z_2 = (a - c) + (b - d)i$

Example 2.1.1:

$(3 + 2i) + (1 - 4i) = (3+1) + (2-4)i = 4 - 2i$

Example 2.1.2:

$(5 - i) - (-2 + 3i) = (5 - (-2)) + (-1 - 3)i = (5 + 2) + (-1 - 3)i = 7 - 4i$

2.2 Multiplication

Multiplication of complex numbers is similar to multiplying two binomials, remembering that $i^2 = -1$.

Let $z_1 = a + bi$ and $z_2 = c + di$.

$z_1 \cdot z_2 = (a + bi)(c + di) = ac + adi + bci + bdi^2$

Since $i^2 = -1$:

$z_1 \cdot z_2 = ac + adi + bci - bd = (ac - bd) + (ad + bc)i$

Example 2.2.1:

$(2 + 3i)(4 - i) = (2)(4) + (2)(-i) + (3i)(4) + (3i)(-i)$

$ = 8 - 2i + 12i - 3i^2$

$ = 8 + 10i - 3(-1)$

$ = 8 + 10i + 3 = 11 + 10i$

2.3 Complex Conjugate

The complex conjugate of a complex number $z = a + bi$ is denoted as $\bar{z}$ (or $z^*$) and is obtained by changing the sign of the imaginary part:

$\bar{z} = a - bi$

The product of a complex number and its conjugate is always a real number:

$z \cdot \bar{z} = (a + bi)(a - bi) = a^2 - (bi)^2 = a^2 - b^2i^2 = a^2 + b^2$

Example 2.3.1:

The conjugate of $3 + 4i$ is $3 - 4i$.

$(3 + 4i)(3 - 4i) = 3^2 + 4^2 = 9 + 16 = 25$

2.4 Division

To divide complex numbers, we use the complex conjugate of the denominator to eliminate the imaginary part from the denominator. This process is similar to rationalizing the denominator with square roots.

Let $z_1 = a + bi$ and $z_2 = c + di$. To compute $\frac{z_1}{z_2}$:

$\frac{a + bi}{c + di} = \frac{a + bi}{c + di} \cdot \frac{c - di}{c - di} = \frac{(a + bi)(c - di)}{c^2 + d^2}$

Example 2.4.1:

$\frac{2 + 3i}{1 - 2i} = \frac{2 + 3i}{1 - 2i} \cdot \frac{1 + 2i}{1 + 2i}$

$ = \frac{2 + 4i + 3i + 6i^2}{1^2 + (-2)^2}$

$ = \frac{2 + 7i - 6}{5}$

$ = \frac{-4 + 7i}{5} = -\frac{4}{5} + \frac{7}{5}i$

Chapter 3: The Complex Plane (Argand Diagram)

Just as real numbers can be represented on a number line, complex numbers can be represented as points in a two-dimensional plane called the complex plane or Argand diagram.

• The horizontal axis is the real axis, representing the real part ($a$).
• The vertical axis is the imaginary axis, representing the imaginary part ($b$).

A complex number $z = a + bi$ is plotted as the point $(a, b)$ in the complex plane.

Example 3.1.1: Plotting Complex Numbers

• $z_1 = 3 + 2i$ is plotted at $(3, 2)$.
• $z_2 = -2 + i$ is plotted at $(-2, 1)$.
• $z_3 = -4 - 3i$ is plotted at $(-4, -3)$.
• $z_4 = 5i$ is plotted at $(0, 5)$.
• $z_5 = -6$ is plotted at $(-6, 0)$.

3.2 Modulus (Absolute Value) of a Complex Number

The modulus (or absolute value) of a complex number $z = a + bi$ is the distance from the origin $(0,0)$ to the point $(a,b)$ in the complex plane. It is denoted as $|z|$.

Using the Pythagorean theorem:

$|z| = \sqrt{a^2 + b^2}$

Example 3.2.1:

Find the modulus of $z = 3 + 4i$.

$|z| = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5$

3.3 Argument (Angle) of a Complex Number

The argument of a complex number $z = a + bi$ is the angle $\theta$ (theta) that the line segment from the origin to the point $(a,b)$ makes with the positive real axis. It is typically measured in radians and restricted to the interval $(-\pi, \pi]$ or $[0, 2\pi)$.

The argument can be found using trigonometric functions:

$\tan \theta = \frac{b}{a}$

However, care must be taken to determine the correct quadrant for $\theta$ based on the signs of $a$ and $b$. The `atan2(b, a)` function is often used in programming to get the correct angle directly.

Example 3.3.1:

Find the argument of $z = 1 + i$.

• $a = 1, b = 1$. The point $(1,1)$ is in the first quadrant.
• $\tan \theta = \frac{1}{1} = 1$
• $\theta = \arctan(1) = \frac{\pi}{4}$ radians (or $45^\circ$)

Example 3.3.2:

Find the argument of $z = -1 + i$.

• $a = -1, b = 1$. The point $(-1,1)$ is in the second quadrant.
• $\tan \theta = \frac{1}{-1} = -1$
• Using `atan2(1, -1)` or considering the quadrant, $\theta = \frac{3\pi}{4}$ radians (or $135^\circ$)

Chapter 4: Polar Form and Euler's Formula

4.1 Polar Form (Trigonometric Form)

A complex number $z = a + bi$ can also be expressed in polar form (or trigonometric form), which uses its modulus $r = |z|$ and argument $\theta = \arg(z)$.

From the complex plane, we can see that:

• $a = r \cos \theta$
• $b = r \sin \theta$

Substituting these into $z = a + bi$ gives:

$z = r \cos \theta + (r \sin \theta)i = r(\cos \theta + i \sin \theta)$

Example 4.1.1: Converting to Polar Form

Convert $z = 1 + \sqrt{3}i$ to polar form.

• Modulus: $r = |z| = \sqrt{1^2 + (\sqrt{3})^2} = \sqrt{1 + 3} = \sqrt{4} = 2$
• Argument: $\tan \theta = \frac{\sqrt{3}}{1} = \sqrt{3}$. Since $(1, \sqrt{3})$ is in the first quadrant, $\theta = \frac{\pi}{3}$.
• So, $z = 2(\cos \frac{\pi}{3} + i \sin \frac{\pi}{3})$

4.2 Euler's Formula

One of the most beautiful and profound formulas in mathematics, Euler's Formula, connects complex exponentials with trigonometric functions:

$e^{i\theta} = \cos \theta + i \sin \theta$

Where $e$ is Euler's number (the base of the natural logarithm, approximately 2.71828).

Using Euler's formula, the polar form of a complex number can be written as the exponential form:

$z = re^{i\theta}$

Example 4.2.1: Converting to Exponential Form

Convert $z = 1 + \sqrt{3}i$ to exponential form (from Example 4.1.1, we found $r=2$ and $\theta=\frac{\pi}{3}$).

$z = 2e^{i\frac{\pi}{3}}$

4.3 De Moivre's Theorem

De Moivre's Theorem is incredibly useful for finding powers and roots of complex numbers. For any integer $n$:

$(r(\cos \theta + i \sin \theta))^n = r^n (\cos (n\theta) + i \sin (n\theta))$

In exponential form, this is even simpler:

$(re^{i\theta})^n = r^n e^{in\theta}$

Example 4.3.1: Powers of Complex Numbers

Calculate $(1 + i)^4$.

• First, convert $1+i$ to polar form: $r = \sqrt{1^2 + 1^2} = \sqrt{2}$, $\theta = \frac{\pi}{4}$.
• So, $1 + i = \sqrt{2}(\cos \frac{\pi}{4} + i \sin \frac{\pi}{4})$.
• Using De Moivre's Theorem with $n=4$:
• $(\sqrt{2})^4 (\cos (4 \cdot \frac{\pi}{4}) + i \sin (4 \cdot \frac{\pi}{4}))$
• $= 4 (\cos \pi + i \sin \pi)$
• $= 4 (-1 + i \cdot 0) = -4$

Chapter 5: Applications of Complex Numbers

Complex numbers are far from abstract curiosities; they are fundamental to many areas of science and engineering:

• Electrical Engineering: Used to analyze AC circuits. Impedance, voltage, and current are often represented as complex numbers, simplifying calculations involving phase shifts.
• Physics: Quantum mechanics uses complex numbers extensively to describe wave functions. They are also used in optics, electromagnetism, and fluid dynamics.
• Signal Processing: Fourier analysis, which decomposes signals into their constituent frequencies, relies heavily on complex exponentials.
• Control Systems: Used in stability analysis and design of control systems, often visualized using pole-zero plots in the complex plane.
• Fractals: The Mandelbrot set and Julia sets are generated using iterative functions in the complex plane, producing stunning visual patterns.
• Computer Graphics: Transformations like rotations and scaling in 2D graphics can be elegantly represented and performed using complex number multiplication.

Practice Problems (with Solutions)