45.1 Cartesian Complex Numbers

A complex number is a number that can be expressed in the form $a + bi$, where $a$ and $b$ are real numbers, and $i$ is the imaginary unit, satisfying the equation $i^2 = -1$. Complex numbers extend the concept of the one-dimensional real number line to a two-dimensional complex plane.

In the expression $z = a + bi$:

• $a$ is called the real part of $z$, denoted as $\text{Re}(z)$.
• $b$ is called the imaginary part of $z$, denoted as $\text{Im}(z)$.
• The imaginary unit $i$ is defined as $i = \sqrt{-1}$.

A complex number with $b=0$ is a real number ($a + 0i = a$), and a complex number with $a=0$ is a purely imaginary number ($0 + bi = bi$).

Complex Conjugate

The complex conjugate of a complex number $z = a + bi$ is denoted by $\bar{z}$ (or $z^*$) and is defined as $\bar{z} = a - bi$. The sign of the imaginary part is flipped. Complex conjugates are particularly useful in division of complex numbers and finding roots of polynomials.

For example, if $z = 3 + 4i$, then $\bar{z} = 3 - 4i$.

45.2 Argand Diagram

The Argand diagram (also known as the complex plane) provides a graphical representation of complex numbers. It is a two-dimensional plot where the horizontal axis represents the real part ($\text{Re}(z)$) and the vertical axis represents the imaginary part ($\text{Im}(z)$) of a complex number.

A complex number $z = x + iy$ is plotted as a point $(x, y)$ in this plane. Alternatively, it can be represented as a vector from the origin $(0,0)$ to the point $(x,y)$.

The Argand diagram is crucial for visualizing complex number operations like addition, subtraction, multiplication, and for understanding the geometric interpretation of properties like modulus and argument.

45.3 Addition & Subtraction

Addition and subtraction of complex numbers in Cartesian form are performed by separately adding or subtracting their real and imaginary parts, similar to vector addition/subtraction.

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

Addition:

Subtraction:

Geometrically, adding complex numbers on an Argand diagram is equivalent to vector addition using the parallelogram rule or head-to-tail rule.

45.4 Multiplication & Division

Multiplication and division of complex numbers involve slightly more steps than addition and subtraction.

Multiplication:

To multiply two complex numbers $z_1 = a + bi$ and $z_2 = c + di$, we use the distributive property (similar to FOIL method for binomials) and remember that $i^2 = -1$:

Division:

To divide complex numbers $z_1 = a + bi$ by $z_2 = c + di$, we multiply the numerator and the denominator by the complex conjugate of the denominator. This eliminates the imaginary part from the denominator, resulting in a real number in the denominator:

45.5 Complex Equations

Solving equations involving complex numbers often means finding the unknown complex number(s) that satisfy the given equation. This can involve algebraic manipulation, equating real and imaginary parts, or using properties of complex numbers like conjugates.

Equating Real and Imaginary Parts:

If two complex numbers are equal, then their real parts must be equal, and their imaginary parts must be equal. If $a + bi = c + di$, then $a = c$ and $b = d$. This property is fundamental for solving many complex equations.

Quadratic Equations:

Quadratic equations of the form $ax^2 + bx + c = 0$ can have complex roots if the discriminant ($b^2 - 4ac$) is negative. The roots will be complex conjugates.

If $b^2 - 4ac = -k$ (where $k > 0$), then $\sqrt{b^2 - 4ac} = \sqrt{-k} = i\sqrt{k}$.

Solving Equations with Conjugates:

Equations involving complex conjugates often require substituting $z = x + iy$ and $\bar{z} = x - iy$ and then equating real and imaginary parts to form a system of simultaneous equations.

45.6 Polar Form

The polar form of a complex number represents it using its distance from the origin (modulus) and the angle it makes with the positive real axis (argument) in the Argand diagram. This form is often more convenient for multiplication, division, and powers of complex numbers.

A complex number $z = x + iy$ can be expressed in polar form as $z = r(\cos \theta + i \sin \theta)$, often abbreviated as $z = r \text{ cis } \theta$.

Using Euler's formula ($e^{i\theta} = \cos \theta + i \sin \theta$), the exponential form is $z = re^{i\theta}$.

• Modulus ($r$ or $|z|$): The distance from the origin to the point $(x, y)$ in the Argand diagram. It is calculated as: $$ r = |z| = \sqrt{x^2 + y^2} $$
• Argument ($\theta$ or $\text{arg}(z)$): The angle (in radians or degrees) measured counter-clockwise from the positive real axis to the line segment connecting the origin to the point $(x, y)$. It is calculated as: $$ \theta = \arctan\left(\frac{y}{x}\right) $$ When calculating $\theta$, always consider the quadrant of $(x, y)$ to ensure the correct angle. The function Math.atan2(y, x) directly gives the argument in the range $(-\pi, \pi]$ radians. The principal argument is usually taken in the range $(-\pi, \pi]$ or $[0, 2\pi)$.

45.7 Multiplication & Division (Polar)

Operations with complex numbers in polar form are significantly simpler than in Cartesian form.

Let $z_1 = r_1 (\cos \theta_1 + i \sin \theta_1)$ and $z_2 = r_2 (\cos \theta_2 + i \sin \theta_2)$.

Multiplication:

To multiply two complex numbers in polar form, you multiply their moduli and add their arguments:

In exponential form: $z_1 z_2 = (r_1 e^{i\theta_1})(r_2 e^{i\theta_2}) = r_1 r_2 e^{i(\theta_1 + \theta_2)}$

Division:

To divide two complex numbers in polar form, you divide their moduli and subtract their arguments:

In exponential form: $ \frac{z_1}{z_2} = \frac{r_1}{r_2} e^{i(\theta_1 - \theta_2)} $

De Moivre's Theorem:

De Moivre's Theorem is a powerful tool for finding powers and roots of complex numbers in polar form:

And for roots: $$ (r (\cos \theta + i \sin \theta))^{1/n} = r^{1/n} \left(\cos\left(\frac{\theta + 2k\pi}{n}\right) + i \sin\left(\frac{\theta + 2k\pi}{n}\right)\right) $$ where $k = 0, 1, 2, \dots, n-1$ to find $n$ distinct roots.

45.8 Applications

Complex numbers are not just abstract mathematical concepts; they have wide-ranging practical applications in various fields of science and engineering.

• Electrical Engineering (AC Circuits): Complex numbers are indispensable for analyzing alternating current (AC) circuits. Voltage and current can be represented as complex phasors, simplifying calculations involving impedance, admittance, and power.
• Signal Processing: In digital signal processing, complex numbers are used in Fourier analysis (e.g., Fast Fourier Transform - FFT) to decompose signals into their constituent frequencies, which is fundamental for audio processing, image compression, and telecommunications.
• Control Systems: Complex numbers are used in control theory to analyze the stability and performance of feedback control systems, particularly in the s-plane for Laplace transforms.
• Fluid Dynamics: Complex analysis can be used to model two-dimensional fluid flow.
• Quantum Mechanics: Complex numbers are fundamental to the mathematical formulation of quantum mechanics, where wave functions are inherently complex-valued.
• Aerodynamics: Used in conformal mapping to transform complex shapes (like airfoils) into simpler geometries for analysis.
• Graphics and Image Processing: Complex numbers are used in various transformations and fractal generation (e.e., Mandelbrot set).

The ability of complex numbers to represent both magnitude and phase makes them a powerful tool for describing oscillating and rotating phenomena found across many scientific and engineering disciplines.

46.1 De Moivre's Theorem Introduction

De Moivre's Theorem is a powerful mathematical formula that connects complex numbers, trigonometry, and powers. It states that for any real number $x$ and integer $n$, it holds that: $$ (\cos x + i \sin x)^n = \cos(nx) + i \sin(nx) $$

This theorem is incredibly useful for finding powers and roots of complex numbers, especially when they are expressed in polar form. It simplifies what would otherwise be very tedious algebraic expansions.

It can also be extended to rational exponents ($n$ can be a fraction), which is essential for finding roots of complex numbers.

46.2 Powers of Complex Numbers

De Moivre's Theorem is primarily used to raise complex numbers to integer powers. If a complex number is given in polar form $z = r(\cos \theta + i \sin \theta)$, then raising it to the power of $n$ (where $n$ is an integer) is straightforward:

This means you raise the modulus to the power of $n$ and multiply the argument by $n$.

If the complex number is initially in Cartesian form ($a + bi$), you must first convert it to polar form before applying De Moivre's Theorem.

46.3 Roots of Complex Numbers

De Moivre's Theorem can also be used to find the $n$-th roots of a complex number. Unlike powers, a complex number will have $n$ distinct $n$-th roots. The formula for the $n$-th roots of $z = r(\cos \theta + i \sin \theta)$ is:

Where $k = 0, 1, 2, \dots, n-1$. Each value of $k$ yields a different root. The addition of $2k\pi$ (or $360k^\circ$) accounts for the multiple rotations around the Argand diagram, leading to distinct arguments for each root. These roots are geometrically spaced equally around a circle in the Argand diagram.

46.4 Exponential Form

The exponential form of a complex number is $z = re^{i\theta}$, where $r$ is the modulus and $\theta$ is the argument in radians. This form is derived directly from Euler's Formula: $$ e^{i\theta} = \cos \theta + i \sin \theta $$

The exponential form is extremely concise and makes multiplication, division, and applying De Moivre's Theorem even simpler due to the properties of exponents:

• Multiplication: $z_1 z_2 = (r_1 e^{i\theta_1})(r_2 e^{i\theta_2}) = r_1 r_2 e^{i(\theta_1 + \theta_2)}$
• Division: $\frac{z_1}{z_2} = \frac{r_1 e^{i\theta_1}}{r_2 e^{i\theta_2}} = \frac{r_1}{r_2} e^{i(\theta_1 - \theta_2)}$
• Powers (De Moivre's): $z^n = (re^{i\theta})^n = r^n e^{in\theta}$

46.5 Locus Problems Intro

A locus in the complex plane (Argand diagram) is the set of all points that satisfy a given condition or equation involving complex numbers. These problems often involve interpreting geometric shapes such as circles, straight lines, or regions.

Common forms of locus equations include:

• $|z - z_0| = r$: Represents a circle centered at $z_0$ with radius $r$.
• $|z - z_1| = |z - z_2|$: Represents the perpendicular bisector of the line segment joining $z_1$ and $z_2$.
• $\text{arg}(z - z_0) = \alpha$: Represents a ray starting from $z_0$ at an angle $\alpha$ with the positive real axis.

Solving locus problems typically involves translating the complex number equation into Cartesian coordinates (by substituting $z = x + iy$) and then simplifying to recognize the geometric shape.