Comprehensive Study Notes on Complex Numbers

Historical Context and Introduction to Complex Numbers

• The real number system possesses inherent limitations that prevented the solution of certain algebraic expressions, specifically quadratic equations with negative discriminants.
• These limitations were resolved through sequential conceptual and mechanical advancements in mathematics.
• The resolution of these limitations led to the introduction of imaginary numbers and the notation $i^2 = -1$ or $i = \sqrt{-1}$.
• The extension of the real number system to incorporate imaginary numbers created the complex number system.
• Numbers are fundamentally classified into two main divisions:
  • Real numbers: Includes integers, rational numbers, irrational numbers, and surds.
  • Imaginary numbers: Specialized numbers used predominantly in advanced science and engineering disciplines.
• Together, real numbers and imaginary numbers form the basis of all complex numbers.

Definition and Forms of Complex Numbers

• A complex number is an expression of the standard form $a + bi$, where $a$ and $b$ are real numbers.
• In the expression $a + bi$:
  • The value $a$ represents the real part.
  • The value $b$ represents the imaginary part.
• Special cases of complex numbers include:
  • Purely imaginary numbers: Occur when $a = 0$, reducing the complex number to $ib$ (or $bi$).
  • Pure real numbers: Occur when $b = 0$, reducing the complex number to $a$.
• Denotational conventions:
  • Complex numbers are denoted by the letter $Z$, where $Z = a + bi$.
  • Complex numbers can also be written in coordinate or ordered pair form as $Z = (a, b)$.
• Fundamental principle: Every real number is a complex number whose imaginary part is equal to $0$.

Fundamental Properties of Complex Numbers

• Equality of Complex Numbers:
  • Two complex numbers $a + bi$ and $c + di$ are equal if and only if their real parts are equal and their imaginary parts are equal:     $a + bi = c + di \iff a = c \text{ and } b = d$
  • Application Example: Given $x - 2 + 4yi = 3 + 12i$, solving for $x$ and $y$ yields:     $x - 2 = 3 \implies x = 5$$4y = 12 \implies y = 3$
• Vanishing Complex Number Property:
  • If a complex number equals zero, its real part and imaginary part must independently equal zero:     $a + ib = 0 \implies a = 0 \text{ and } b = 0$
  • Proof / Derivation:     $a + ib = 0$$a = -ib$$a^2 = (-ib)^2$$a^2 = -b^2$$a^2 + b^2 = 0$     Because $a$ and $b$ are real, $a^2 \ge 0$ and $b^2 \ge 0$. Thus, $a^2 + b^2 = 0$ is possible if and only if $a = 0$ and b = 0$.\n\n# Basic Algebraic Operations on Complex Numbers\n\n* Addition:\n * Given Z_1 = a_1 + b_1 i$and$Z_2 = a_2 + b_2 i:\n    Z_1 + Z_2 = (a_1 + b_1 i) + (a_2 + b_2 i) = (a_1 + a_2) + i(b_1 + b_2)\n* Subtraction:\n * Given Z_1 = a_1 + b_1 i$and$Z_2 = a_2 + b_2 i:\n    Z_1 - Z_2 = (a_1 + b_1 i) - (a_2 + b_2 i) = (a_1 - a_2) + i(b_1 - b_2)\n* Multiplication:\n * Expand using the distributive law and substitute i^2 = -1:\n    Z_1 Z_2 = (a_1 + b_1 i)(a_2 + b_2 i)\n    Z_1 Z_2 = a_1 a_2 + a_1 b_2 i + b_1 a_2 i + b_1 b_2 i^2\n    Z_1 Z_2 = (a_1 a_2 - b_1 b_2) + i(a_1 b_2 + b_1 a_2)\n* Division:\n * To evaluate \frac{Z_1}{Z_2} = \frac{a_1 + b_1 i}{a_2 + b_2 i}$, multiply both numerator and denominator by the complex conjugate of the denominator ($a_2 - b_2 i) to rationalize the denominator:\n    \frac{Z_1}{Z_2} = \frac{(a_1 + b_1 i)(a_2 - b_2 i)}{(a_2 + b_2 i)(a_2 - b_2 i)}\n    \frac{Z_1}{Z_2} = \frac{(a_1 a_2 + b_1 b_2) + i(b_1 a_2 - a_1 b_2)}{a_2^2 + b_2^2}\n    \frac{Z_1}{Z_2} = \frac{a_1 a_2 + b_1 b_2}{a_2^2 + b_2^2} + i\,\frac{b_1 a_2 - a_1 b_2}{a_2^2 + b_2^2}\n* All final algebraic results should generally be simplified into standard form a + ib.\n\n# Worked Examples of Basic Algebraic Operations\n\n* Example 1 (Addition and Subtraction):\n * Perform addition and subtraction on 3 + 4i$and$2 - 7i:\n * Addition: (3 + 4i) + (2 - 7i) = (3 + 2) + i(4 - 7) = 5 - 3i\n * Subtraction: (3 + 4i) - (2 - 7i) = (3 - 2) + i(4 - (-7)) = 1 + 11i\n* Example 2 (Multiplication):\n * Calculate the product of 3 + 4i$and$2 - 7i:\n    (3 + 4i)(2 - 7i) = 6 - 21i + 8i - 28i^2 = 6 - 13i - 28(-1) = 6 + 28 - 13i = 34 - 13i\n* Example 3 (Division):\n * Divide 3 + 4i$by$2 - 7i:\n    \frac{3 + 4i}{2 - 7i} = \frac{(3 + 4i)(2 + 7i)}{(2 - 7i)(2 + 7i)} = \frac{6 + 21i + 8i + 28i^2}{4 - 49i^2} = \frac{6 - 28 + i(21 + 8)}{4 + 49} = \frac{-22 + 29i}{53} = -\frac{22}{53} + i\,\frac{29}{53}\n* Example 4 (Simplification to a + ib Form):\n * Express \frac{(2 + i)(1 - i)}{4 - 3i}$in the form$a + ib:\n * Step 1: Simplify the numerator:\n      (2 + i)(1 - i) = 2 - 2i + i - i^2 = 2 - i - (-1) = 3 - i\n * Step 2: Rationalize the expression:\n      \frac{3 - i}{4 - 3i} = \frac{(3 - i)(4 + 3i)}{(4 - 3i)(4 + 3i)} = \frac{12 + 9i - 4i - 3i^2}{16 - 9i^2} = \frac{12 + 3 + i(9 - 4)}{16 + 9} = \frac{15 + 5i}{25} = \frac{15}{25} + i\,\frac{5}{25} = \frac{3}{5} + i\,\frac{1}{5}\n* Example 5 (Separating into Real and Imaginary Parts):\n * Separate \frac{1 + 4i}{3 + i} into its real and imaginary parts:\n    \frac{1 + 4i}{3 + i} = \frac{(1 + 4i)(3 - i)}{(3 + i)(3 - i)} = \frac{3 - i + 12i - 4i^2}{9 - i^2} = \frac{3 + 4 + i(12 - 1)}{9 + 1} = \frac{7 + 11i}{10} = \frac{7}{10} + i\,\frac{11}{10}\n * Real part a = \frac{7}{10}$, Imaginary part$b = \frac{11}{10}.\n\n# Extraction of Square Roots of a Complex Number\n\n* Step-by-step methodology for finding \sqrt{Z}:\n * Set a + ib = \sqrt{Z}.\n * Square both sides to produce a^2 - b^2 + 2abi = Z.\n * Equate respective real and imaginary parts to set up a system of simultaneous equations.\n * Solve for real constants a$and$b.\n* Example 6 (Square Root Calculation):\n * Extract the square root of 21 - 20i:\n * Let a + ib = \sqrt{21 - 20i}.\n * Squaring both sides yields:\n      (a + ib)^2 = 21 - 20i\n      a^2 - b^2 + 2abi = 21 - 20i\n * Equating real and imaginary parts:\n      a^2 - b^2 = 21 \quad \text{--- (Equation 1)}\n      2ab = -20 \implies b = -\frac{10}{a} \quad \text{--- (Equation 2)}\n * Substitute Equation 2 into Equation 1:\n      a^2 - \left(-\frac{10}{a}\right)^2 = 21\n      a^2 - \frac{100}{a^2} = 21\n      a^4 - 21a^2 - 100 = 0\n * Factoring the quadratic in terms of a^2:\n      (a^2 - 25)(a^2 + 4) = 0\n      a^2 = 25 \text{ or } a^2 = -4\n * Since a$must be a real number,$a^2 = -4$is rejected (as it yields imaginary$a = \pm 2i).\n * Therefore, a^2 = 25 \implies a = 5 \text{ or } a = -5$.
  • Calculating corresponding values for $b$ via Equation 2:
    • If $a = 5$, $b = -\frac{10}{5} = -2$
    • If $a = -5$, $b = -\frac{10}{-5} = 2$
  • The two square roots of $21 - 20i$ are $5 - 2i$ and $-5 + 2i$.

Factorization of Complex Expressions

• Real quadratic sums of squares $a^2 + b^2$ cannot be factored over real numbers, but can be factored over complex numbers using i^2 = -1$.\n* Example 7 (Sum of Squares Factorization):\n * Factorise a^2 + b^2:\n    a^2 + b^2 = a^2 - (-b^2)\n    a^2 + b^2 = a^2 - (i^2 b^2)\n    a^2 + b^2 = (a)^2 - (ib)^2\n    a^2 + b^2 = (a + ib)(a - ib)\n\n# Additive and Multiplicative Inverses\n\n* Additive Inverse:\n * For any complex number Z = a + ib$, its additive inverse is$-Z = -(a + ib) = -a - ib$.
  • Fundamental Property: $Z + (-Z) = 0$.
• Multiplicative Inverse:
  • For a complex number $a + ib$, its multiplicative inverse $x + iy$ satisfies:     $(x + iy)(a + ib) = 1$
  • Derivation of components:     $x + iy = \frac{1}{a + ib} = \frac{1}{a + ib} \times \frac{a - ib}{a - ib} = \frac{a - ib}{a^2 + b^2} = \frac{a}{a^2 + b^2} - i\,\frac{b}{a^2 + b^2}$
  • Separating real and imaginary parts:     $x = \frac{a}{a^2 + b^2}$$y = -\frac{b}{a^2 + b^2}$
  • In coordinate form, the multiplicative inverse of $(a, b)$ is $\left(\frac{a}{a^2 + b^2}, -\frac{b}{a^2 + b^2}\right)$.

Conjugate of a Complex Number

• Definition:
  • Two complex numbers are conjugates of each other if their real parts are identical and their imaginary parts differ only in sign.
  • If $Z = a + bi$, its conjugate is denoted as $\bar{Z}$ and is defined as \bar{Z} = a - bi$.\n* Theorem 1 (Properties of Conjugates):\n * For any complex numbers Z_1$and$Z_2:\n * \overline{Z_1 + Z_2} = \bar{Z}_1 + \bar{Z}_2\n * \overline{Z_1 - Z_2} = \bar{Z}_1 - \bar{Z}_2\n * \overline{Z_1 Z_2} = \bar{Z}_1 \times \bar{Z}_2\n * \overline{\left(\frac{Z_1}{Z_2}\right)} = \frac{\bar{Z}_1}{\bar{Z}_2}\n* Example 11 (Finding Complex Conjugates):\n * Find the conjugate of 2i(-3 + 8i):\n * Expansion Method:\n      2i(-3 + 8i) = -6i + 16i^2 = -16 - 6i\n      \text{Conjugate } = -16 + 6i\n * Theorem Method:\n      \overline{2i(-3 + 8i)} = \overline{2i} \times \overline{(-3 + 8i)} = (-2i)(-3 - 8i) = 6i + 16i^2 = -16 + 6i\n\n# Argand Plane (Complex Plane)\n\n* Origin and Naming:\n * Named after the French mathematician Jean-Robert Argand and the German mathematician Carl Friedrich Gauss.\n * Also referred to as the Gauss Plane or Complex Plane.\n* Geometrical Structure:\n * Represents complex numbers visually on a two-dimensional Cartesian plane (x-y plane).\n * Real Axis (x$--axis): Horizontal axis representing real parts$a.\n * Imaginary Axis (y$--axis): Vertical axis representing imaginary parts$b$.
• Point Mapping:
  • Every complex number $Z = a + ib$ corresponds uniquely to an ordered pair $P(a, b)$ in the plane.

Modulus of a Complex Number

• Definition:
  • The modulus (or absolute value) of $Z = a + ib$ represents the distance from the origin to the point $(a, b)$ in the Argand plane.
  • Denoted by $r$, $|Z|$, or $|a + ib|$.
  • Formula: $r = |Z| = |a + ib| = \sqrt{a^2 + b^2}$.
• Theorem 2 (Properties of Modulus):
  • For any complex numbers $Z_1$ and $Z_2$:
  • $|Z_1 \times Z_2| = |Z_1| \times |Z_2|$
  • $|Z_1 \div Z_2| = |Z_1| \div |Z_2|$
  • $|Z_1 + Z_2| \le |Z_1| + |Z_2|$ (Triangle Inequality)
  • $|Z_1 - Z_2| \ge |Z_1| - |Z_2|$

Polar Form of Complex Numbers

• Trigonometric/Polar Definition:
  • A complex number $Z = a + ib$ in polar form is expressed as:     $Z = r(\cos(\theta) + i\sin(\theta)) \quad \text{or} \quad Z = r\,\text{Cis}(\theta)$
  • $r$ = modulus or absolute value of $Z$.
  • $\theta$ = argument or amplitude of $Z$, denoted as $\theta = \arg(Z)$, representing the directed angle from the positive real axis.
• Polar Multiplication and Division Laws:
  • Given $Z_1 = r_1(\cos(\theta_1) + i\sin(\theta_1))$ and $Z_2 = r_2(\cos(\theta_2) + i\sin(\theta_2))$:
  • Product: $Z_1 Z_2 = r_1 r_2 (\cos(\theta_1 + \theta_2) + i\sin(\theta_1 + \theta_2))$
  • Quotient: $\frac{Z_1}{Z_2} = \frac{r_1}{r_2} (\cos(\theta_1 - \theta_2) + i\sin(\theta_1 - \theta_2))$
• Negative Power in Polar Form:
  • $Z^{-n} = \frac{1}{r^n} \left(\frac{1}{\cos(n\theta) + i\sin(n\theta)}\right)$
• Example 12 (Conversion to Polar Form):
  • Express $1 + i\sqrt{3}$ in polar form:
  • $a = 1, b = \sqrt{3}$
  • Modulus $r = \sqrt{1^2 + (\sqrt{3})^2} = \sqrt{1 + 3} = 2$
  • Argument $\tan(\theta) = \frac{b}{a} = \sqrt{3} \implies \theta = 60^\circ$
  • Polar form: $1 + i\sqrt{3} = 2(\cos(60^\circ) + i\sin(60^\circ)) = 2\,\text{Cis}(60^\circ)$
• Example 13 (Conversion to Rectangular Form):
  • Express $4(\cos(225^\circ) + i\sin(225^\circ))$ in rectangular form $a + bi$:
  • $\cos(225^\circ) = -\frac{1}{\sqrt{2}}$
  • $\sin(225^\circ) = -\frac{1}{\sqrt{2}}$
  • Expansion: $4\left(-\frac{1}{\sqrt{2}} - i\,\frac{1}{\sqrt{2}}\right) = -2\sqrt{2} - 2i\sqrt{2}$
• Example 14 (Modulus and Argument Evaluation):
  • Find the modulus and argument of $(4 + 7i)(3 - 2i)$:
  • Expansion: $(4 + 7i)(3 - 2i) = 12 - 8i + 21i - 14i^2 = 26 + 13i$
  • Modulus $r = \sqrt{26^2 + 13^2} = \sqrt{676 + 169} = \sqrt{845} \approx 29.1$
  • Argument $\tan(\theta) = \frac{13}{26} = \frac{1}{2} \implies \theta = 26.57^\circ$

De Moivre's Theorem and Roots of Unity

• De Moivre's Theorem Statement:
  • If $Z = r(\cos(\theta) + i\sin(\theta))$ is a complex number and $n$ is a positive integer, then:     $Z^n = [r(\cos(\theta) + i\sin(\theta))]^n = r^n (\cos(n\theta) + i\sin(n\theta))$
  • Facilitates computation of powers, $n$th roots, and roots of unity.
• Roots of Unity:
  • Solutions to the algebraic equation $Z^n = 1$ for positive integer $n$.
  • Polar representation of equation: $Z^n = \cos(0 + 2k\pi) + i\sin(0 + 2k\pi) = e^{i2k\pi}$, where $k = 0, 1, 2, \dots$
  • General formula for $n$th roots of unity using De Moivre's Theorem:     $z = \cos\left(\frac{2k\pi}{n}\right) + i\sin\left(\frac{2k\pi}{n}\right) = e^{i\,\frac{2\pi k}{n}}, \quad k = 0, 1, 2, \dots, n - 1$
• Key Properties of $n$th Roots of Unity:
  1. All $n$ roots form a Geometric Progression (GP).
  2. The sum of the $n$ roots of unity is equal to zero. (The sum of the $n$ roots of any complex number is also zero).
  3. The product of the $n$ roots of unity is equal to (-1)^{n-1}$.\n 4. All n$roots lie on the circumference of a unit circle ($\text{radius} = 1$) centered at the origin, dividing the circle into$n$equal parts to form a regular polygon of$n sides.\n 5. The n$th roots of unity are written as$1, w, w^2, \dots, w^{n-1}$, where$w = \text{Cis}\left(\frac{2\pi}{n}\right).\n* Example 15 (Cube Roots of Unity):\n * Calculate 1^{\frac{1}{3}}:\n * Equation: z^3 = 1 = \cos(0 + 2k\pi) + i\sin(0 + 2k\pi) = e^{i2k\pi}\n * General root: z = \cos\left(\frac{2k\pi}{3}\right) + i\sin\left(\frac{2k\pi}{3}\right) = e^{i\,\frac{2k\pi}{3}}$for$k = 0, 1, 2\n * Evaluating roots:\n * For k = 0$:$z = \cos(0) + i\sin(0) = 1\n * For k = 1$:$z = \cos\left(\frac{2\pi}{3}\right) + i\sin\left(\frac{2\pi}{3}\right) = \cos\left(\pi - \frac{\pi}{3}\right) + i\sin\left(\pi - \frac{\pi}{3}\right) = -\cos\left(\frac{\pi}{3}\right) + i\sin\left(\frac{\pi}{3}\right) = -\frac{1}{2} + i\,\frac{\sqrt{3}}{2}\n * For k = 2$:$z = \cos\left(\frac{4\pi}{3}\right) + i\sin\left(\frac{4\pi}{3}\right) = \cos\left(\pi + \frac{\pi}{3}\right) + i\sin\left(\pi + \frac{\pi}{3}\right) = -\cos\left(\frac{\pi}{3}\right) - i\sin\left(\frac{\pi}{3}\right) = -\frac{1}{2} - i\,\frac{\sqrt{3}}{2}\n * The cube roots of unity are 1$,$-\frac{1}{2} + i\,\frac{\sqrt{3}}{2}$, and$-\frac{1}{2} - i\,\frac{\sqrt{3}}{2}.\n\n# Worked Examples using De Moivre's Theorem\n\n* Example 16 (Higher Powers of Complex Numbers):\n * Compute Z^6$given$Z = 2 + 2i:\n * Conversion to polar form:\n      r = \sqrt{2^2 + 2^2} = \sqrt{8} = 2\sqrt{2}\n      \theta = \arctan\left(\frac{2}{2}\right) = \arctan(1) = 45^\circ\n      Z = 2\sqrt{2}(\cos(45^\circ) + i\sin(45^\circ))\n * Applying De Moivre's Theorem:\n      Z^6 = (2 + 2i)^6 = (2\sqrt{2})^6 [\cos(6 \times 45^\circ) + i\sin(6 \times 45^\circ)]\n      Z^6 = 512 [\cos(270^\circ) + i\sin(270^\circ)] = 512 (0 + i(-1)) = -512i\n* Example 17 (Evaluating Complex Powers and Expressions):\n * (i) Compute (1 + i\sqrt{3})^3:\n * 1 + i\sqrt{3} = 2\left(\cos\left(\frac{\pi}{3}\right) + i\sin\left(\frac{\pi}{3}\right)\right)\n * (1 + i\sqrt{3})^3 = 2^3 \left[\cos\left(3 \times \frac{\pi}{3}\right) + i\sin\left(3 \times \frac{\pi}{3}\right)\right] = 8[\cos(\pi) + i\sin(\pi)] = 8(-1 + 0) = -8\n * (ii) Compute (1 - i)^8:\n * 1 - i = \sqrt{2}\left(\frac{1}{\sqrt{2}} - i\,\frac{1}{\sqrt{2}}\right) = \sqrt{2}\left(\cos\left(\frac{\pi}{4}\right) - i\sin\left(\frac{\pi}{4}\right)\right)\n * (1 - i)^8 = (\sqrt{2})^8 \left[\cos\left(8 \times \frac{\pi}{4}\right) - i\sin\left(8 \times \frac{\pi}{4}\right)\right] = 16[\cos(2\pi) - i\sin(2\pi)] = 16(1 - 0) = 16\n * (iii) Compute \left(\frac{\sqrt{3}}{2} + \frac{i}{2}\right)^5 - \left(\frac{\sqrt{3}}{2} - \frac{i}{2}\right)^5:\n * Express terms in polar form:\n      \frac{\sqrt{3}}{2} + \frac{i}{2} = \cos\left(\frac{\pi}{6}\right) + i\sin\left(\frac{\pi}{6}\right)\n      \frac{\sqrt{3}}{2} - \frac{i}{2} = \cos\left(\frac{\pi}{6}\right) - i\sin\left(\frac{\pi}{6}\right)\n * Apply De Moivre's Theorem:\n      \left(\frac{\sqrt{3}}{2} + \frac{i}{2}\right)^5 = \cos\left(\frac{5\pi}{6}\right) + i\sin\left(\frac{5\pi}{6}\right)\n      \left(\frac{\sqrt{3}}{2} - \frac{i}{2}\right)^5 = \cos\left(\frac{5\pi}{6}\right) - i\sin\left(\frac{5\pi}{6}\right)\n * Subtract equations:\n      \left[\cos\left(\frac{5\pi}{6}\right) + i\sin\left(\frac{5\pi}{6}\right)\right] - \left[\cos\left(\frac{5\pi}{6}\right) - i\sin\left(\frac{5\pi}{6}\right)\right] = 2i\sin\left(\frac{5\pi}{6}\right)\n      2i\sin\left(\frac{5\pi}{6}\right) = 2i\left(\frac{1}{2}\right) = i\n\n# Comprehensive Practice Exercises\n\n* Problem Set 1 (Modulus Calculation):\n * Find the magnitude (modulus) of:\n * (a) (-2, -1)\n * (b) \frac{1 + 2i}{2 - i}\n * (c) \frac{(3 - 5i)(1 + i)}{4 + 2i}\n* Problem Set 2 (Polar Representation):\n * Express the following complex numbers in polar form:\n * (a) 2 + 2\sqrt{3}i\n * (b) \frac{1 + 2i}{1 - 3i}\n * (c) \left(\frac{2 + i}{3 - i}\right)^2\n* Problem Set 3 (Standard Form Conversion):\n * Write each complex number in standard form a + bi:\n * (a) 4\,\text{Cis}(230^\circ)\n * (b) 2\,\text{Cis}(-30^\circ)\n * (c) 12\,\text{Cis}(420^\circ)\n* Problem Set 4 (Modulus and Principal Argument):\n * Find the magnitude and principal argument of:\n * (a) 5 - 7i\n * (b) (5 - 7i)(8 + 5i)\n * (c) \frac{1 + i}{1 - i}\n* Problem Set 5 (Determining Complex Number Z):\n * Find Z given:\n * (a) |Z| = 8\sqrt{2}$,$\arg(Z) = \frac{\pi}{4}\n * (b) |Z| = 5$,$\arg(Z) = \frac{\pi}{2}\n * (c) |Z| = \frac{1}{3}$,$\arg(Z) = \frac{\pi}{3}\n* Problem Set 6 (Conjugate and Modulus):\n * Find the conjugate and modulus of:\n * (a) \frac{1 + i}{1 - i}\n * (b) \frac{1 + 2i}{2 - i}\n * (c) -\frac{2}{3} - \frac{4}{9}i\n* Problem Set 7 (Inverses):\n * Find the additive and multiplicative inverse of:\n * (a) (-3, 2)\n * (b) \frac{1 + 2i}{2 - i}\n * (c) (5 - 7i)(8 + 5i)\n* Problem Set 8 (Algebraic Proofs):\n * If Z = 2 + 3i$, prove that$Z\bar{Z} = 13.\n* Problem Set 9 (Trigonometric Form):\n * Write the following complex numbers in polar (trigonometric) form:\n * (a) -\sqrt{2} + \sqrt{6}i\n * (b) -1 + \sqrt{3}i\n* Problem Set 10 (Modulus Proofs):\n * Show that \left|\frac{1 + 2i}{2 - i}\right| = 1.\n* Problem Set 11 (Factorization):\n * Factorise each expression over complex numbers:\n * (a) 36a^2 + 100b^2\n * (b) 2x^2 + 5y^2\n * (c) 9a^2 + 64b^2\n* Problem Set 12 (Division and Simplification):\n * Perform the division and write the result in standard form a + bi:\n * (a) \frac{8 - 7i}{1 - 2i}\n * (b) \frac{1}{(4 - 5i)^2}\n * (c) \frac{2i}{2 + i} + \frac{5}{2 - i}\n * (d) \frac{(2 - 3i)(5i)}{2 + 3i}\n* Problem Set 13 (Imaginary Unit Simplification):\n * Simplify and write in standard form:\n * (a) 4i^2 - 7i^7\n * (b) (\sqrt{-2})^8\n * (c) -5i^9\n* Problem Set 14 (De Moivre's Powers):\n * Use De Moivre's Theorem to evaluate powers in standard form:\n * (a) (i + \sqrt{3})^3\n * (b) (-1 + i)^{10}\n * (c) \left[\cos\left(\frac{5\pi}{4}\right) + i\sin\left(\frac{5\pi}{4}\right)\right]^{10}\n * (d) [5(\cos(20^\circ) + i\sin(20^\circ))]^3\n* Problem Set 15 (Root Extraction):\n * Find:\n * (a) The fifth roots of 16\left(\cos\left(\frac{4\pi}{3}\right) + i\sin\left(\frac{4\pi}{3}\right)\right)\n * (b) The fourth roots of 1\n* Problem Set 16 (Polynomial Equations):\n * Find all solutions of the equations:\n * (a) x^4 - i = 0\n * (b) x^5 - 243 = 0$$