# 2.2 Complex numbers ## Detailed Theory: Complex Numbers ### **1. Introduction to Complex Numbers** #### **1.1 The Need for Complex Numbers** The equation $$x^2 + 1 = 0$$ has no real solution because there is no real number whose square is negative. This limitation of real numbers led to the invention of complex numbers. #### **1.2 Definition of Imaginary Unit** The imaginary unit $$i$$ is defined as: $$i = \sqrt{-1}$$ From this definition, we get: $$i^2 = -1$$ #### **1.3 Powers of** $$i$$ The powers of $$i$$ follow a cyclic pattern with period 4: $$i^1 = i$$ $$i^2 = -1$$ $$i^3 = i^2 \cdot i = -i$$ $$i^4 = i^2 \cdot i^2 = (-1)(-1) = 1$$ For any integer $$n$$, we can compute $$i^n$$ by dividing $$n$$ by 4 and using the remainder: $$i^n = i^{4k + r} = i^r$$ where $$r = 0, 1, 2, 3$$ **Examples:** $$i^{17} = i^{4\cdot4 + 1} = i^1 = i$$ $$i^{-3} = \frac{1}{i^3} = \frac{1}{-i} = \frac{i}{-i^2} = \frac{i}{-(-1)} = i$$ *** ### **2. Definition and Representation of Complex Numbers** #### **2.1 Standard Form** A complex number $$z$$ is expressed in the standard form as: $$z = a + bi$$ where $$a$$ and $$b$$ are real numbers, and $$i$$ is the imaginary unit. **Components:** $$a$$ is called the **real part** of $$z$$, denoted $$\text{Re}(z) = a$$ $$b$$ is called the **imaginary part** of $$z$$, denoted $$\text{Im}(z) = b$$ #### **2.2 Set Notation** The set of all complex numbers is denoted by: $$\mathbb{C} = {a + bi : a, b \in \mathbb{R}}$$ #### **2.3 Equality of Complex Numbers** Two complex numbers $$z\_1 = a + bi$$ and $$z\_2 = c + di$$ are equal if and only if: $$a = c$$ and $$b = d$$ That is, their real parts are equal AND their imaginary parts are equal. #### **2.4 Special Types of Complex Numbers** **a) Purely Real Number** If $$b = 0$$, then $$z = a$$ is a purely real number. **Example:** $$z = 5$$, which is $$5 + 0i$$ **b) Purely Imaginary Number** If $$a = 0$$, then $$z = bi$$ is a purely imaginary number. **Example:** $$z = 3i$$, which is $$0 + 3i$$ **c) Zero Complex Number** If $$a = 0$$ and $$b = 0$$, then $$z = 0$$ is the zero complex number. **d) Complex Conjugate** For $$z = a + bi$$, its complex conjugate is: $$\overline{z} = a - bi$$ **Properties:** $$\overline{\overline{z}} = z$$ If $$z$$ is purely real, then $$\overline{z} = z$$ If $$z$$ is purely imaginary, then $$\overline{z} = -z$$ #### **2.5 Modulus (Absolute Value)** For $$z = a + bi$$, the modulus of $$z$$ is: $$|z| = \sqrt{a^2 + b^2}$$ The modulus is always a non-negative real number. **Geometric interpretation:** $$|z|$$ represents the distance from the origin to the point $$(a, b)$$ in the complex plane. *** ### **3. Geometric Representation (Argand Plane)** #### **3.1 The Complex Plane** The complex plane (Argand plane) is a Cartesian plane where: The horizontal axis is the **real axis** The vertical axis is the **imaginary axis** The complex number $$z = a + bi$$ is represented by the point $$(a, b)$$ #### **3.2 Polar Form of Complex Numbers** Instead of using rectangular coordinates $$(a, b)$$, we can represent a complex number using polar coordinates: $$z = r(\cos \theta + i \sin \theta)$$ where: $$r = |z| = \sqrt{a^2 + b^2}$$ (modulus) $$\theta = \arg(z) = \tan^{-1}\left(\frac{b}{a}\right)$$ (argument) **Note:** The argument $$\theta$$ is determined up to multiples of $$2\pi$$. The **principal argument** is usually taken in the interval $$(-\pi, \pi]$$ or $$\[0, 2\pi)$$. #### **3.3 Euler's Formula** Euler's formula provides an elegant representation: $$e^{i\theta} = \cos \theta + i \sin \theta$$ Using this, the polar form becomes: $$z = re^{i\theta}$$ This is called the **exponential form** of a complex number. *** ### **4. Operations on Complex Numbers** #### **4.1 Addition** For $$z\_1 = a + bi$$ and $$z\_2 = c + di$$: $$z\_1 + z\_2 = (a + c) + (b + d)i$$ **Geometric interpretation:** Vector addition in the complex plane. **Properties:** Commutative: $$z\_1 + z\_2 = z\_2 + z\_1$$ Associative: $$(z\_1 + z\_2) + z\_3 = z\_1 + (z\_2 + z\_3)$$ Additive identity: $$z + 0 = z$$ Additive inverse: For $$z = a + bi$$, $$-z = -a - bi$$ #### **4.2 Subtraction** For $$z\_1 = a + bi$$ and $$z\_2 = c + di$$: $$z\_1 - z\_2 = (a - c) + (b - d)i$$ #### **4.3 Multiplication** **a) Using Standard Form** For $$z\_1 = a + bi$$ and $$z\_2 = c + di$$: $$z\_1 \cdot z\_2 = (a + bi)(c + di)$$ $$= ac + adi + bci + bdi^2$$ $$= ac + (ad + bc)i + bd(-1)$$ $$= (ac - bd) + (ad + bc)i$$ **b) Using Polar Form** For $$z\_1 = r\_1(\cos \theta\_1 + i \sin \theta\_1)$$ and $$z\_2 = r\_2(\cos \theta\_2 + i \sin \theta\_2)$$: $$z\_1 \cdot z\_2 = r\_1 r\_2\[\cos(\theta\_1 + \theta\_2) + i \sin(\theta\_1 + \theta\_2)]$$ Using exponential form: $$z\_1 \cdot z\_2 = r\_1 e^{i\theta\_1} \cdot r\_2 e^{i\theta\_2} = r\_1 r\_2 e^{i(\theta\_1 + \theta\_2)}$$ **Geometric interpretation:** Multiply the moduli, add the arguments. **Properties:** Commutative: $$z\_1 \cdot z\_2 = z\_2 \cdot z\_1$$ Associative: $$(z\_1 \cdot z\_2) \cdot z\_3 = z\_1 \cdot (z\_2 \cdot z\_3)$$ Multiplicative identity: $$z \cdot 1 = z$$ Distributive over addition: $$z\_1(z\_2 + z\_3) = z\_1 z\_2 + z\_1 z\_3$$ #### **4.4 Division** **a) Using Standard Form** For $$z\_1 = a + bi$$ and $$z\_2 = c + di \neq 0$$: $$\frac{z\_1}{z\_2} = \frac{a + bi}{c + di}$$ Multiply numerator and denominator by the conjugate of the denominator: $$= \frac{(a + bi)(c - di)}{(c + di)(c - di)}$$ $$= \frac{(ac + bd) + (bc - ad)i}{c^2 + d^2}$$ $$= \frac{ac + bd}{c^2 + d^2} + \frac{bc - ad}{c^2 + d^2} i$$ **b) Using Polar Form** For $$z\_1 = r\_1(\cos \theta\_1 + i \sin \theta\_1)$$ and $$z\_2 = r\_2(\cos \theta\_2 + i \sin \theta\_2) \neq 0$$: $$\frac{z\_1}{z\_2} = \frac{r\_1}{r\_2}\[\cos(\theta\_1 - \theta\_2) + i \sin(\theta\_1 - \theta\_2)]$$ Using exponential form: $$\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)}$$ **Geometric interpretation:** Divide the moduli, subtract the arguments. #### **4.5 Properties of Conjugates** For complex numbers $$z$$, $$z\_1$$, and $$z\_2$$: 1. $$\overline{z\_1 + z\_2} = \overline{z\_1} + \overline{z\_2}$$ 2. $$\overline{z\_1 - z\_2} = \overline{z\_1} - \overline{z\_2}$$ 3. $$\overline{z\_1 \cdot z\_2} = \overline{z\_1} \cdot \overline{z\_2}$$ 4. $$\overline{\left(\frac{z\_1}{z\_2}\right)} = \frac{\overline{z\_1}}{\overline{z\_2}}$$ for $$z\_2 \neq 0$$ 5. $$z + \overline{z} = 2\text{Re}(z)$$ 6. $$z - \overline{z} = 2i\text{Im}(z)$$ 7. $$z \cdot \overline{z} = |z|^2$$ 8. $$\overline{\overline{z}} = z$$ #### **4.6 Properties of Modulus** For complex numbers $$z$$, $$z\_1$$, and $$z\_2$$: 1. $$|z| \geq 0$$ and $$|z| = 0$$ if and only if $$z = 0$$ 2. $$|z| = |\overline{z}|$$ 3. $$|z\_1 \cdot z\_2| = |z\_1| \cdot |z\_2|$$ 4. $$\left|\frac{z\_1}{z\_2}\right| = \frac{|z\_1|}{|z\_2|}$$ for $$z\_2 \neq 0$$ 5. $$|z\_1 + z\_2| \leq |z\_1| + |z\_2|$$ (Triangle Inequality) 6. $$||z\_1| - |z\_2|| \leq |z\_1 - z\_2|$$ 7. $$|z\_1 + z\_2|^2 + |z\_1 - z\_2|^2 = 2(|z\_1|^2 + |z\_2|^2)$$ (Parallelogram Law) *** ### **5. De Moivre's Theorem and Applications** #### **5.1 De Moivre's Theorem** For any integer $$n$$ and any real number $$\theta$$: $$(\cos \theta + i \sin \theta)^n = \cos(n\theta) + i \sin(n\theta)$$ Using exponential form: $$(e^{i\theta})^n = e^{in\theta}$$ #### **5.2 Finding** $$n$$**-th Powers** To compute $$(a + bi)^n$$: **Step 1:** Convert to polar form: $$a + bi = r(\cos \theta + i \sin \theta)$$ **Step 2:** Apply De Moivre's theorem: $$(a + bi)^n = r^n\[\cos(n\theta) + i \sin(n\theta)]$$ **Step 3:** Convert back to standard form if needed. **Example:** Find $$(1 + i)^8$$ **Step 1:** Convert to polar form: $$r = \sqrt{1^2 + 1^2} = \sqrt{2}$$ $$\theta = \frac{\pi}{4}$$ (since $$a = 1$$, $$b = 1$$) So $$1 + i = \sqrt{2}(\cos \frac{\pi}{4} + i \sin \frac{\pi}{4})$$ **Step 2:** Apply De Moivre: $$(1 + i)^8 = (\sqrt{2})^8\[\cos(8 \cdot \frac{\pi}{4}) + i \sin(8 \cdot \frac{\pi}{4})]$$ $$= 2^4\[\cos(2\pi) + i \sin(2\pi)]$$ $$= 16\[1 + 0i] = 16$$ #### **5.3 Finding** $$n$$**-th Roots** The $$n$$-th roots of a complex number $$z = r(\cos \theta + i \sin \theta)$$ are given by: $$z^{1/n} = r^{1/n}\left\[\cos\left(\frac{\theta + 2k\pi}{n}\right) + i \sin\left(\frac{\theta + 2k\pi}{n}\right)\right]$$ for $$k = 0, 1, 2, \ldots, n-1$$ **Properties:** 1. There are exactly $$n$$ distinct $$n$$-th roots of any non-zero complex number. 2. In the complex plane, the $$n$$-th roots are equally spaced on a circle of radius $$r^{1/n}$$. **Example:** Find the cube roots of $$8i$$ **Step 1:** Convert $$8i$$ to polar form: $$8i = 8(\cos \frac{\pi}{2} + i \sin \frac{\pi}{2})$$ since $$8i = 0 + 8i$$ **Step 2:** Apply formula with $$n = 3$$: For $$k = 0$$: $$z\_0 = 8^{1/3}\[\cos(\frac{\pi/2}{3}) + i \sin(\frac{\pi/2}{3})] = 2(\cos \frac{\pi}{6} + i \sin \frac{\pi}{6})$$ $$= 2\left(\frac{\sqrt{3}}{2} + \frac{1}{2}i\right) = \sqrt{3} + i$$ For $$k = 1$$: $$z\_1 = 2\[\cos(\frac{\pi/2 + 2\pi}{3}) + i \sin(\frac{\pi/2 + 2\pi}{3})] = 2(\cos \frac{5\pi}{6} + i \sin \frac{5\pi}{6})$$ $$= 2\left(-\frac{\sqrt{3}}{2} + \frac{1}{2}i\right) = -\sqrt{3} + i$$ For $$k = 2$$: $$z\_2 = 2\[\cos(\frac{\pi/2 + 4\pi}{3}) + i \sin(\frac{\pi/2 + 4\pi}{3})] = 2(\cos \frac{3\pi}{2} + i \sin \frac{3\pi}{2})$$ $$= 2(0 - i) = -2i$$ The cube roots are: $$\sqrt{3} + i$$, $$-\sqrt{3} + i$$, and $$-2i$$ #### **5.4 Applications of De Moivre's Theorem** **a) Expressing** $$\cos(n\theta)$$ **and** $$\sin(n\theta)$$ Using the binomial expansion and De Moivre's theorem: $$\cos(n\theta) = \text{Re}\[(\cos \theta + i \sin \theta)^n]$$ $$\sin(n\theta) = \text{Im}\[(\cos \theta + i \sin \theta)^n]$$ **Example:** Express $$\cos(3\theta)$$ in terms of $$\cos \theta$$ $$(\cos \theta + i \sin \theta)^3 = \cos^3 \theta + 3i\cos^2 \theta \sin \theta - 3\cos \theta \sin^2 \theta - i\sin^3 \theta$$ By De Moivre: $$(\cos \theta + i \sin \theta)^3 = \cos(3\theta) + i \sin(3\theta)$$ Equating real parts: $$\cos(3\theta) = \cos^3 \theta - 3\cos \theta \sin^2 \theta$$ Using $$\sin^2 \theta = 1 - \cos^2 \theta$$: $$\cos(3\theta) = \cos^3 \theta - 3\cos \theta (1 - \cos^2 \theta) = 4\cos^3 \theta - 3\cos \theta$$ **b) Summing Series** De Moivre's theorem can be used to sum trigonometric series using geometric series formula. **Example:** Sum $$S = \cos \theta + \cos 2\theta + \cdots + \cos n\theta$$ Consider $$C = \cos \theta + \cos 2\theta + \cdots + \cos n\theta$$ and $$S = \sin \theta + \sin 2\theta + \cdots + \sin n\theta$$ Then $$C + iS = e^{i\theta} + e^{i2\theta} + \cdots + e^{in\theta}$$ This is a geometric series with first term $$e^{i\theta}$$, ratio $$e^{i\theta}$$, and $$n$$ terms. *** ### **6. Complex Numbers in Quadratic Equations** #### **6.1 Solving Quadratic Equations with Real Coefficients** For a quadratic equation with real coefficients: $$ax^2 + bx + c = 0$$ with $$a \neq 0$$ The discriminant is: $$\Delta = b^2 - 4ac$$ **Cases:** 1. If $$\Delta > 0$$: Two distinct real roots 2. If $$\Delta = 0$$: One real root (double root) 3. If $$\Delta < 0$$: Two complex conjugate roots #### **6.2 Complex Roots Formula** When $$\Delta < 0$$, we can write $$\Delta = -D$$ where $$D > 0$$ Then the roots are: $$x = \frac{-b \pm i\sqrt{D}}{2a}$$ These are complex conjugates of each other. **Example:** Solve $$x^2 - 2x + 5 = 0$$ Here $$a=1$$, $$b=-2$$, $$c=5$$ $$\Delta = (-2)^2 - 4(1)(5) = 4 - 20 = -16$$ Since $$\Delta < 0$$, roots are complex: $$x = \frac{2 \pm i\sqrt{16}}{2} = \frac{2 \pm 4i}{2} = 1 \pm 2i$$ Roots: $$1 + 2i$$ and $$1 - 2i$$ (complex conjugates) #### **6.3 Sum and Product of Roots** For $$ax^2 + bx + c = 0$$ with roots $$\alpha$$ and $$\beta$$: Sum of roots: $$\alpha + \beta = -\frac{b}{a}$$ Product of roots: $$\alpha\beta = \frac{c}{a}$$ This holds even when roots are complex. **Verification of previous example:** Sum: $$(1+2i) + (1-2i) = 2 = -\frac{-2}{1} \quad \checkmark$$ Product: $$(1+2i)(1-2i) = 1 - 4i^2 = 1 + 4 = 5 = \frac{5}{1} \quad \checkmark$$ #### **6.4 Quadratic with Given Complex Roots** To form a quadratic equation with real coefficients having roots $$\alpha$$ and $$\beta$$: If roots are complex conjugates, use: $$x^2 - (\alpha + \beta)x + \alpha\beta = 0$$ **Example:** Form quadratic with roots $$2+3i$$ and $$2-3i$$ Sum: $$(2+3i) + (2-3i) = 4$$ Product: $$(2+3i)(2-3i) = 4 - 9i^2 = 4 + 9 = 13$$ Equation: $$x^2 - 4x + 13 = 0$$ *** ### **7. Locus Problems in Complex Plane** #### **7.1 Basic Locus Concepts** A locus in the complex plane is a set of points satisfying certain conditions. **Common conditions:** 1. $$|z - z\_0| = r$$: Circle with center $$z\_0$$ and radius $$r$$ 2. $$|z - z\_1| = |z - z\_2|$$: Perpendicular bisector of segment joining $$z\_1$$ and $$z\_2$$ 3. $$\text{Re}(z) = c$$: Vertical line through $$x = c$$ 4. $$\text{Im}(z) = c$$: Horizontal line through $$y = c$$ 5. $$\arg(z - z\_0) = \theta$$: Ray from $$z\_0$$ making angle $$\theta$$ with positive real axis #### **7.2 Circle Loci** **a) Standard Circle** $$|z - z\_0| = r$$ represents a circle with center at $$z\_0$$ and radius $$r$$. If $$z\_0 = a + bi$$ and $$z = x + yi$$, then: $$|(x + yi) - (a + bi)| = r$$ $$|(x-a) + (y-b)i| = r$$ $$(x-a)^2 + (y-b)^2 = r^2$$ **b) Circle in Different Forms** 1. $$|z| = r$$: Circle centered at origin with radius $$r$$ 2. $$|z - i| = 2$$: Circle centered at $$i$$ with radius $$2$$ 3. $$|z - 1| = |z + i|$$: Set of points equidistant from $$1$$ and $$-i$$, which is the perpendicular bisector #### **7.3 Line Loci** **a) Perpendicular Bisector** $$|z - z\_1| = |z - z\_2|$$ represents the perpendicular bisector of the segment joining $$z\_1$$ and $$z\_2$$. **Example:** $$|z - 1| = |z + i|$$ Let $$z = x + yi$$: $$|(x-1) + yi| = |x + (y+1)i|$$ $$\sqrt{(x-1)^2 + y^2} = \sqrt{x^2 + (y+1)^2}$$ Squaring both sides: $$(x-1)^2 + y^2 = x^2 + (y+1)^2$$ $$x^2 - 2x + 1 + y^2 = x^2 + y^2 + 2y + 1$$ $$-2x = 2y$$ $$y = -x$$ This is a straight line through origin with slope $$-1$$. **b) Lines Parallel to Axes** $$\text{Re}(z) = k$$: Vertical line $$x = k$$ $$\text{Im}(z) = k$$: Horizontal line $$y = k$$ **c) Lines at an Angle** $$\arg(z - z\_0) = \theta$$: Ray starting at $$z\_0$$ and making angle $$\theta$$ with positive real axis. **Example:** $$\arg(z - 1) = \frac{\pi}{4}$$ represents the ray starting at $$1$$ and making $$45^\circ$$ angle with positive real axis. #### **7.4 Ellipse and Hyperbola Loci** **a) Ellipse** $$|z - z\_1| + |z - z\_2| = 2a$$ (with $$2a > |z\_1 - z\_2|$$) represents an ellipse with foci at $$z\_1$$ and $$z\_2$$, and major axis length $$2a$$. **b) Hyperbola** $$||z - z\_1| - |z - z\_2|| = 2a$$ (with $$2a < |z\_1 - z\_2|$$) represents a hyperbola with foci at $$z\_1$$ and $$z\_2$$. *** ### **8. Applications and Solved Examples** #### **Example 1:** Simplify Complex Expression Simplify: $$\frac{(1+i)^3}{(1-i)^2}$$ **Solution:** **Step 1:** Compute $$(1+i)^3$$: $$1+i$$ in polar form: $$\sqrt{2}(\cos \frac{\pi}{4} + i \sin \frac{\pi}{4})$$ By De Moivre: $$(1+i)^3 = (\sqrt{2})^3(\cos \frac{3\pi}{4} + i \sin \frac{3\pi}{4}) = 2\sqrt{2}(-\frac{1}{\sqrt{2}} + \frac{1}{\sqrt{2}}i)$$ $$= -2 + 2i$$ **Step 2:** Compute $$(1-i)^2$$: $$1-i = \sqrt{2}(\cos(-\frac{\pi}{4}) + i \sin(-\frac{\pi}{4}))$$ $$(1-i)^2 = 2(\cos(-\frac{\pi}{2}) + i \sin(-\frac{\pi}{2})) = 2(0 - i) = -2i$$ **Step 3:** Divide: $$\frac{-2 + 2i}{-2i} = \frac{-2(1 - i)}{-2i} = \frac{1 - i}{i}$$ Multiply numerator and denominator by $$-i$$: $$= \frac{(1-i)(-i)}{i(-i)} = \frac{-i + i^2}{-i^2} = \frac{-i - 1}{1} = -1 - i$$ #### **Example 2:** Solve Complex Equation Solve: $$z^2 + 2z + 5 = 0$$ **Solution:** Using quadratic formula: $$z = \frac{-2 \pm \sqrt{4 - 20}}{2} = \frac{-2 \pm \sqrt{-16}}{2} = \frac{-2 \pm 4i}{2} = -1 \pm 2i$$ Roots: $$-1 + 2i$$ and $$-1 - 2i$$ #### **Example 3:** Find Locus Find locus of $$z$$ satisfying $$|z-2| = 3|z+1|$$ **Solution:** Let $$z = x + yi$$: $$|(x-2) + yi| = 3|(x+1) + yi|$$ $$\sqrt{(x-2)^2 + y^2} = 3\sqrt{(x+1)^2 + y^2}$$ Square both sides: $$(x-2)^2 + y^2 = 9\[(x+1)^2 + y^2]$$ $$x^2 - 4x + 4 + y^2 = 9(x^2 + 2x + 1 + y^2)$$ $$x^2 - 4x + 4 + y^2 = 9x^2 + 18x + 9 + 9y^2$$ $$0 = 8x^2 + 22x + 5 + 8y^2$$ Divide by 8: $$x^2 + \frac{11}{4}x + \frac{5}{8} + y^2 = 0$$ Complete the square for $$x$$: $$(x^2 + \frac{11}{4}x + \frac{121}{64}) + y^2 = -\frac{5}{8} + \frac{121}{64}$$ $$(x + \frac{11}{8})^2 + y^2 = \frac{-40 + 121}{64} = \frac{81}{64}$$ $$(x + \frac{11}{8})^2 + y^2 = (\frac{9}{8})^2$$ This is a circle with center $$(-\frac{11}{8}, 0)$$ and radius $$\frac{9}{8}$$ #### **Example 4:** Find Modulus and Argument For $$z = 1 - i\sqrt{3}$$, find $$|z|$$ and $$\arg(z)$$ **Solution:** $$|z| = \sqrt{1^2 + (-\sqrt{3})^2} = \sqrt{1 + 3} = 2$$ For argument: $$\tan \theta = \frac{-\sqrt{3}}{1} = -\sqrt{3}$$ Since point is in 4th quadrant (positive real, negative imaginary): $$\theta = -\frac{\pi}{3}$$ or $$\frac{5\pi}{3}$$ Principal argument usually taken in $$(-\pi, \pi]$$, so $$\arg(z) = -\frac{\pi}{3}$$ #### **Example 5:** Find Cube Roots of Unity Find the cube roots of 1. **Solution:** Write 1 in polar form: $$1 = \cos 0 + i \sin 0$$ Cube roots: $$1^{1/3} = \cos\left(\frac{0 + 2k\pi}{3}\right) + i \sin\left(\frac{0 + 2k\pi}{3}\right)$$ for $$k = 0, 1, 2$$ For $$k=0$$: $$z\_0 = \cos 0 + i \sin 0 = 1$$ For $$k=1$$: $$z\_1 = \cos\left(\frac{2\pi}{3}\right) + i \sin\left(\frac{2\pi}{3}\right) = -\frac{1}{2} + \frac{\sqrt{3}}{2}i$$ For $$k=2$$: $$z\_2 = \cos\left(\frac{4\pi}{3}\right) + i \sin\left(\frac{4\pi}{3}\right) = -\frac{1}{2} - \frac{\sqrt{3}}{2}i$$ The cube roots of unity are: $$1$$, $$-\frac{1}{2} + \frac{\sqrt{3}}{2}i$$, and $$-\frac{1}{2} - \frac{\sqrt{3}}{2}i$$ Note: $$\omega = -\frac{1}{2} + \frac{\sqrt{3}}{2}i$$ is often denoted as the primitive cube root of unity, and satisfies $$\omega^3 = 1$$ and $$1 + \omega + \omega^2 = 0$$ *** ### **9. Important Formulas and Theorems** #### **9.1 Basic Identities** 1. $$i^2 = -1$$ 2. $$i^3 = -i$$ 3. $$i^4 = 1$$ 4. $$\frac{1}{i} = -i$$ 5. $$(a+bi)(a-bi) = a^2 + b^2$$ #### **9.2 Polar Form Relationships** 1. $$r = |z| = \sqrt{a^2 + b^2}$$ 2. $$\theta = \arg(z) = \tan^{-1}\left(\frac{b}{a}\right)$$ (with quadrant consideration) 3. $$a = r\cos\theta$$ 4. $$b = r\sin\theta$$ 5. $$z = r(\cos\theta + i\sin\theta) = re^{i\theta}$$ #### **9.3 De Moivre's Theorem** For any integer $$n$$: $$(\cos\theta + i\sin\theta)^n = \cos(n\theta) + i\sin(n\theta)$$ #### **9.4** $$n$$**-th Roots Formula** For $$z = r(\cos\theta + i\sin\theta) \neq 0$$, the $$n$$-th roots are: $$z\_k = r^{1/n}\left\[\cos\left(\frac{\theta + 2k\pi}{n}\right) + i\sin\left(\frac{\theta + 2k\pi}{n}\right)\right]$$ for $$k = 0, 1, 2, \ldots, n-1$$ #### **9.5 Cube Roots of Unity** The cube roots of 1 are: $$1$$, $$\omega = -\frac{1}{2} + \frac{\sqrt{3}}{2}i$$, and $$\omega^2 = -\frac{1}{2} - \frac{\sqrt{3}}{2}i$$ Properties: $$\omega^3 = 1$$ $$1 + \omega + \omega^2 = 0$$ $$\omega^2 = \overline{\omega}$$ #### **9.6 Quadratic Formula with Complex Roots** For $$ax^2 + bx + c = 0$$ with real coefficients: If $$\Delta = b^2 - 4ac < 0$$, then roots are: $$x = \frac{-b \pm i\sqrt{-\Delta}}{2a}$$ #### **9.7 Triangle Inequality** For any complex numbers $$z\_1$$ and $$z\_2$$: $$|z\_1 + z\_2| \leq |z\_1| + |z\_2|$$ $$||z\_1| - |z\_2|| \leq |z\_1 - z\_2|$$ *** ### **10. Exam Tips and Common Pitfalls** #### **10.1 Common Mistakes to Avoid** 1. **Incorrect simplification of powers of** $$i$$: Remember the cyclic nature $$i, -1, -i, 1$$ 2. **Forgetting to rationalize denominators**: Always write complex numbers in standard form $$a+bi$$ 3. **Incorrect argument calculation**: Remember to consider the quadrant of the complex number 4. **Misapplying De Moivre's theorem**: It applies to polar/exponential form, not directly to standard form 5. **Ignoring complex conjugate pairs**: For polynomials with real coefficients, complex roots occur in conjugate pairs #### **10.2 Problem-Solving Strategies** 1. **For computations**: Convert to polar form if multiplication/division/powers are involved 2. **For equations**: Use algebraic manipulation, conjugate properties, or geometric interpretation 3. **For locus problems**: Substitute $$z = x + yi$$ and use coordinate geometry 4. **For inequalities**: Use modulus properties and triangle inequality 5. **For root-finding**: Use polar form and De Moivre's theorem for $$n$$-th roots #### **10.3 Verification Techniques** 1. **Check conjugates**: For real-coefficient polynomials, verify roots are conjugates 2. **Check modulus**: Use $$|z|^2 = z\overline{z}$$ to verify computations 3. **Check special values**: Test $$z=0$$, $$z=1$$, $$z=i$$ when applicable 4. **Check dimensions**: Ensure real and imaginary parts are correctly identified *** This comprehensive theory covers all aspects of complex numbers with detailed explanations and examples, providing complete preparation for the entrance examination.