2.2 Complex Numbers, Modulus, Conjugates & De Moivre's Theorem

Complex Numbers, Modulus, Conjugates & De Moivre's Theorem

In the Intermediate Mathematics syllabus and the UET ECAT entry test, complex numbers form the cornerstone of algebraic analysis. The ECAT Mathematics section allocates a substantial portion of its 30 questions to evaluating complex expressions, calculating moduli under tight time limits, and exploiting the symmetries of roots of unity. Mastering algebraic operations, trigonometric polar conversions, and De Moivre's identities allows candidates to solve problems in under 60 seconds without long-form arithmetic.

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1. The Complex Number System and Fundamental Operations

A complex number $z$ is defined as an ordered pair of real numbers $(a, b)$ or as an algebraic expression:

$z = a + bi$

where $a, b \in \mathbb{R}$, and $i = \sqrt{-1}$ is the imaginary unit satisfying $i^2 = -1$. The term $a = \text{Re}(z)$ represents the real part, and $b = \text{Im}(z)$ represents the imaginary part. If $a = 0$ with $b \neq 0$, $z$ is termed purely imaginary; if $b = 0$, $z$ is purely real.

Powers of the Imaginary Unit $i$

The cyclic behavior of $i^n$ repeats in cycles of 4:

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

For any integer $n$, write $n = 4k + r$ where $r \in \{0, 1, 2, 3\}$. Then:

$i^n = i^{4k+r} = (i^4)^k \cdot i^r = 1^k \cdot i^r = i^r$

ECAT Trap Alert: When simplifying negative powers of $i$, such as $i^{-35}$, express the exponent as $-35 = -36 + 1 = 4(-9) + 1$, yielding $i^{-35} = i^1 = i$. Alternatively, multiply numerator and denominator: $\frac{1}{i^{35}} = \frac{1}{i^{32} \cdot i^3} = \frac{1}{-i} = \frac{i}{-i^2} = i$.

Basic Algebraic Operations

For two complex numbers $z_1 = a_1 + b_1 i$ and $z_2 = a_2 + b_2 i$:

1. Equality: $z_1 = z_2 \iff a_1 = a_2$ and $b_1 = b_2$.
2. Addition & Subtraction: $z_1 \pm z_2 = (a_1 \pm a_2) + (b_1 \pm b_2)i$.
3. Multiplication: $z_1 z_2 = (a_1 a_2 - b_1 b_2) + (a_1 b_2 + a_2 b_1)i$.
4. Division: Computed by multiplying the numerator and denominator by the conjugate of the denominator:

$\frac{z_1}{z_2} = \frac{(a_1 + b_1 i)(a_2 - b_2 i)}{(a_2 + b_2 i)(a_2 - b_2 i)} = \frac{(a_1 a_2 + b_1 b_2) + (a_2 b_1 - a_1 b_2)i}{a_2^2 + b_2^2}$

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2. Complex Conjugates & Algebraic Properties

The complex conjugate of $z = a + bi$ is denoted by $\overline{z}$ and defined as:

$\overline{z} = a - bi$

Geometrically, $\overline{z}$ is the reflection of the point $(a, b)$ across the real axis (the x-axis) in the Argand plane.

Conjugate Properties Table

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3. Modulus, Distance, and Geometric Inequalities

The modulus (or absolute value) of $z = a + bi$ is the Euclidean distance from the origin $(0, 0)$ to $(a, b)$ in the complex plane:

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

Core Properties of Modulus

1. $|z| \ge 0$, and $|z| = 0 \iff z = 0$.
2. $|z| = |\overline{z}| = |-z| = |-\overline{z}|$.
3. $|z_1 z_2| = |z_1| |z_2|$ (multiplicative property extends to any finite product: $|z_1 z_2 \dots z_n| = |z_1||z_2|\dots|z_n|$).
4. $\left|\frac{z_1}{z_2}\right| = \frac{|z_1|}{|z_2|}$ for $z_2 \neq 0$.
5. $|z^n| = |z|^n$ for $n \in \mathbb{Z}^+$.

The Triangle Inequalities

For any two complex numbers $z_1, z_2 \in \mathbb{C}$:

$|z_1 + z_2| \le |z_1| + |z_2|$

Equality holds if and only if $z_1$ and $z_2$ lie on the same ray from the origin, meaning $\arg(z_1) = \arg(z_2)$ (or one of them is zero).

The lower bound (reverse triangle inequality) establishes:

$|z_1 - z_2| \ge \bigl| |z_1| - |z_2| \bigr|$

Combining these yields the double inequality defining the bounds of a sum:

$\bigl| |z_1| - |z_2| \bigr| \le |z_1 + z_2| \le |z_1| + |z_2|$

ECAT Shortcut: To find the maximum and minimum values of $|z + z_0|$ given $|z| = r$, use:

• $|z + z_0|_{\max} = r + |z_0|$
• $|z + z_0|_{\min} = \bigl| r - |z_0| \bigr|$ For example, if $|z| = 3$, the maximum value of $|z + 4 - 3i|$ is $3 + |4 - 3i| = 3 + 5 = 8$, and the minimum is $|3 - 5| = 2$.

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4. Polar Form, Euler's Formula & Argument

Let $P(a, b)$ be the representation of $z = a + bi$ on the Argand plane. Let $r = |z| = \sqrt{a^2 + b^2}$, and let $\theta$ be the directed angle from the positive real axis to the segment $OP$:

$a = r\cos\theta, \quad b = r\sin\theta$

Thus, the polar form of $z$ is:

$z = r(\cos\theta + i\sin\theta) = r\text{ cis }\theta$

Using Euler's Formula, $e^{i\theta} = \cos\theta + i\sin\theta$, we write the exponential form:

$z = r e^{i\theta}$

Determining the Principal Argument $\text{Arg}(z)$

The principal argument $\theta = \text{Arg}(z)$ is restricted to the interval $(-\pi, \pi]$ (some texts use $[0, 2\pi)$ instead; this guide uses $(-\pi, \pi]$):

• Quadrant I ($a > 0, b > 0$): $\theta = \tan^{-1}\left(\frac{b}{a}\right)$
• Quadrant II ($a < 0, b > 0$): $\theta = \pi - \tan^{-1}\left(\frac{|b|}{|a|}\right)$
• Quadrant III ($a < 0, b < 0$): $\theta = -\pi + \tan^{-1}\left(\frac{|b|}{|a|}\right)$
• Quadrant IV ($a > 0, b < 0$): $\theta = -\tan^{-1}\left(\frac{|b|}{|a|}\right)$

Multiplication and Division in Polar Form

• Product: $z_1 z_2 = r_1 r_2 e^{i(\theta_1 + \theta_2)} = r_1 r_2 [\cos(\theta_1 + \theta_2) + i\sin(\theta_1 + \theta_2)]$
• Quotient: $\frac{z_1}{z_2} = \frac{r_1}{r_2} e^{i(\theta_1 - \theta_2)} = \frac{r_1}{r_2} [\cos(\theta_1 - \theta_2) + i\sin(\theta_1 - \theta_2)]$
• Therefore: $\arg(z_1 z_2) = \arg(z_1) + \arg(z_2)$ and $\arg\left(\frac{z_1}{z_2}\right) = \arg(z_1) - \arg(z_2)$.

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5. De Moivre's Theorem

De Moivre's Theorem connects complex arithmetic with trigonometry:

For any real number $\theta$ and any integer $n$:

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

If $n = \frac{p}{q}$ is a rational number ($q > 0$), then one of the values of $(\cos\theta + i\sin\theta)^{p/q}$ is $\cos\left(\frac{p\theta}{q}\right) + i\sin\left(\frac{p\theta}{q}\right)$. The complete set of $q$ distinct roots is given by:

$z_k = \cos\left(\frac{\theta + 2k\pi}{q}\right) + i\sin\left(\frac{\theta + 2k\pi}{q}\right), \quad k = 0, 1, 2, \dots, q - 1$

Worked Example: High Power Simplification

Evaluate $\left(\frac{1 + i}{\sqrt{2}}\right)^{100}$:

1. Convert $\frac{1 + i}{\sqrt{2}}$ to polar form: $r = \sqrt{\left(\frac{1}{\sqrt{2}}\right)^2 + \left(\frac{1}{\sqrt{2}}\right)^2} = 1$. The argument is $\theta = \frac{\pi}{4}$.
2. Thus, $\frac{1+i}{\sqrt{2}} = \cos\left(\frac{\pi}{4}\right) + i\sin\left(\frac{\pi}{4}\right) = e^{i\pi/4}$.
3. Apply De Moivre's Theorem:

$\left(e^{i\pi/4}\right)^{100} = e^{i(100\pi/4)} = e^{i(25\pi)} = \cos(25\pi) + i\sin(25\pi) = -1 + 0 = -1$

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6. Cube Roots of Unity ($1, \omega, \omega^2$)

Consider the equation $z^3 = 1$. In factorized form:

$z^3 - 1 = 0 \implies (z - 1)(z^2 + z + 1) = 0$

Solving gives:

1. $z - 1 = 0 \implies z = 1$ (the real cube root of unity).
2. $z^2 + z + 1 = 0 \implies z = \frac{-1 \pm \sqrt{1 - 4}}{2} = \frac{-1 \pm i\sqrt{3}}{2}$.

We designate:

$\omega = \frac{-1 + i\sqrt{3}}{2}, \quad \omega^2 = \frac{-1 - i\sqrt{3}}{2}$

Notice that if we square $\omega$:

$\omega^2 = \left(\frac{-1 + i\sqrt{3}}{2}\right)^2 = \frac{1 - 2i\sqrt{3} + 3i^2}{4} = \frac{1 - 2i\sqrt{3} - 3}{4} = \frac{-2 - 2i\sqrt{3}}{4} = \frac{-1 - i\sqrt{3}}{2}$

Thus, each complex root is the square of the other, and each is also the complex conjugate of the other: $\overline{\omega} = \omega^2$ and $\overline{\omega^2} = \omega$.

Fundamental Properties of Cube Roots of Unity

1. Sum of Roots: $1 + \omega + \omega^2 = 1 + \left(\frac{-1+i\sqrt{3}}{2}\right) + \left(\frac{-1-i\sqrt{3}}{2}\right) = 0$.
   • Immediate derivations:
     • $1 + \omega = -\omega^2$
     • $1 + \omega^2 = -\omega$
     • $\omega + \omega^2 = -1$
2. Product of Roots: $1 \cdot \omega \cdot \omega^2 = \omega^3 = 1$.
   • Periodic powers: $\omega^{3k} = 1$, $\omega^{3k+1} = \omega$, $\omega^{3k+2} = \omega^2$.
   • Inverse: $\omega^{-1} = \omega^2$ and $\omega^{-2} = \omega$.
3. Geometric Symmetries:
   • In polar form, $1 = e^{i 0}$, $\omega = e^{i 2\pi/3}$, $\omega^2 = e^{i 4\pi/3}$.
   • The points representing $1, \omega, \omega^2$ on the Argand plane lie on the unit circle $|z| = 1$ and form the vertices of an equilateral triangle with side length $\sqrt{3}$.

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7. Fourth Roots of Unity ($1, -1, i, -i$)

Consider the polynomial equation $z^4 = 1$:

$z^4 - 1 = 0 \implies (z^2 - 1)(z^2 + 1) = 0 \implies (z - 1)(z + 1)(z - i)(z + i) = 0$

The four distinct roots are:

$z \in \{1, -1, i, -i\}$

Properties of Fourth Roots of Unity

1. Sum of Roots: $1 + (-1) + i + (-i) = 0$.
2. Product of Roots: $(1)(-1)(i)(-i) = -1 \cdot (-i^2) = -1 \cdot (1) = -1$.
3. Geometric Configuration: The roots lie on the unit circle at arguments $0, \frac{\pi}{2}, \pi, \frac{3\pi}{2}$. They form the vertices of a square inscribed inside $|z| = 1$, with diagonal length 2 and side length $\sqrt{2}$.
4. Cyclic Group: Under complex multiplication, $\{1, -1, i, -i\}$ forms a finite cyclic group generated by either $i$ or $-i$ ($i^1 = i, i^2 = -1, i^3 = -i, i^4 = 1$).

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8. High-Speed ECAT Algebraic Shortcuts

Shortcut 1: Rapid Modulus of Multi-Factor Fractions

Never expand complex expressions when asked for their modulus. Distribute the modulus operator directly.

Problem: Find $|z|$ if $z = \frac{(2 + i\sqrt{5})^3 (3 - 4i)}{(1 - i\sqrt{3})^4}$.

Solution in 15 seconds:

• $|2 + i\sqrt{5}| = \sqrt{2^2 + (\sqrt{5})^2} = \sqrt{4 + 5} = \sqrt{9} = 3$.
• $|3 - 4i| = \sqrt{3^2 + (-4)^2} = \sqrt{9 + 16} = 5$.
• $|1 - i\sqrt{3}| = \sqrt{1^2 + (-\sqrt{3})^2} = \sqrt{1 + 3} = 2$.
• Apply the power and quotient rules:

$|z| = \frac{|2 + i\sqrt{5}|^3 \cdot |3 - 4i|}{|1 - i\sqrt{3}|^4} = \frac{3^3 \cdot 5}{2^4} = \frac{27 \cdot 5}{16} = \frac{135}{16}$

Shortcut 2: Powers of $(1 \pm i)$

Recognize that $(1 + i)^2 = 1 + 2i - 1 = 2i$ and $(1 - i)^2 = 1 - 2i - 1 = -2i$.

Therefore, for any even power $2n$:

$(1 + i)^{2n} = [(1 + i)^2]^n = (2i)^n = 2^n i^n$

Example: $(1 + i)^{18} = (2i)^9 = 2^9 i^9 = 512 i$.

Shortcut 3: Expressions Involving $\omega$

When simplifying expressions of the form $(a + b\omega + c\omega^2)$:

• Replace constants using $1 = -\omega - \omega^2$ or combinations like $a + b\omega + c\omega^2 = (a - c) + (b - c)\omega$ because $c\omega^2 = -c - c\omega$.
• For example: $(1 - \omega + \omega^2)^5$. Since $1 + \omega^2 = -\omega$, the interior becomes $(-\omega - \omega)^5 = (-2\omega)^5 = -32\omega^5 = -32\omega^2$.