1.11 Complex Numbers Basic Operations (HL)

1. Introduction and Definition

Within the real number system, square roots of negative numbers such as $\sqrt{-1}$, $\sqrt{-4}$, or $\sqrt{-9}$ are undefined. To resolve this, an imaginary unit $i$ is defined with the property $i^2 = -1$.

Using this definition, negative roots can be expressed as follows:

  • $\sqrt{-4}$ is written as $2i$
  • $\sqrt{-9}$ is written as $3i$
  • $\sqrt{-5}$ is written as $\sqrt{5}i$

Evaluating the quadratic equation $x^2 - 4x + 13 = 0$ yields a discriminant of $\Delta = -36$, indicating no real solutions exist. However, by substituting $\sqrt{\Delta} = i\sqrt{36} = 6i$, valid solutions are formed:

$x = \frac{4 \pm \sqrt{\Delta}}{2} = \frac{4 \pm 6i}{2} = 2 \pm 3i$

In general, for any quadratic where $\Delta < 0$, the complex roots are calculated using $x = \frac{-b \pm i\sqrt{|\Delta|}}{2a}$.

The Definition of a Complex Number

A numerical value $z$ structured as $z = x + yi$, where $x, y \in \mathbb{R}$, is defined as a complex number.

  • The real part of $z$ is $x$, denoted as $\text{Re}(z) = x$.
  • The imaginary part of $z$ is $y$, denoted as $\text{Im}(z) = y$.

The set containing all complex numbers is designated by the symbol $\mathbb{C}$. Any real number $x$ can also be classified as a complex number of the form $x + 0i$, meaning its imaginary component is zero.

2. The Conjugate ($\overline{z}$) and Modulus ($|z|$)

The Conjugate: The conjugate of a complex number $z = x + yi$ is defined as $\overline{z} = x - yi$.
The Modulus: The modulus of $z = x + yi$ represents its magnitude and is calculated by $|z| = \sqrt{x^2 + y^2}$.

EXAMPLE 1

Given the complex number $z = 3 + 4i$, the components are $\text{Re}(z) = 3$, $\text{Im}(z) = 4$, and the conjugate is $\overline{z} = 3 - 4i$.

Complex number $z$ Real part $\text{Re}(z)$ Imaginary part $\text{Im}(z)$ Conjugate $\overline{z}$
$2+3i$$2$$3$$2-3i$
$2-3i$$2$$-3$$2+3i$
$-2+3i$$-2$$3$$-2-3i$
$1+i$$1$$1$$1-i$
$3i$$0$$3$$-3i$
$2$$2$$0$$2$
$\frac{2+3i}{4}$$\frac{1}{2}$$\frac{3}{4}$$\frac{2-3i}{4}$

Notice on Modulus: The expressions $z$, $\overline{z}$, $-z$, and $-\overline{z}$ all share the identical modulus value of $\sqrt{x^2 + y^2}$.

For instance, the numbers $3+4i$, $3-4i$, $-3-4i$, and $-3+4i$ all have a calculated modulus of $\sqrt{3^2 + 4^2} = \sqrt{25} = 5$. The modulus concept generalizes the absolute value operation used for real numbers (e.g., $|3|=3$ and $|-3|=3$).

3. Equality: $z_1 = z_2$

Equality between two complex numbers is established strictly when their respective real and imaginary parts are identical. For $z_1 = x_1 + y_1i$ and $z_2 = x_2 + y_2i$:

$z_1 = z_2 \Leftrightarrow \begin{cases} x_1 = x_2 \\ y_1 = y_2 \end{cases}$

This demonstrates that an equation involving complex numbers functions effectively as a system of two simultaneous real equations.

EXAMPLE 2

Determine the values of parameters $a$ and $b$ given the equality of $z_1 = 3 + 4i$ and $z_2 = a + (3b - 2)i$.

Solution:

$z_1 = z_2 \Leftrightarrow \begin{cases} 3 = a \\ 4 = 3b - 2 \end{cases} \Leftrightarrow \begin{cases} a = 3 \\ b = 2 \end{cases}$.

4. Addition, Subtraction, Multiplication, and Division

Arithmetic operations on complex numbers adhere to standard algebraic laws, with the critical substitution rule that $i^2 = -1$.

EXAMPLE 3

Perform the four primary operations on the complex numbers $z = 7 + 4i$ and $w = 2 + 3i$.

Addition: $z + w = (7 + 4i) + (2 + 3i) = \mathbf{9 + 7i}$ (Combine real parts and imaginary parts).
Subtraction: $z - w = (7 + 4i) - (2 + 3i) = \mathbf{5 + i}$.
Multiplication: $z \cdot w = (7 + 4i)(2 + 3i) = 14 + 21i + 8i + 12i^2$
$= 14 + 29i - 12 = \mathbf{2 + 29i}$.
Division: The fraction $\frac{z}{w} = \frac{7+4i}{2+3i}$ must be simplified to the standard $x+yi$ format. This is achieved by multiplying the numerator and denominator by the conjugate of the denominator, $\overline{w} = 2 - 3i$.
$\frac{z}{w} = \frac{7+4i}{2+3i} \cdot \frac{2-3i}{2-3i} = \frac{14 - 21i + 8i - 12i^2}{13} = \frac{26 - 13i}{13} = \mathbf{2 - i}$.

NOTICE: Modulus Squared Property

$|z|^2 = z \cdot \overline{z}$

Both sides of this equation resolve to $x^2 + y^2$. For $z = x + yi$:

  • Squaring the modulus yields $|z|^2 = x^2 + y^2$.
  • Multiplying by the conjugate yields $z\overline{z} = (x + yi)(x - yi) = x^2 - y^2i^2 = x^2 + y^2$.

EXAMPLE 4 (Multiplication by a Conjugate)

Multiplying any complex number by its conjugate strictly produces a real number corresponding to the square of its modulus.

  • $(3+4i)(3-4i) = 9 + 16 = \mathbf{25}$
  • $(1+i)(1-i) = 1 + 1 = \mathbf{2}$
  • $(2-i)(2+i) = 4 + 1 = \mathbf{5}$

EXAMPLE 5 (Powers of $i$)

The consecutive powers of the imaginary unit $i$ follow a repeating sequence of four values:

  • $i^0 = 1$
  • $i^1 = i$
  • $i^2 = -1$
  • $i^3 = -i$
  • $i^4 = 1$
  • $i^5 = i$
  • $i^6 = -1$
  • $i^7 = -i$

Using this cyclical pattern, higher powers can be simplified easily. For example, $i^{35} = i^{32+3} = i^3 = \mathbf{-i}$ (since 32 is an exact multiple of 4).

EXAMPLE 6

Evaluate the expressions (a) $z = (2+i)^3$ and (b) $w = \frac{(2+i)^3}{1-i}$.

Solution:

(a) Expanding the cubic binomial: $z = (2+i)^3 = 2^3 + 3(2^2)i + 3(2)i^2 + i^3$
$= 8 + 12i + 6(-1) - i = \mathbf{2 + 11i}$.
Alternatively, this can be computed as $(2+i)^2(2+i) = (4+4i-1)(2+i) = (3+4i)(2+i) = \mathbf{2+11i}$.
(b) Substituting the result from part (a): $w = \frac{2+11i}{1-i}$.
Multiply by the conjugate of the denominator: $\frac{2+11i}{1-i} \cdot \frac{1+i}{1+i} = \frac{2 + 2i + 11i + 11i^2}{2}$
$= \frac{-9 + 13i}{2} = \mathbf{-\frac{9}{2} + \frac{13}{2}i}$.

EXAMPLE 7

Determine the complex number $z$ that satisfies the equation $z(1-i) = 2+11i$.

Solution:

Method A (Analytical Expansion): Define $z = x + yi$.
$(x+yi)(1-i) = 2+11i$
$x - xi + yi - yi^2 = 2+11i$
$(x+y) + (y-x)i = 2+11i$
Equating the real and imaginary parts creates the system $\begin{cases} x+y=2 \\ y-x=11 \end{cases}$.
Solving this system yields $x = -9/2$ and $y = 13/2$. The resulting number is $z = \mathbf{-\frac{9}{2} + \frac{13}{2}i}$.
Method B (Algebraic Isolation): Treat the expression like a standard linear equation $ax=b$.
$z = \frac{2+11i}{1-i} = \mathbf{-\frac{9}{2} + \frac{13}{2}i}$ (The division logic matches Example 6).

1.13 The Complex Plane (HL)

1. Cartesian Representation

Complex numbers of the form $z = x + yi$ are mapped onto a two-dimensional Cartesian coordinate system, known as the complex plane or Argand diagram.

  • The value $z = x + yi$ correlates to the coordinate point $(x, y)$.
  • The real part defines the position on the horizontal x-axis.
  • The imaginary part defines the position on the vertical y-axis.
  • The modulus $|z| = \sqrt{x^2 + y^2}$ geometrically represents the direct distance from the origin to the point.

Notice: A complex number $z$ can be equivalently conceptualized as a directional vector originating from $(0,0)$ and terminating at $(x, y)$.

EXAMPLE 1

Plotting various complex numbers demonstrates that the modulus remains a strict measure of distance. For instance, the modulus of $3+4i$ is $\sqrt{25}=5$, and the modulus of $-5-2i$ is $\sqrt{29}$. Furthermore, pure real numbers like $3$ and pure imaginary numbers like $4i$ fall precisely on the axes.

EXAMPLE 2 (Geometrical Symmetries)

The geometric representations of a complex number, its conjugate, and its negative exhibit specific spatial symmetries across the coordinate plane.

Taking the reference point $z = 3 + 4i$:
The conjugate $\overline{z} = 3 - 4i$ is a reflection of $z$ strictly across the horizontal real axis.
The opposite $-z = -3 - 4i$ is a reflection of $z$ strictly across the origin $(0,0)$.
The negated conjugate $-\overline{z} = -3 + 4i$ is a reflection of $z$ strictly across the vertical imaginary axis.

2. The Polar Form (Modulus-Argument Form)

An alternative method for describing a spatial position on the complex plane utilizes polar coordinates denoted as $(r, \theta)$.

  • The variable $r$ denotes the length of the vector, which is mathematically equivalent to the modulus $|z|$.
  • The variable $\theta$ denotes the angle formed between the positive x-axis and the vector. This is defined as the argument of $z$, formatted as $\arg(z) = \theta$.
$\cos\theta = \frac{x}{r}$ $\sin\theta = \frac{y}{r}$ $\tan\theta = \frac{y}{x}$

Rearranging these trigonometric ratios establishes the relationships $x = r\cos\theta$ and $y = r\sin\theta$.

Applying these substitutions, a complex number can be expressed in its Polar Form:

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

Note: To maintain consistency, the principal argument $\theta$ is restricted to the interval $-\pi < \theta \le \pi$ (equivalent to $-180^\circ < \theta \le 180^\circ$).

3. Transformations Between Forms

EXAMPLE 3: Cartesian to Polar

Convert the complex numbers $z = 1 + \sqrt{3}i$ and $w = 3 + 4i$ into their polar equivalents.

Analysis for $z = 1 + \sqrt{3}i$:
Calculate modulus: $r = \sqrt{1^2 + (\sqrt{3})^2} = \sqrt{4} = 2$.
Determine argument: $\tan\theta = \frac{\sqrt{3}}{1} = \sqrt{3}$. Because both components are positive (1st quadrant), the angle is $\theta = \frac{\pi}{3}$.
The polar form is: $z = \mathbf{2(\cos\frac{\pi}{3} + i\sin\frac{\pi}{3})}$.
Analysis for $w = 3 + 4i$:
Calculate modulus: $r = \sqrt{3^2 + 4^2} = 5$.
Determine argument: $\tan\theta = \frac{4}{3}$. Located in the 1st quadrant, the angle is evaluated as $\theta \approx 0.927$ radians.
The polar form is: $w = \mathbf{5(\cos 0.927 + i\sin 0.927)}$.

EXAMPLE 4: Quadrant Analysis

Determine the polar representations for $z_1 = 1+i$, $z_2 = -1-i$, $z_3 = 1-i$, and $z_4 = -1+i$, observing their respective spatial quadrants.

The modulus for every variant remains $r = \sqrt{1^2+1^2} = \sqrt{2}$.

For $z_1 = 1+i$: $\tan\theta = 1/1 = 1$. Located in the 1st quadrant, $\theta = \pi/4$. Result: $\mathbf{\sqrt{2}(\cos\frac{\pi}{4} + i\sin\frac{\pi}{4})}$.
For $z_2 = -1-i$: $\tan\theta = -1/-1 = 1$. Located in the 3rd quadrant, $\theta = 5\pi/4$. Result: $\mathbf{\sqrt{2}(\cos\frac{5\pi}{4} + i\sin\frac{5\pi}{4})}$.
For $z_3 = 1-i$: $\tan\theta = -1/1 = -1$. Located in the 4th quadrant, $\theta = -\pi/4$. Result: $\mathbf{\sqrt{2}(\cos(-\frac{\pi}{4}) + i\sin(-\frac{\pi}{4}))}$.
For $z_4 = -1+i$: $\tan\theta = 1/-1 = -1$. Located in the 2nd quadrant, $\theta = 3\pi/4$. Result: $\mathbf{\sqrt{2}(\cos\frac{3\pi}{4} + i\sin\frac{3\pi}{4})}$.

Polar to Cartesian Transformation

Transforming a polar expression back into Cartesian coordinates requires simply executing the trigonometric multiplication: $z = r\cos\theta + ir\sin\theta$.

As a demonstration, evaluating $z = 2(\cos\frac{\pi}{3} + i\sin\frac{\pi}{3})$ produces $2(\frac{1}{2} + i\frac{\sqrt{3}}{2})$, which simplifies to the Cartesian form $\mathbf{1 + \sqrt{3}i}$.

4. Alternative Notations: CIS and Euler's Form

The CIS Abbreviation: The trigonometric formulation $z = r(\cos\theta + i\sin\theta)$ is frequently shortened using the acronym "cis". This creates the format $z = r\text{cis}\theta$. For instance, the expression $2(\cos\frac{\pi}{3} + i\sin\frac{\pi}{3})$ is compactly written as $2\text{cis}\frac{\pi}{3}$.

Euler's Form: A fundamental mathematical identity defines $e^{i\theta} = \cos\theta + i\sin\theta$. Applying this identity to the polar equation creates the exponential format $z = re^{i\theta}$. Under this notation, the previous example is written as $2e^{i\frac{\pi}{3}}$.

EXAMPLE 5

Express the following complex numbers across all four recognized formatting standards:

Cartesian Polar form CIS form Euler form
$1+i$$2(\cos\frac{\pi}{4}+i\sin\frac{\pi}{4})$$\sqrt{2}\text{cis}\frac{\pi}{4}$$\sqrt{2}e^{i\frac{\pi}{4}}$
$3+4i$$5[\cos(0.927)+i\sin(0.927)]$$5\text{cis}(0.927)$$5e^{0.927i}$
$3-4i$$5[\cos(-0.927)+i\sin(-0.927)]$$5\text{cis}(-0.927)$$5e^{-0.927i}$

Crucial Observations for Polar Form

  • Any complex number possessing a modulus of 1 simplifies strictly to $z = \text{cis}\theta$, as $|z| = \sqrt{\cos^2\theta + \sin^2\theta} = 1$.
  • Positive real numbers reside at an argument of $0$ (e.g., $3 \rightarrow 3\text{cis}0$).
  • Negative real numbers reside at an argument of $\pi$ (e.g., $-2 \rightarrow 2\text{cis}\pi$).
  • Positive imaginary numbers reside at an argument of $\pi/2$ (e.g., $i \rightarrow \text{cis}\frac{\pi}{2}$).
  • Negative imaginary numbers reside at an argument of $-\pi/2$ (e.g., $-3i \rightarrow 3\text{cis}(-\frac{\pi}{2})$).
  • The conjugate of $z = r\text{cis}\theta$ must maintain a positive scalar before the imaginary unit, which is achieved by negating the angle: $\overline{z} = r[\cos(-\theta) + i\sin(-\theta)]$. Consequently, if $\arg(z) = \theta$, then $\arg(\overline{z}) = -\theta$.