1.11 Complex Numbers Basic OperationsHL Only

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:

  • instead of $\sqrt{-4}$ we write $2i$.
  • instead of $\sqrt{-9}$ we write $3i$.
  • instead of $\sqrt{-5}$ we write $\sqrt{5}i$.

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

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

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

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 $\overline{z}$
$2+3i$ $2$ $3$ $2-3i$
$2-3i$ $2$ $-3$ $2+3i$
$-2+3i$ $-2$ $3$ $-2-3i$
$-2-3i$ $-2$ $-3$ $-2+3i$
$1+i$ $1$ $1$ $1-i$
$3i$ $0$ $3$ $-3i$
$2$ $2$ $0$ $2$
$\dfrac{2+3i}{4}$ $\dfrac{1}{2}$ $\dfrac{3}{4}$ $\dfrac{2-3i}{4}$

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 $z_2 = x_2 + y_2i$:

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

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:

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

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: Combine real parts and imaginary parts directly.

    $$z + w = (7 + 4i) + (2 + 3i) = 9 + 7i$$

  • Subtraction:

    $$z - w = (7 + 4i) - (2 + 3i) = 5 + i$$

  • Multiplication:

    $$\begin{aligned} z \cdot w &= (7 + 4i)(2 + 3i) \\ &= 14 + 21i + 8i + 12i^2 \\ &= 14 + 29i - 12 \\ &= 2 + 29i \end{aligned}$$

  • Division: The fraction $\dfrac{z}{w} = \dfrac{7+4i}{2+3i}$ must be simplified to the standard $x+yi$ format. This is achieved by multiplying the numerator and denominator by the conjugate of the denominator, $\overline{w} = 2 - 3i$.

    $$\begin{aligned} \dfrac{z}{w} &= \dfrac{7+4i}{2+3i} \cdot \dfrac{2-3i}{2-3i} \\ &= \dfrac{14 - 21i + 8i - 12i^2}{2^2 + 3^2} \\ &= \dfrac{14 - 13i + 12}{13} \\ &= \dfrac{26 - 13i}{13} \\ &= 2 - i \end{aligned}$$

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 $|z|^2 = (\sqrt{x^2 + y^2})^2 = x^2 + y^2$.
  • 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.

  • $(3+4i)(3-4i) = 3^2 + 4^2 = 9 + 16 = 25$
  • $(1+i)(1-i) = 1^2 + 1^2 = 1 + 1 = 2$
  • $(2-i)(2+i) = 2^2 + (-1)^2 = 4 + 1 = 5$

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 by finding the remainder when the exponent is divided by 4. For example:

$$i^{35} = i^{32+3} = i^{32} \cdot i^3 = (i^4)^8 \cdot i^3 = 1 \cdot (-i) = -i$$

EXAMPLE 6

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

Solution:

  • (a) Expanding the cubic binomial:

    $$\begin{aligned} z = (2+i)^3 &= 2^3 + 3(2^2)i + 3(2)i^2 + i^3 \\ &= 8 + 12i + 6(-1) - i \\ &= 2 + 11i \end{aligned}$$

    Alternatively, this can be computed iteratively: $(2+i)^2(2+i) = (4+4i-1)(2+i) = (3+4i)(2+i) = 2+11i$.

  • (b) Substituting the result from part (a): $w = \dfrac{2+11i}{1-i}$.
    Multiply by the conjugate of the denominator:

    $$\begin{aligned} w &= \dfrac{2+11i}{1-i} \cdot \dfrac{1+i}{1+i} \\ &= \dfrac{2 + 2i + 11i + 11i^2}{1^2 + 1^2} \\ &= \dfrac{-9 + 13i}{2} \\ &= -\dfrac{9}{2} + \dfrac{13}{2}i \end{aligned}$$

EXAMPLE 7

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

Solution:

  • Method A (Analytical Expansion): Define $z = x + yi$.

    $$\begin{aligned} (x+yi)(1-i) &= 2+11i \\ x - xi + yi - yi^2 &= 2+11i \\ (x+y) + (y-x)i &= 2+11i \end{aligned}$$

    Equating the real and imaginary parts creates the linear system:

    $$\begin{cases} x+y=2 \\ y-x=11 \end{cases}$$

    Solving this system yields $x = -\dfrac{9}{2}$ and $y = \dfrac{13}{2}$. The resulting number is $z = -\dfrac{9}{2} + \dfrac{13}{2}i$.

  • Method B (Algebraic Isolation): Treat the expression like a standard linear equation $ax=b$.

    $$z = \dfrac{2+11i}{1-i} = -\dfrac{9}{2} + \dfrac{13}{2}i$$

    (The division logic precisely matches the steps performed in Example 6).

1.13 The Complex PlaneHL Only

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 directly 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 from the origin.

  • The modulus of $3+4i$ is $\sqrt{3^2 + 4^2} = \sqrt{25} = 5$.
  • The modulus of $-5-2i$ is $\sqrt{(-5)^2 + (-2)^2} = \sqrt{29}$.
  • Pure real numbers fall precisely on the x-axis: $|3| = 3$ and $|-5| = 5$.
  • Pure imaginary numbers fall precisely on the y-axis: $|4i| = 4$.

EXAMPLE 2 (Geometrical Symmetries)

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

Re
Im
$z = 3 + 4i$
$\overline{z} = 3 - 4i$
$-z = -3 - 4i$
$-\overline{z} = -3 + 4i$
  • 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 $\arg(z) = \theta$.

$$\cos\theta = \dfrac{x}{r} \qquad \sin\theta = \dfrac{y}{r} \qquad \tan\theta = \dfrac{y}{x}$$

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: $\tan\theta = \dfrac{\sqrt{3}}{1} = \sqrt{3}$. Because both components are positive (1st quadrant), the angle is exactly $\theta = \dfrac{\pi}{3}$.
    The polar form is:

    $$z = 2\left(\cos\dfrac{\pi}{3} + i\sin\dfrac{\pi}{3}\right)$$

  • Analysis for $w = 3 + 4i$:
    Calculate modulus: $r = \sqrt{3^2 + 4^2} = 5$.
    Determine argument: $\tan\theta = \dfrac{4}{3}$. Located in the 1st quadrant, the angle is evaluated as $\theta \approx 0.927$ radians.
    The polar form is:

    $$w = 5(\cos 0.927 + i\sin 0.927)$$

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 $z_1 = 1+i$: $\tan\theta = \dfrac{1}{1} = 1$. Located in the 1st quadrant, $\theta = \dfrac{\pi}{4}$.
    Result: $\sqrt{2}\left(\cos\dfrac{\pi}{4} + i\sin\dfrac{\pi}{4}\right)$.
  • For $z_2 = -1-i$: $\tan\theta = \dfrac{-1}{-1} = 1$. Located in the 3rd quadrant, $\theta = -\dfrac{3\pi}{4}$ (or $\dfrac{5\pi}{4}$ depending on interval).
    Result: $\sqrt{2}\left(\cos\dfrac{5\pi}{4} + i\sin\dfrac{5\pi}{4}\right)$.
  • For $z_3 = 1-i$: $\tan\theta = \dfrac{-1}{1} = -1$. Located in the 4th quadrant, $\theta = -\dfrac{\pi}{4}$.
    Result: $\sqrt{2}\left(\cos\left(-\dfrac{\pi}{4}\right) + i\sin\left(-\dfrac{\pi}{4}\right)\right)$.
  • For $z_4 = -1+i$: $\tan\theta = \dfrac{1}{-1} = -1$. Located in the 2nd quadrant, $\theta = \dfrac{3\pi}{4}$.
    Result: $\sqrt{2}\left(\cos\dfrac{3\pi}{4} + i\sin\dfrac{3\pi}{4}\right)$.

EXAMPLE 5: Polar to Cartesian Transformation

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

As a demonstration, evaluating $z = 2\left(\cos\dfrac{\pi}{3} + i\sin\dfrac{\pi}{3}\right)$ produces:

$$z = 2\left(\dfrac{1}{2} + i\dfrac{\sqrt{3}}{2}\right)$$

Which directly simplifies to the Cartesian form $1 + \sqrt{3}i$.

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 $2\left(\cos\dfrac{\pi}{3} + i\sin\dfrac{\pi}{3}\right)$ is compactly written as $2\text{cis}\dfrac{\pi}{3}$.

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 6

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

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

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., $3 \implies 3\text{cis}0$).
  • Negative real numbers reside at an argument of $\pi$ (e.g., $-2 \implies 2\text{cis}\pi$).
  • Positive imaginary numbers reside at an argument of $\pi/2$ (e.g., $i \implies \text{cis}\dfrac{\pi}{2}$).
  • Negative imaginary numbers reside at an argument of $-\pi/2$ (e.g., $-3i \implies 3\text{cis}\left(-\dfrac{\pi}{2}\right)$).
  • 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 $\arg(\overline{z}) = -\theta$.