What Are Complex Numbers? Definition, Formulas & Examples

A complex number is an ordered pair of real numbers combined by the rule of addition and multiplication, constructed to resolve the equation $x^2 + 1 = 0$ within the field of numbers. This construction results in a number system that extends the real numbers by introducing a distinct unit, commonly denoted $i$, satisfying $i^2 = -1$.

Algebraic Structure of Complex Numbers and Cartesian Representation

The set of complex numbers is defined as $\mathbb{C} = \{z = a + ib : a, b \in \mathbb{R} \}$, where $a$ is termed the real part and $b$ the imaginary part of $z$, with $i$ being the solution to $i^2 = -1$. The notations $\mathrm{Re}(z) = a$ and $\mathrm{Im}(z) = b$ are used to designate the real and imaginary parts, respectively.

If $b = 0$, then $z$ is called a purely real number. If $a = 0$ and $b \ne 0$, $z$ is a purely imaginary number. Every real number $r$ is a complex number with $r = r + i \cdot 0$. Every imaginary number $bi$ is a complex number with $0 + ib$.

Addition of complex numbers follows the rule $(a+ib) + (c+id) = (a+c) + i(b+d)$. Multiplication is executed by the distributive law as $(a+ib)(c+id) = ac + i(ad+bc) + i^2bd$; substituting $i^2 = -1$, this simplifies to $(ac-bd) + i(ad+bc)$.

Existence and Algebra of the Imaginary Unit $i$

The unit $i$ is defined as the root of the polynomial $x^2 + 1 = 0$, so $i^2 = -1$. The powers of $i$ are determined as follows: $i^1 = i$, $i^2 = -1$, $i^3 = i \cdot i^2 = i \cdot (-1) = -i$, $i^4 = (i^2)^2 = (-1)^2 = 1$, and the sequence repeats with period 4. Thus, for any integer $n \geq 0$: $i^{4n} = 1$, $i^{4n+1} = i$, $i^{4n+2} = -1$, $i^{4n+3} = -i$.

Conjugate, Modulus, and Argument for Complex Numbers

Given $z = a + ib$, the conjugate of $z$ is defined as $\overline{z} = a - ib$. The modulus (or absolute value) of $z$ is defined as $|z| = \sqrt{a^2 + b^2}$, and the argument (principal value, for $z \neq 0$) is $\arg(z) = \arctan \left(\frac{b}{a}\right)$, with the value restricted to $(-\pi, \pi]$ in usual conventions.

Key properties include: $\overline{z_1 + z_2} = \overline{z_1} + \overline{z_2}$, $\overline{z_1 z_2} = \overline{z_1} \cdot \overline{z_2}$, $|\overline{z}| = |z|$, and $z \cdot \overline{z} = a^2 + b^2 = |z|^2$.

Polar and Exponential Forms of Complex Numbers

For a nonzero complex number $z = a + ib$, let $r = |z| = \sqrt{a^2 + b^2}$ and $\theta = \arg(z)$. Then $z$ can be represented in polar or trigonometric form as $z = r(\cos \theta + i \sin \theta)$. By Euler's formula, $z = r e^{i \theta}$. One may convert from Cartesian to polar form by extracting $r$ and $\theta$ as precise values: $r = \sqrt{a^2 + b^2}$ and, if $a > 0$, $\theta = \arctan (b/a)$; otherwise, quadrant corrections are required.

Further reading: Polar and Euler Forms of Complex Numbers

Algebraic Operations: Addition, Subtraction, Multiplication, and Division

Given $z_1 = a + ib$ and $z_2 = c + id$, then:

Addition: $z_1 + z_2 = (a + c) + i(b + d)$

Multiplication: \[ z_1 z_2 = (a + ib)(c + id) \] \[ = a \cdot c + a \cdot id + ib \cdot c + ib \cdot id \] \[ = ac + i a d + i b c + i^2 b d \] \[ = ac + i (ad + bc) + (-1) b d \] \[ = (ac - bd) + i (ad + bc) \]

Subtraction: $z_1 - z_2 = (a - c) + i(b - d)$

Division: To compute $\cfrac{z_1}{z_2}$, multiply numerator and denominator by $\overline{z_2}$: \[ \frac{a + ib}{c + id} \cdot \frac{c - id}{c - id} = \frac{(a + ib)(c - id)}{(c + id)(c - id)} \] \[ = \frac{ac - a id + ib c - ib id}{c^2 + (id)c - (id)c - (id)^2} \] \[ = \frac{ac - a i d + i b c - i^2 b d}{c^2 - i^2 d^2} \] \[ = \frac{ac - a i d + i b c + b d}{c^2 + d^2} \] \[ = \frac{(ac + b d) + i (b c - a d)}{c^2 + d^2} \] So, \[ \frac{z_1}{z_2} = \frac{ac + bd}{c^2 + d^2} + i \cdot \frac{b c - a d}{c^2 + d^2} \]

For visualised geometry on the Argand plane, see: Geometry of Complex Numbers

Conjugate Multiplication and Division Simplification

Consideration of the product $(a + ib)(a - ib)$ yields \[ (a + ib)(a - ib) = a^2 - i a b + i a b - i^2 b^2 = a^2 - i a b + i a b + b^2 \] The cross terms $-iab + iab$ cancel, and since $i^2 = -1$, we obtain $a^2 + b^2$. Thus, the product of a complex number with its conjugate is the square of its modulus: \[ z \overline{z} = (a+ib)(a-ib) = a^2 + b^2 = |z|^2 \]

Properties of Modulus, Argument, and Conjugate

For complex numbers $z_1$ and $z_2$ (with $z_2 \ne 0$), the following properties hold: $\overline{z_1 \pm z_2} = \overline{z_1} \pm \overline{z_2}$, $\overline{z_1 z_2} = \overline{z_1} \overline{z_2}$, $\overline{\frac{z_1}{z_2}} = \frac{\overline{z_1}}{\overline{z_2}}$, $\overline{\overline{z}} = z$, $|z_1 z_2| = |z_1| |z_2|$, $\left|\frac{z_1}{z_2}\right| = \frac{|z_1|}{|z_2|}$, $\arg(z_1 z_2) = \arg(z_1) + \arg(z_2)$, $\arg\left(\frac{z_1}{z_2}\right) = \arg(z_1) - \arg(z_2)$.

Further details on the modulus are provided in: Modulus of Complex Numbers

Stepwise Examples of Complex Number Operations

Example 1: Addition
Given $z_1 = 3 + 4i$, $z_2 = 2 - 5i$. Add $z_1$ and $z_2$.
Substitution: $(3 + 4i) + (2 - 5i)$.
Combine terms: $3 + 2 = 5$, $4i + (-5i) = -i$.
Result: $z_1 + z_2 = 5 - i$.

Example 2: Multiplication
Given $z_1 = 1 + 2i$, $z_2 = 3 - i$. Compute $z_1 z_2$.
Substitution: $(1 + 2i)(3 - i)$.
Multiply: $1 \cdot 3 = 3$, $1 \cdot (-i) = -i$, $2i \cdot 3 = 6i$, $2i \cdot (-i) = -2i^2$.
Sum: $3 - i + 6i - 2i^2$.
Since $i^2 = -1$, $-2i^2 = 2$.
Final combination: $3 + 2 + 5i = 5 + 5i$.
Result: $z_1 z_2 = 5 + 5i$.

Example 3: Division
Given $z_1 = 2 + 3i$, $z_2 = 1 - 2i$. Compute $\frac{z_1}{z_2}$.
Conjugate of denominator: $\overline{z_2} = 1 + 2i$.
Numerator: $(2 + 3i)(1 + 2i) = 2(1) + 2(2i) + 3i(1) + 3i(2i) = 2 + 4i + 3i + 6i^2 = 2 + 7i + 6(-1) = (2 - 6) + 7i = -4 + 7i$.
Denominator: $(1 - 2i)(1 + 2i) = 1(1) + 1(2i) - 2i(1) - 2i(2i) = 1 + 2i - 2i - 4i^2 = 1 + (2i - 2i) - 4(-1) = 1 + 4 = 5$.
Result: $\frac{z_1}{z_2} = \frac{-4}{5} + i \frac{7}{5}$.

Worked Example: Cartesian to Polar Conversion

Given $z = -1 + \sqrt{3}i$. First, compute $r = |z| = \sqrt{(-1)^2 + (\sqrt{3})^2} = \sqrt{1 + 3} = 2$. Next, compute the argument: $\theta = \arctan\left(\frac{\sqrt{3}}{-1}\right) = \arctan(-\sqrt{3})$. Since the real part is negative and the imaginary part is positive, $z$ is in the second quadrant. Thus, $\theta = \pi - \frac{\pi}{3} = \frac{2\pi}{3}$.
Result: $z = 2 \left[\cos \left(\frac{2\pi}{3}\right) + i \sin \left(\frac{2\pi}{3}\right)\right] = 2 e^{i \frac{2\pi}{3}}$.

Summary of Complex Number Structure and Links

Complex numbers establish an extension of the real numbers that enables solution of all quadratic equations over $\mathbb{C}$. Mastery of their algebraic, polar, and geometric forms is fundamental to higher mathematics, including their application in quadratic equations and analytic geometry. Refer to Understanding Complex Numbers and Complex Numbers and Quadratics for further structured discussion and practice.