Complex Number
A complex number is a number of the form $a + bi$, where $a$ and $b$ are real numbers and $i$ is the imaginary unit satisfying $i^2 = -1$.
Formula
z = a + bi
Definition
A complex number has two parts, a real part and an imaginary part, written $a + bi$ where $i$ is the square root of $-1$; complex numbers were invented to solve equations like $x^2 = -1$, which has no real solution, and once $i = \sqrt{-1}$ is allowed, any polynomial equation can be solved, no matter how difficult. Formally, $z = a + bi$ with $a, b$ real and $i^2 = -1$: addition is $(a+bi) + (c+di) = (a+c) + (b+d)i$, multiplication is $(a+bi)(c+di) = (ac-bd) + (ad+bc)i$, and the modulus is $|z| = \sqrt{a^2 + b^2}$, extending the real number line to a 2D plane. The complex numbers $\mathbb{C}$ form an algebraically closed field, every non-constant polynomial with complex coefficients has a complex root (the Fundamental Theorem of Algebra), and $\mathbb{C}$ is also a complete metric space under $|z|$, the unique (up to isomorphism) algebraic closure of $\mathbb{R}$.
Example
$3 + 4i$ has real part $3$ and imaginary part $4$; $5 + 0i$ is just the real number $5$, and $0 + 7i = 7i$ is a pure imaginary number. Solving $x^2 + 4 = 0$ gives $x^2 = -4$, so $x = \pm 2i$, and multiplying $(2+3i)(1-i) = 2 - 2i + 3i - 3i^2 = 5 + i$. Euler's formula, $e^{i\theta} = \cos(\theta) + i\sin(\theta)$, represents complex numbers in polar form $z = re^{i\theta}$ where $r = |z|$ and $\theta = \arg(z)$, so multiplying $z_1 z_2$ multiplies the moduli and adds the arguments.
Key Insight
The complex plane (Argand diagram) represents $a + bi$ as the point $(a, b)$, with the modulus $|z|$ as the distance from the origin, unifying algebra, geometry, and trigonometry in one picture. The algebraic closure of $\mathbb{R}$ is $\mathbb{C}$, reached by adjoining just a single element, $i$, which is exceptional: for other fields like $\mathbb{Q}$, the algebraic closure is infinite-dimensional, making $\mathbb{C}$ uniquely "complete" in both an algebraic and metric sense.