Complex Number
Functions & Advanced AlgebraA 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 is a number that has two parts: a regular (real) part and an imaginary part. It is written as $a + bi$, where $i$ is the square root of $-1$.
Example
$3 + 4i$ is a complex number. Here, $3$ is the real part and $4$ is the imaginary part. $5 + 0i$ is just $5$ (a real number). $0 + 7i = 7i$ (a pure imaginary number).
Key Insight
Complex numbers were invented to solve equations like $x^2 = -1$, which has no real solution. Once we allow $i = \sqrt{-1}$, we can solve any polynomial equation, no matter how difficult.
Definition
A complex number $z = a + bi$, where $a, b$ are real and $i^2 = -1$. Add: $(a+bi) + (c+di) = (a+c) + (b+d)i$. Multiply: $(a+bi)(c+di) = (ac-bd) + (ad+bc)i$. The modulus $|z| = \sqrt{a^2 + b^2}$. Complex numbers extend the real number line to a $2$D plane.
Example
Solve $x^2 + 4 = 0$: $x^2 = -4$, $x = \pm\sqrt{-4} = \pm 2i$. These are complex solutions. Multiply $(2+3i)(1-i) = 2 - 2i + 3i - 3i^2 = 2 + i - 3(-1) = 5 + i$.
Key Insight
The complex plane (Argand diagram) represents $a + bi$ as the point $(a, b)$. The modulus $|z|$ is the distance from the origin. Complex numbers unify algebra, geometry, and trigonometry.
Definition
The complex numbers $\mathbb{C}$ form an algebraically closed field: every non-constant polynomial with complex coefficients has a complex root (Fundamental Theorem of Algebra). $\mathbb{C}$ is also a complete metric space under $|z|$. The field $\mathbb{C}$ is the unique (up to isomorphism) algebraic closure of $\mathbb{R}$.
Example
Euler's formula: $e^{i\theta} = \cos(\theta) + i\sin(\theta)$. This represents complex numbers in polar form: $z = re^{i\theta}$ where $r = |z|$ and $\theta = \arg(z)$. Multiplication: $z_1 z_2$ has modulus $r_1 r_2$ and argument $\theta_1 + \theta_2$.
Key Insight
The algebraic closure of $\mathbb{R}$ is $\mathbb{C}$, reached by adjoining just one element ($i$). This is exceptional: for other fields (like $\mathbb{Q}$), the algebraic closure is infinite-dimensional. $\mathbb{C}$ is uniquely "complete" in both an algebraic and metric sense.