Complex Numbers

1. Introduction

For most of your mathematical life you have been told that you "cannot take the square root of a negative number." That is true within the real numbers — no real number squared gives $-1$, because squaring always produces something $\ge 0$. But mathematicians discovered that if you simply invent a new number whose square is $-1$ and play by the usual algebraic rules, something remarkable happens: every polynomial equation becomes solvable, and a whole new, internally consistent number system opens up. That invented number is the imaginary unit $i$, and the system built from it is the complex numbers.

The name "imaginary" is an unfortunate historical accident — complex numbers are no less real than negative numbers or fractions, both of which were once viewed with suspicion. They are indispensable in electrical engineering (where current and voltage are tracked with complex amplitudes), in physics (quantum mechanics is written in complex numbers), in signal processing, and in pure mathematics. Even problems that only mention real numbers, like factoring $x^2 + 1$, are illuminated by allowing complex answers.

In this article we define $i$, build the arithmetic of complex numbers (addition, subtraction, multiplication, and the clever conjugate trick for division), uncover the elegant four-step cycle of the powers of $i$, learn to picture complex numbers as points in a plane with a measurable distance from the origin called the modulus, and solve quadratics with negative discriminants. By the end, complex arithmetic should feel as routine as working with binomials.

Complex numbers unify algebra and geometry: every point in the plane corresponds to one complex number, and every complex number to one point. This geometric viewpoint will become essential when you study polar form and Euler's formula in later courses, but even now the Argand diagram gives you a visual check on whether your arithmetic makes sense.

2. Core Concepts

2.1 The imaginary unit

The imaginary unit is defined by the single rule

$i^{2} = -1, \qquad \text{equivalently } i = \sqrt{-1}.$

From this one definition everything else follows. For instance, the square root of any negative number can now be written in terms of $i$: $\sqrt{-9} = \sqrt{9}\,\sqrt{-1} = 3i$, and $\sqrt{-7} = i\sqrt{7}$. A useful safety note: always pull out the $i$ first, because the careless step $\sqrt{-4}\,\sqrt{-9} = \sqrt{36} = 6$ is wrong; correctly, $\sqrt{-4}\,\sqrt{-9} = (2i)(3i) = 6i^2 = -6$.

2.2 Complex numbers: real and imaginary parts

A complex number is any expression of the form

$z = a + bi,$

where $a$ and $b$ are real numbers. We call $a$ the real part, written $\operatorname{Re}(z) = a$, and $b$ the imaginary part, written $\operatorname{Im}(z) = b$ (note: the imaginary part is the real number $b$, not $bi$). When $b = 0$ the number is purely real, so every real number is also complex — the reals sit inside the complex numbers. When $a = 0$ and $b \neq 0$, the number $bi$ is called purely imaginary.

Two complex numbers are equal exactly when their real parts match and their imaginary parts match: $a + bi = c + di$ iff $a = c$ and $b = d$. This "equate parts" principle is used constantly to solve equations.

2.3 The set $\mathbb{C}$ and closure under operations

The collection of all complex numbers is denoted $\mathbb{C}$. A remarkable fact: if you add, subtract, multiply, or divide (except by zero) any two complex numbers, the result is always another complex number in standard form $a + bi$. The number system is closed under these operations — you never need to invent a new kind of number to finish a calculation.

2.4 Addition and subtraction

Because the real and imaginary parts live on independent axes, we combine them separately, exactly like collecting like terms:

$(a + bi) + (c + di) = (a + c) + (b + d)i,$ $(a + bi) - (c + di) = (a - c) + (b - d)i.$

For example $(3 + 2i) + (1 - 5i) = 4 - 3i$. You never mix a real part with an imaginary part, just as you would never combine $3$ and $2x$ into one term.

2.5 Multiplication

Multiply complex numbers exactly as you multiply binomials (FOIL), then replace $i^2$ with $-1$ and recombine:

$(a + bi)(c + di) = ac + adi + bci + bd\,i^{2} = (ac - bd) + (ad + bc)i.$

The $-bd$ appears precisely because $i^2 = -1$. Worked instance:

$(2 + 3i)(1 - 4i) = 2 - 8i + 3i - 12i^{2} = 2 - 5i + 12 = 14 - 5i.$

2.6 Deriving the multiplication formula

Start from $(a + bi)(c + di)$ and expand with FOIL. The only step that differs from real binomials is replacing $i^2$ with $-1$:

$ac + adi + bci + bdi^2 = ac + (ad + bc)i + bd(-1) = (ac - bd) + (ad + bc)i.$

Memorizing the pattern "real part is $ac - bd$, imaginary part is $ad + bc$" speeds up hand calculations.

Shortcut for squaring: $(a + bi)^2 = (a^2 - b^2) + 2abi$. Apply directly instead of full FOIL when the two factors are identical.

2.7 Powers of $i$

The powers of $i$ cycle through four values forever:

$i^{1} = i,\quad i^{2} = -1,\quad i^{3} = i^{2}\cdot i = -i,\quad i^{4} = (i^{2})^{2} = 1,$

and then $i^{5} = i^{4}\cdot i = i$ restarts the cycle. So to evaluate any power, divide the exponent by $4$ and use only the remainder: remainder $0 \to 1$, $1 \to i$, $2 \to -1$, $3 \to -i$.

For negative exponents, use $i^{-1} = \dfrac{1}{i}$. Multiply top and bottom by $i$: $\dfrac{1}{i} \cdot \dfrac{i}{i} = \dfrac{i}{i^2} = \dfrac{i}{-1} = -i$.

2.8 The complex conjugate

The conjugate of $z = a + bi$ is $\bar{z} = a - bi$ — flip the sign of the imaginary part. The conjugate is special because multiplying a number by its conjugate erases $i$ entirely:

$(a + bi)(a - bi) = a^{2} - (bi)^{2} = a^{2} - b^{2}i^{2} = a^{2} + b^{2},$

a non-negative real number. This is the key to division: to simplify a quotient, multiply the numerator and denominator by the conjugate of the denominator, turning the denominator real.

2.9 The complex plane and modulus

We can picture $a + bi$ as the point $(a, b)$ in a plane (the Argand diagram), with the horizontal axis for the real part and the vertical axis for the imaginary part. The distance from the origin to that point is the modulus:

$|a + bi| = \sqrt{a^{2} + b^{2}},$

which comes directly from the Pythagorean theorem. The modulus measures the "size" of a complex number; for a real number it reduces to the ordinary absolute value. Notice also that $z\bar{z} = a^2 + b^2 = |z|^2$, linking conjugates and modulus.

2.10 Properties of conjugation

Conjugation distributes over arithmetic:

$\overline{z_1 + z_2} = \bar{z}_1 + \bar{z}_2, \qquad \overline{z_1 z_2} = \bar{z}_1 \cdot \bar{z}_2, \qquad \overline{\bar{z}} = z.$

These properties make conjugates powerful tools for simplifying expressions and proving identities.

2.11 Complex roots of quadratics

When the discriminant $b^2 - 4ac < 0$ of $ax^2 + bx + c = 0$, the quadratic formula produces a negative under the square root. Writing $\sqrt{-D} = i\sqrt{D}$ gives two complex conjugate roots. For real coefficients, complex roots always come in conjugate pairs: if $a + bi$ is a root, so is $a - bi$.

2.12 The Fundamental Theorem of Algebra (preview)

Every polynomial of degree $n$ with complex coefficients has exactly $n$ complex roots (counting multiplicity). This is why extending to $\mathbb{C}$ "completes" algebra — no polynomial is left without a solution.

2.13 Factoring with complex numbers

Polynomials that do not factor over the reals may factor over the complexes. For example, $x^2 + 1 = (x - i)(x + i)$ and $x^2 + 4 = (x - 2i)(x + 2i)$. Each quadratic $x^2 + k$ with $k > 0$ splits into conjugate linear factors. This is why the quadratic formula always works in $\mathbb{C}$: even when $D < 0$, the roots $x = \dfrac{-b \pm i\sqrt{|D|}}{2a}$ are valid complex numbers.