What Are Complex Numbers?

What are complex numbers? They are numbers of the form a + bi, where a and b are real numbers, and i is the imaginary unit defined by i² = −1. The real part is a, the imaginary part is b (not bi, just the coefficient). For instance, 4 + 3i has real part 4 and imaginary part 3. Without them, the equation x² = −1 has no solution among real numbers. The complex number definition extends the one-dimensional number line into a two-dimensional plane, letting you express rotation and oscillation algebraically. This is a foundational tool in mathematics and engineering.

The imaginary unit i is the engine behind everything. Its defining property, i² = −1, is axiomatic and has not changed since Rafael Bombelli's 1572 rules. When you square i, you get −1; when you square 2i, you get −4. This single rule lets you take square roots of negative numbers: √(−9) = 3i. A pure imaginary number, like 5i, has a real part of zero (0 + 5i). The imaginary unit is not a placeholder or a trick; it is a consistent, self-contained number that behaves under the same algebraic laws as real numbers, once you remember i² replaces −1 in any simplification.

The complex plane, also called the Argand diagram, maps each complex number a + bi to a point (a, b) in a two-dimensional coordinate system. The horizontal axis is the real axis, the vertical axis is the imaginary axis. The number 3 + 4i sits three units right and four units up from the origin. This geometric view is not decoration; it reveals properties like modulus (distance from origin, √(a² + b²)) and argument (angle θ from the positive real axis). The conjugate, a − bi, is the mirror image across the real axis. This plane is a coordinate system like any other, but it carries algebraic meaning that a plain xy-graph does not.

The Origins in 16th-Century Algebra

Complex numbers emerged from a practical dead end. In 1545, Girolamo Cardano published his full cubic equation method in Ars Magna, but he hit a wall: the casus irreducibilis, such as x³ = 15x + 4, required square roots of negative numbers in intermediate steps, even though the final answer was a real number (here, 4). Cardano himself balked. In 1572, Rafael Bombelli provided the first rigorous arithmetic for these 'imaginary' numbers, defining rules for adding, subtracting, and multiplying them, and showing the intermediate terms cancel.

Formalization and Geometric Insight

Leonhard Euler later introduced the symbol i, and Carl Friedrich Gauss formalized the term 'complex number' and proved the fundamental theorem of algebra: every polynomial has a root in the complex plane. Jean-Robert Argand published the geometric interpretation in 1806, giving us the Argand diagram. For the original texts and dates, the MacTutor History of Mathematics archive on Cardano and Bombelli is the authoritative source.

Complex numbers are load-bearing in science and engineering. In electrical engineering, they model alternating current (AC) circuits, where voltage and current are out of phase; a phasor is a complex number representing amplitude and phase. In signal processing, the Fourier transform uses complex exponentials to decompose waveforms into frequencies. Quantum mechanics uses complex probability amplitudes, where the real part is the observable and the imaginary part is a mathematical convenience. In control theory, they analyze system stability via poles in the complex plane. Without them, modern technologies, from radios to MRI machines, would not function. They are also used in fluid dynamics and fractal geometry, where the Mandelbrot set is defined entirely by iterating complex functions. In short, any problem involving rotation, waves, or oscillation finds a natural home in the complex number system.

Many people dismiss complex numbers because of the word 'imaginary.' That name is historical baggage, not a verdict on their reality. They are as real as any number: a consistent tool with rules that never contradict. Another error is thinking a complex number cannot be purely real or purely imaginary. In fact, 5 is a complex number (5 + 0i), and so is 3i (0 + 3i). The complex plane is not the same as a 2D coordinate plane, though it looks similar; the difference lies in the operations. Multiplication by i corresponds to a 90-degree rotation, which has no analogue in a standard xy-graph. The conjugate, a − bi, is not optional; use it for division, because multiplying numerator and denominator by the conjugate clears the imaginary part from the denominator.

Addition, Subtraction, and Multiplication

Manual calculation follows rules you already know from algebra. Addition and subtraction combine like terms: (a + bi) + (c + di) = (a + c) + (b + d)i. Subtraction requires distributing the minus sign to both terms: (a + bi) − (c + di) = (a − c) + (b − d)i. Multiplication uses FOIL, then replaces i² with −1: (a + bi)(c + di) = (ac − bd) + (ad + bc)i.

Division and the Conjugate

Division is the tricky one: multiply top and bottom by the conjugate of the denominator. For (2 + 3i) / (1 − i), multiply by (1 + i)/(1 + i), giving (2 + 3i)(1 + i) / (1 + 1) = (−1 + 5i) / 2 = −0.5 + 2.5i. A common failure is forgetting the sign change in the conjugate; another is assuming arctan(b/a) gives the argument, which only works in the first quadrant. For other quadrants, add or subtract π based on the signs of a and b.

The standard complex number definition, as used in OpenStax Precalculus 2e section 3.1, is a number of the form a + bi, where a and b are real and i = √−1, so i² = −1. The imaginary part is b, the coefficient of i, not the bi term itself. A pure imaginary number is bi with a = 0. In the complex plane, the real axis is horizontal and the imaginary axis is vertical. This definition is not just a formality; it dictates how you plot, add, and multiply. When you see 'complex number,' you should immediately think 'two-part number with a real and an imaginary component.' This is the foundation for polar form, re^(iθ), and Euler's formula, which connects exponentiation to trigonometry.

The complex plane rewards geometric thinking. The modulus, |a + bi| = √(a² + b²), is the distance from the origin; the argument, θ = arctan(b/a) adjusted for quadrant, is the angle from the positive real axis. Multiplication of two complex numbers adds their arguments and multiplies their moduli, this is why powers and roots become intuitive here. The conjugate, a − bi, reflects across the real axis, and multiplying by it yields a real number: (a + bi)(a − bi) = a² + b². When you take the square root of a complex number, you get two answers, but a calculator returns only the principal root, so a problem asking for 'all roots' requires you to add 2πk for integer k. Draw the point on the complex plane before doing algebra. This habit prevents quadrant errors that plague textbook formulas.

The imaginary unit i is often the first hurdle in learning what are complex numbers. Students ask, 'How can a square be negative?' The answer is that i is defined to be that way, and the definition is consistent. In physics, the imaginary part is a mathematical convenience: the real part is what you measure, the imaginary part encodes phase or rotation. In AC circuits, a voltage is written as V = V₀e^(iωt), and the real part is the actual voltage; the imaginary part is not observable directly but carries the phase information. In quantum mechanics, the wavefunction is complex, and only its modulus squared, |ψ|², is observable. Calling i 'fake' because it is not on the number line misses the point: it lives on a separate axis, and that axis is what makes rotation algebraically simple.

Before you rely on a calculator for complex arithmetic, know its conventions. The TI-84 Plus CE has a complex mode (a+bi or re^θi), but its angle output range differs from a Casio; one may return angles in (−π, π] and another in [0, 2π). This is a common source of error when you compare answers with a classmate. Also, when a problem asks for 'all roots,' the calculator gives you only the principal value; you must add 2πk to find the rest. If you are using Python's cmath or MATLAB, the argument function returns angles in different ranges, so check the documentation. The definition of i² = −1 never changes, but the software conventions do, and they are not universal.

Here is the honest caveat: complex numbers are not something you will 'see' in daily life, and most people never need to compute with them after a final exam. The value is in the framework, once you accept that numbers can have two parts, problems in electronics, physics, and even cartography become simpler, not harder. But if you are a student, the single most useful habit is to draw the complex plane before starting any problem, because the quadrant rules for the argument are where most errors originate. A textbook formula like arctan(b/a) is a trap outside the first quadrant; use the signs of a and b to adjust by π, and you will never lose points to a sign slip again.