Complex Numbers

Definition: A complex number is a number of the form a + bi, where a and b are real numbers and i, the imaginary unit, is defined by i = √-1 (equivalently, i² = -1); a is called the real part and b the imaginary part.

How It Works

A complex number is any number of the form a + bi, where a and b are real numbers and i, the imaginary unit, is defined by i = √-1 (equivalently, i² = -1).
a is the real part of the number, written Re(z), and b is the imaginary part, written Im(z); note that b is itself an ordinary real number — it’s the coefficient attached to i that makes the term “imaginary.”
The complex plane plots every complex number a + bi as a single point (a, b): the horizontal axis is the real axis, the vertical axis is the imaginary axis, turning an abstract pair of numbers into a concrete geometric picture.
Equivalently, a complex number can be pictured as a vector from the origin out to that point, with its own length and direction, exactly like a 2D vector in physics.
Addition works component-wise, just like adding vectors: (a + bi) + (c + di) = (a + c) + (b + d)i — add the real parts, add the imaginary parts, done.
Multiplication expands like ordinary algebra (FOIL), with one extra simplification at the end: (a + bi)(c + di) = ac + adi + bci + bdi², and since i² = -1, that last term becomes -bd, giving (ac - bd) + (ad + bc)i.
The modulus (or magnitude), written |a + bi|, equals √(a² + b²) — the point’s straight-line distance from the origin, computed with the exact same Pythagorean formula used for a 2D vector’s length, because geometrically it is one.
The argument, written arg(z), is the angle the vector makes with the positive real axis, typically computed as atan2(b, a) rather than plain arctan(b/a), so the correct quadrant is picked up automatically.
Modulus and argument together give a complex number’s polar form, a second way to describe the same point; multiplying two complex numbers multiplies their moduli and adds their arguments, a shortcut that comes back later in this note.
Complex numbers exist because real numbers alone cannot solve every polynomial equation: x² + 1 = 0 has no real solution, since no real number squares to -1, yet it has two perfectly good complex solutions, x = i and x = -i.
The Fundamental Theorem of Algebra guarantees that once complex numbers are allowed, every polynomial equation of degree n has exactly n roots; real numbers alone offer no such guarantee, which is precisely why complex numbers were ultimately necessary, not optional.

Illustration

The widget below plots a complex number z = a + bi as a vector from the origin — drag the sliders to change a and b — alongside a fixed marker at i = 0 + 1i and a live second vector showing the product z × i.

Watch the second vector as you drag: it is always exactly 90° counterclockwise from z, at the same distance from the origin. That is not a coincidence, and it takes only a few lines of algebra to confirm it. Starting from z = a + bi and multiplying by i:

z × i = (a + bi) × i = ai + bi² = ai + b(-1) = -b + ai

So the point (a, b) lands at (-b, a). Compare that to the standard formula for rotating any point 90° counterclockwise about the origin, (x, y) → (-y, x): plugging in x = a and y = b gives exactly (-b, a) — the same point, term for term. Multiplying by i and rotating 90° counterclockwise are, algebraically, the same operation.

Re Im -5 5 5 -5 i z z×i

z = 3.0 + 2.0i |z| = 3.61 arg(z) = 33.69° (0.59 rad) z × i = -2.0 + 3.0i (90° CCW from z)

Drag the sliders to move z = a + bi (its vector, in front). A fixed marker shows i itself, and a second vector shows z × i live — always exactly 90° counterclockwise from z, at the same distance from the origin.

a 3.0 b 2.0

var aIn = document.getElementById(‘cplx-a’); var bIn = document.getElementById(‘cplx-b’); var aOut = document.getElementById(‘cplx-a-out’); var bOut = document.getElementById(‘cplx-b-out’); var resetBtn = document.getElementById(‘cplx-reset’);

function fmt(n) { return (Math.round(n * 100) / 100).toFixed(2).replace(/.00$/, ‘.0’); }

function signedIm(re, im) { var sign = im < 0 ? ’-’ : ’+’; return fmt(re) + ’ ’ + sign + ’ ’ + fmt(Math.abs(im)) + ‘i’; }

function update() { var a = parseFloat(aIn.value); var b = parseFloat(bIn.value); aOut.textContent = fmt(a); bOut.textContent = fmt(b);

var pz = toPx(a, b);
lineZ.setAttribute('x2', pz[0]);
lineZ.setAttribute('y2', pz[1]);
pointZ.setAttribute('cx', pz[0]);
pointZ.setAttribute('cy', pz[1]);
labelZ.setAttribute('x', pz[0] + 8);
labelZ.setAttribute('y', pz[1] - 8);

// z times i: (a + bi)*i = -b + ai, i.e. the point (a,b) rotated 90 degrees CCW.
var wRe = -b, wIm = a;
var pw = toPx(wRe, wIm);
lineW.setAttribute('x2', pw[0]);
lineW.setAttribute('y2', pw[1]);
pointW.setAttribute('cx', pw[0]);
pointW.setAttribute('cy', pw[1]);
labelW.setAttribute('x', pw[0] + 8);
labelW.setAttribute('y', pw[1] - 8);

var mod = Math.sqrt(a * a + b * b);
var argRad = Math.atan2(b, a);
var argDeg = argRad * 180 / Math.PI;

zTxt.textContent = 'z = ' + signedIm(a, b);
modTxt.textContent = '|z| = ' + fmt(mod);
argTxt.textContent = 'arg(z) = ' + fmt(argDeg) + '° (' + fmt(argRad) + ' rad)';
wTxt.textContent = 'z × i = ' + signedIm(wRe, wIm) + '  (90° CCW from z)';

}

[aIn, bIn].forEach(function (el) { el.addEventListener(‘input’, update); }); resetBtn.addEventListener(‘click’, function () { aIn.value = 3; bIn.value = 2; update(); });

update(); })();

Under the Hood

Complex addition, multiplication, and modulus, stated precisely:

(a + bi) + (c + di) = (a + c) + (b + d)i

(a + bi)(c + di) = ac + adi + bci + bdi²      expand like any binomial (FOIL)
                 = ac + adi + bci - bd         substitute i² = -1
                 = (ac - bd) + (ad + bc)i      collect real and imaginary terms

|a + bi| = √(a² + b²)                         modulus: Pythagorean distance from the origin

As a cross-check on the rotation shown above: multiplying two complex numbers in polar form multiplies their moduli and adds their arguments. Since i itself has modulus |i| = √(0² + 1²) = 1 and argument arg(i) = 90°, multiplying any z by i leaves its length unchanged (multiplying by 1) and adds exactly 90° to its angle — the identical rotation, confirmed a second, independent way.

Worked Example 1: Multiplying two complex numbers. Given: (2 + 3i)(1 - 4i). Step 1: expand with FOIL: 2(1) + 2(-4i) + 3i(1) + 3i(-4i) = 2 - 8i + 3i - 12i². Step 2: substitute i² = -1, so -12i² becomes +12: 2 - 8i + 3i + 12. Step 3: combine real and imaginary terms: (2 + 12) + (-8 + 3)i. Answer: 14 - 5i (matches the formula directly: a=2, b=3, c=1, d=-4 gives (ac-bd) + (ad+bc)i = (2+12) + (-8+3)i = 14 - 5i).

Worked Example 2: Computing a modulus. Given: z = 5 - 12i. Step 1: |z| = √(a² + b²) = √(5² + (-12)²). Step 2: = √(25 + 144) = √169. Answer: |z| = 13 (a 5-12-13 right triangle in disguise).

Worked Example 3: A quadratic with complex roots. Given: x² + 2x + 5 = 0 (a=1, b=2, c=5). Step 1: discriminant = b² - 4ac = 4 - 20 = -16. Step 2: since the discriminant is negative, √-16 = √16 · √-1 = 4i. Step 3: x = (-b ± √disc) / 2a = (-2 ± 4i) / 2. Answer: x = -1 ± 2i (check: (-1+2i)² + 2(-1+2i) + 5 = (1 - 4i - 4) + (-2 + 4i) + 5 = (-3-4i) + (-2+4i) + 5 = 0, confirming both roots).

History

Complex numbers first forced their way into mathematics not through quadratics, which mathematicians were content to simply call unsolvable, but through Gerolamo Cardano’s and Rafael Bombelli’s 16th-century work on cubic equations.
Cardano’s formula for solving cubics, published in his 1545 Ars Magna, sometimes requires taking the square root of a negative number as an intermediate step even when the cubic’s three roots are all perfectly real, a snag later named the casus irreducibilis.
Rafael Bombelli picked up the problem around 1572 in his book L’Algebra, working out formal rules for adding and multiplying these “impossible” square roots and showing the imaginary parts cancel out cleanly, leaving the correct real answer behind — proof these numbers were doing real mathematical work, not just nonsense to be discarded.
The term “imaginary” itself was coined somewhat dismissively by René Descartes in his 1637 La Géométrie, reflecting the era’s lingering suspicion that these numbers were a bookkeeping trick rather than genuine quantities.
Leonhard Euler introduced the now-standard symbol i for √-1 in the 18th century and connected complex exponentials to trigonometry through what is now called Euler’s formula, e^(iθ) = cos θ + i sin θ.
Caspar Wessel (1799) and Jean-Robert Argand (1806) independently proposed plotting complex numbers as points on a plane, but it was Carl Friedrich Gauss, in the early 19th century, who popularized that geometric picture widely and coined the term “complex number” itself, which finally made them feel legitimate rather than fictional.

Why It Matters

Electrical engineers analyze AC circuits almost entirely in complex numbers: impedance (resistance generalized to include capacitors and inductors) is a complex quantity, turning Ohm’s law into simple complex multiplication instead of a differential equation (engineers write j instead of i, since i is reserved for current, but the math is identical).
Signal processing and the Fourier transform are built on complex exponentials, decomposing audio, images, and radio signals into frequency components that are most naturally expressed as complex numbers.
Quantum mechanics is fundamentally complex-valued: the Schrödinger equation contains i explicitly, and a particle’s wavefunction is a complex quantity whose squared magnitude gives a real, measurable probability.
Control systems engineers judge whether a feedback system is stable by checking where its poles land in the complex plane; poles in the left half of the plane mean the system settles down instead of oscillating out of control.
Computer graphics’ most recognizable fractals, including the Mandelbrot and Julia sets, are generated purely by iterating simple complex-number functions like z → z² + c.
Structural and mechanical engineers analyzing vibration rely on eigenvalues of real-valued matrices that are often complex, with each one encoding both a natural frequency and a damping rate in a single number.

Common Pitfalls

Assuming complex numbers are not “real” or lack practical use — they are a standard, essential tool in electrical engineering, signal processing, quantum mechanics, and control theory, not just an abstract curiosity.
Sign errors when multiplying: forgetting that i² = -1 and leaving a term like bdi² unsimplified, or mishandling the negative sign it introduces once it is substituted in.
Confusing the modulus |z| (a single real, non-negative number describing distance from the origin) with the complex number z itself, which still has two independent components.
Forgetting that complex roots of a real-coefficient polynomial always come in conjugate pairs, a + bi together with a - bi, never alone.
Treating i as an ordinary unknown that can be squared without consequence, rather than a fixed, agreed-upon value (-1) that must always be substituted in once i² appears.
Mixing up degrees and radians when reporting an argument, especially before feeding it into a formula like Euler’s, which expects radians.

Comparison

Aspect	Real Numbers	Complex Numbers
Form	a single value, x	a + bi (two components)
Geometric picture	points on a line (the number line)	points on a plane (the complex plane)
Equations fully solvable	linear equations, plus polynomials whose roots happen to be real	every polynomial equation (Fundamental Theorem of Algebra)
x² + 1 = 0	no solution	x = i and x = -i
Ordering	totally ordered (a < b always means something)	no natural ordering — i is neither positive nor negative

FAQ

Is i actually a “real” concept, or just a mathematical trick? It’s as legitimate as any other number system in mathematics — it obeys consistent, well-defined rules, and physical theories built directly on it, from AC circuits to quantum mechanics, make experimentally verified predictions. “Imaginary” is a historical label from an era of skepticism, not a verdict on legitimacy; negative and irrational numbers faced the same suspicion centuries earlier.

Why can’t complex numbers be ordered the way real numbers can? Any ordering consistent with normal arithmetic rules leads to a contradiction. If i were positive, i times i would have to be positive too, but i² = -1 is negative. If i were negative, -i would be positive, and (-i) times (-i) still equals i² = -1, negative again. Either assumption breaks down, so no consistent ordering can be defined.

What’s the actual difference between i and -i? Both square to -1, so algebraically they are mirror images of each other, called complex conjugates; swapping every i for -i throughout a true equation produces another true equation. That symmetry is exactly why complex roots of real polynomials always show up in conjugate pairs — a+bi and a-bi are equally valid solutions, indistinguishable by the algebra alone.

Example

An electrical engineer analyzing an AC circuit with 3 ohms of resistance and 4 ohms of inductive reactance writes its impedance as the single complex number Z = 3 + 4i ohms. The modulus |Z| = √(3² + 4²) = √25 = 5 ohms is the circuit’s effective resistance to current flow, and the argument of Z gives the phase shift between voltage and current — both read directly off the same complex number using the modulus and argument formulas above.

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