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Complex numbers
1572· Mathematics
ProvenTested many times; part of working physics.
Numbers that include the square root of minus one, called i, giving each number two parts: real and 'imaginary'.
See it on the Science map →How it works
No ordinary number squared gives −1, so mathematicians invented one: i. A complex number like 3 + 2i is a point on a flat plane rather than a line. Multiplying by i rotates a point a quarter-turn, so complex numbers naturally describe rotations, oscillations and waves. Rafael Bombelli set out rules for them in 1572. For centuries they were seen as a trick, and 'imaginary' was meant dismissively.
Why it matters
Electrical engineers use them for AC circuits, and quantum mechanics cannot be written without them.
How we know
Mathematically rigorous. Experiments in 2022 tested a real-number-only version of quantum theory, proposed as an alternative, and ruled it out, confirming complex numbers are needed.
Still a mystery
Why nature's deepest theory needs numbers first invented as a calculation shortcut is a standing puzzle.
Connections (3)
▸Complex numbers are required by The wave functionProven
The relationship
A wave function gives each possibility a complex number, which carries both a size and a phase (an angle). When possibilities combine, phases can reinforce or cancel, producing interference. Real numbers alone can't track phase this way. The Schrödinger equation itself contains i, so the wave function is unavoidably complex.
How the link was found
Erwin Schrödinger's 1926 equation built i in; he initially hoped to remove it but could not.
Why it's proven
Interference experiments confirm phase effects. Experiments in 2022 ruled out real-number quantum theory, following a 2021 proposal by Marc-Olivier Renou and colleagues.
Open questions and mysteries
Why nature's basic description needs complex numbers rather than real ones is not understood.
▸Complex numbers are the building blocks of Hilbert space: the stage for quantum statesProven
The relationship
The coordinates of quantum Hilbert space are complex numbers, not ordinary ones. That lets each state carry a phase, which governs how states interfere and how they evolve over time. The quarter-turn rotation built into i gives quantum theory its wave-like behaviour.
How the link was found
Hilbert-space theory developed around 1904–1910; von Neumann specified complex Hilbert space for quantum mechanics in 1927–1932.
Why it's proven
Quantum predictions computed in complex Hilbert space match experiments, and real-number versions were ruled out by experiments in 2022.
Open questions and mysteries
Some physicists have tried quaternions, a larger number system, instead. None are needed so far, and why complex numbers are the right choice is unclear.
▸Complex numbers are a prime example for Why does math describe reality?Open question
The relationship
Complex numbers were invented in the 1500s to solve cubic equations and long dismissed as 'imaginary'. Four centuries later, quantum mechanics turned out to require them. A tool created purely for calculation became the language of nature's most fundamental theory, a striking case of Wigner's puzzle.
How the link was found
Bombelli formalised them in 1572; Schrödinger built them into quantum mechanics in 1926; Wigner discussed quantum mechanics in his 1960 essay.
Is it proven?
The example is documented history. What it means about math and reality is philosophy, not a testable claim.
Open questions and mysteries
Was it luck, or do mathematicians unconsciously pick up patterns from the physical world?