# Complex Number

**Type:** Concept  
**Domain:** Analysis  
**Codex URL:** /mathematics/analysis/complex-number/  
**Entry status:** Live — v1.0 (2026-05-27)

## Summary
a+bi where i²=−1. Forms an algebraically closed field ℂ. Essential to electrical engineering, quantum mechanics, complex analysis.

## Arithmetic
Addition componentwise; multiplication using i²=−1. Forms a field.

## The complex plane & polar form
z=a+bi plotted as (a,b). Polar form z=re^(iθ) connects to Euler's Identity — multiplication = rotate+scale.

## Historical origins — from CUBICS, not quadratics
Cardano and Bombelli (1500s) encountered √(negative) as intermediate steps solving cubic equations, even when final answers were real. This, not x²=−1, was the actual historical origin.

## Fundamental Theorem of Algebra
ℂ is algebraically closed (Gauss, 1799) — every polynomial has a root.

## Quantum mechanics
Wavefunctions are essentially complex-valued. 2021 Nature paper (Renou et al.) provided experimental evidence complex numbers are genuinely necessary, not just convenient, in quantum theory.

## Sources
### Tier 1
- Ahlfors, L.V. (1979). *Complex Analysis*. 3rd ed.
- Needham, T. (1997). *Visual Complex Analysis*.
### Tier 2
- Nahin, P.J. (1998). *An Imaginary Tale*.

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*Mathematics Codex entry v1.0 — added 2026-05-27 — thecodex.expert/mathematics/*