Imaginary numbers

1545–1831

Mathematics — algebra and complex analysis

Multiples of i, the square root of −1, a quantity no real number can supply. They surfaced in Italy in the sixteenth century when Cardano's Ars Magna of 1545 and then Rafael Bombelli's Algebra of 1572 found that solving cubic equations with three perfectly ordinary real roots required passing through square roots of negative numbers. Descartes coined the dismissive name in 1637; Euler introduced the symbol i; and Gauss, Argand and Wessel made them respectable by drawing them as a plane. Combined with reals they give the complex numbers, in which every polynomial has a root.

How to spot them

▸Built on i, defined so that i squared equals −1
▸Descartes named them derisively in 1637; the name stuck
▸Plotted on the vertical axis of the Argand diagram, with reals on the horizontal
▸Engineers write j instead of i, because i already means current
▸The fundamental theorem of algebra: every polynomial of degree n has n roots among them

Key facts

Cardano's 1545 Ars Magna, which reluctantly split 10 into two parts multiplying to 40 (1545)

The first appearance in print, dismissed by Cardano as 'as subtle as it is useless'

Bombelli's 1572 Algebra, which made the cubic formula work by keeping the square roots of negatives (1572)

Rafael Bombelli set out the arithmetic rules that made them usable

Descartes' dismissive 1637 coinage in La Géométrie (1637)

He meant it as a sneer; the name outlived the scorn

The plane in which the horizontal axis is real and the vertical is not (1806)

The Argand diagram, after Jean-Robert Argand; Caspar Wessel had the idea in 1799

The symbol i, introduced by Euler in 1777 (1777)

Leonhard Euler's notation, which Gauss then popularised

The theorem that every polynomial of degree n has exactly n roots, counted with multiplicity (1799)

The fundamental theorem of algebra, proved by Gauss in his doctoral thesis

The letter j, which electrical engineers use because the usual symbol means current

A notational quirk worth knowing for engineering questions

Hamilton's four-dimensional extension, carved into a Dublin bridge in 1843 (1843)

The quaternions, scratched on Broom Bridge by William Rowan Hamilton

Key ideas

Multiplying by i is a quarter turn anticlockwise in the plane. Do it twice and you have turned through 180°, which is why i squared is −1.
The powers of i cycle with period four: i, −1, −i, 1, and back again.
The complex numbers are algebraically closed — every polynomial with complex coefficients factorises completely — which the reals are not.

Quick-fire facts

•i is defined by i² = −1, and its powers cycle i, −1, −i, 1
•Girolamo Cardano's Ars Magna (1545) is the first printed appearance, in the problem of dividing 10 into two parts with product 40
•Rafael Bombelli's Algebra (1572) gave the rules of arithmetic that made them usable in Cardano's cubic formula
•René Descartes coined the adjective in La Géométrie (1637) as a term of abuse
•Euler introduced the symbol i in 1777; Gauss introduced the term 'complex number' in 1831
•Caspar Wessel described the geometric representation in 1799 and Jean-Robert Argand in 1806, but the plane is usually named after Argand
•Gauss's 1799 doctoral thesis proved the fundamental theorem of algebra: an nth-degree polynomial has n complex roots
•Hamilton extended them to the four-dimensional quaternions in 1843, sacrificing commutativity; the eight-dimensional octonions followed
•'Hypercomplex number' is the umbrella term for quaternions, octonions and similar extensions
•They are indispensable in quantum mechanics, where the Schrödinger equation contains i explicitly, and in AC circuit analysis
•The Mandelbrot set lives in the complex plane, defined by iterating z → z² + c

Drill it — 9 cards

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Sources

Plus Magazine, Millennium Mathematics Project, University of Cambridge — An imaginary tale
MacTutor History of Mathematics Archive, University of St Andrews — The fundamental theorem of algebra