Renaissance Journey to the Birth of Imaginary Numbers

Why do square roots of negative numbers appear even when a cubic has real solutions? Follow the Renaissance contests, Bombelli's workable rules, and the later geometric picture.

About 20 min·6 nodes

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The path across the map

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STEP 1 · Mathematician1535Venice· Main activity

Niccolò Tartaglia

A central figure in the Renaissance dispute where a method and a promise of secrecy collided. Tartaglia was wounded during the 1512 Sack of Brescia; details of the injury and resulting stammer come chiefly from his later autobiographical account. With little formal schooling, he became a teacher and mathematical author. A 1535 problem contest against Antonio Fior made his method for one class of cubic equations famous. In 1539 he gave Cardano a versified rule under a pledge of secrecy. After Cardano and Ferrari found evidence of del Ferro’s earlier work, Cardano published the method with attributions in the 1545 Ars Magna; Tartaglia regarded this as betrayal. The episode is more revealing as a conflict among priority, secrecy, and prior work than as a simple hero-and-plagiarist story.

↓10 years later

STEP 2 · Mathematician1545Milan· Main activity

Gerolamo Cardano

A Renaissance physician and mathematician whose 1545 Ars Magna published methods for cubic and quartic equations. The cubic story involved Tartaglia’s secrecy oath, del Ferro’s earlier work, and Ferrari’s collaboration, while calculations with square roots of negative numbers foreshadowed later complex arithmetic.

↓Same year

STEP 3 · Number system1545Nuremberg· Publication

The Complex Plane

The plane of numbers a + bi, combining real and imaginary axes and supplying roots for every non-constant complex polynomial.

↓27 years later

STEP 4 · Mathematician1572Bologna· Main activity

Rafael Bombelli

A Bolognese engineer-mathematician. He took the imaginary numbers Cardano had handled reluctantly and made them into a working arithmetic. His 1572 L'Algebra gave the explicit rules for adding, multiplying, and dividing (a+bi) and (c+di).

↓65 years later

STEP 5 · Mathematician1637Paris· Main activity

René Descartes

A philosopher and mathematician who developed a method of doubt and translated geometric problems into algebraic form. Analytic geometry also had independent contributors, especially Fermat.

↓Next step

STEP 6 · RiverBaghdad· Journey start

The River of Algebra

Building on Babylonian, Greek, Indian, and Chinese equation traditions, al-Khwarizmi gave a systematic ninth-century account of al-jabr. Arabic, Hebrew, and Latin texts moved through several Mediterranean routes into Renaissance work on equations.

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