How were complex numbers discovered?

The discovery of complex numbers, particularly the idea of the square root of a negative number, was a significant departure from traditional mathematical thinking. Edward Frenkel discussed this on the Lex Fridman Podcast.

Initially, the notion that a square root could be taken of a negative number was considered impossible because squaring any real number results in a positive number. The breakthrough came in the 16th century when Italian mathematician Girolamo Cardano, known for his work in algebra, encountered the need to accept the square root of negative numbers (specifically square root of -17) while solving cubic and quartic equations. He described the process as undergoing "mental tortures" but persevered because those numbers conveniently canceled out in his final solutions, suggesting they might have practical use 1 2.

Cardano's introduction of these "imaginary numbers" was initially met with skepticism but eventually proved essential, especially in fields such as quantum mechanics. This ability to think beyond established knowledge and experiment with seemingly illogical ideas was crucial for this mathematical advancement 1 2.

Overall, the discovery underscores how new mathematical concepts often emerge from re-examining and extending existing frameworks.