How Imaginary Numbers Were Invented

0:38

In 1494, Luca Pacioli who is Leonardo da Vinci's math teacher publishes "Summa de Arithmetica," a comprehensive summary of all mathematics known in Renaissance Italy at the time

3:42

For thousands of years, mathematicians were oblivious to the negative solutions to their equations because they were dealing with things in the real world, lengths and areas and volumes.

4:28

In the 11th century, Persian mathematician Omar Khayyam identified 19 different cubic equations, again, keeping all coefficients positive

5:05

Sometime around 1510, Scipione del Ferro finds a method to reliably solve depressed cubics

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Only on his deathbed in 1526 does he let it slip to his student Antonio Fior

6:26

On February 12, 1535, Fior challenges mathematician Niccolo Fontana Tartaglia who has recently moved to Fior's hometown of Venice

10:36

Tartaglia summarizes his method in an algorithm, a set of instructions.

11:25

In Milan, on March 25, 1539 Tartaglia reveals his method to Gerolamo Cardano

12:07

Cardano finds a way to turn any general cubic equation into a depressed cubic

12:43

In 1542, Cardano travels to Bologna and finds the solution to the depressed cubic in del Ferro's old notebook and publishes the full solution to the cubic

13:16

Cardano publishes "Ars Magna," The Great Art, an updated compendium of mathematics

16:19

10 years later, the Italian engineer Rafael Bombelli picks up where Cardano left off.

17:13

Negative areas, which make no sense in reality, must exist as an intermediate step on the way to the solution

17:25

In the 1600s, Francois Viete introduces the modern symbolic notation for algebra

17:39

Rene Descartes makes heavy use of the square roots of negatives, popularizing them as a result

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He calls them imaginary numbers, a name that sticks, which is why Euler later introduces the letter i to represent the square root of negative one

18:20

In 1925, Erwin Schrödinger is searching for a wave equation that governs the behavior of quantum particles building on de Broglie's insight that matter consists of waves