Algebra II - Imaginary Numbers Mathematicians Notes

Luca Pacioli

Published “Summa de Arithmetica,” a comprehensive summary of all mathematics known in Renaissance Italy at the time

Concluded that there is no solution to the cubic equation ax³+bx²+cx+d=0

Leonardo da Vinci’s math teacher

Omar Khayyam

Persian mathematician that identified 19 cubic equations that kept all the coefficients positive

He found numerical solutions to some of of them by considering the intersections of shapes, like hyperbolas and circles, but could not find a general solution

“Maybe one of those who will come after us will succeed in finding it.”

Scipione Del Ferro

A mathematics professor at the University of Bologna that found how to reliably solve depressed cubics, or a subset of cubic equations without the x² term.

Never revealed his solution until his deathbed because his position could be challenged by another mathematician in a duel

Antonio Fior

Student of Scipione del Ferro who was revealed how to solve a depressed cubic

Challenged mathematician Niccolo Fontana Tartaglia for his position, but failed to answer any of Tartaglia’s 30 questions

Niccolo Fontana Tartaglia

A self-taught mathematician whose face was cut open by a French soldier during his childhood, resulting in a stutter and a nickname which meant stutterer in Italian

Was challenged by Antonio Fior for his position, and successfully solved all 30 of Fior’s questions in 2 hours

“I did not deem him capable of finding such a rule of his own.”

Became the second person to solve the depressed cubic by extending the idea of completing the square into 3 dimensions, and summarized his method in a poem

Gerolamo Cardono

Polymath based in Milan that convinced Tartaglia to tell him the secret to solving the depressed cubic, but swore an oath not to reveal it

Discovered how to solve the full cubic equation, but was unable to publish his methods until he found del Ferro’s notebook so as to not violate his oath with Tartaglia

Published “Ars Magna” or “The Great Art,” in which is a chapter containing a geometric proof for each of the 13 arrangements of the cubic equation

Claimed that the square roots of negatives is “as subtle as [they] are useless”

Rafael Bombelli

Italian engineer that let the square roots of negatives be their own new type of number and was able to solve a cubic that was previously deemed unsolvable by abandoning the use of geometric proofs

Francois Viete

Introduced the modern symbolic notation for algebra, making geometry no longer the source of truth

Rene Descartes

Made heavy use of the square roots of negatives and called them imaginary numbers

Leonhard Euler

Introduced the letter “i” to represent the square root of negative one, which formed a complex number when combined with regular numbers

Erwin Schrodinger

Came up with a wave equation that governed the behavior of quantum particles, called the Schrodinger equation, that featured the number i

Freeman Dyson

Physicist who commended Schrodinger’s equation and wrote that his equation meant that nature works with complex numbers rather than real numbers