Our Daily Story #7: Algebraic Equation Owed to the Mathematical Thief

From the previous O.D.S. stories (#3, #4) on Quintic equations (degree 5) by Galois and Abel in the 19th century, we now trace back to the first breakthrough in the 16th century of the Cubic (degree 3) & Quartic (degree 4) equations with radical solution, i.e. expressed by 4 operations (+ – × /) and radical roots {\sqrt{x} , \: \sqrt [n]{x} }.

Example: Since Babylonian time, and in 220 AD China’s Three Kingdoms Period by 趙爽 Zhao Shuang of the state of Wu 吳, we knew the radical solution of Quadratic equations of degree 2 :
ax^2 + bx + c = 0

can be expressed in radical form with the coefficients a, b, c:

\boxed{x= \frac{-b \pm \sqrt{b^{2}-4ac} }{2a}}

Are there radical solutions for Cubic equation (degree 3) and Quartic equations (degree 4) ? We had to wait till the European Renaissance Period in the 16th century Italy.

If not for Cardano,  the 16th century man who stole the secretive radical solution of cubic equation from Tartaglia, Algebra would not have had progressed so rapidly since then, especially the discovery of Complex number i = \sqrt {-1} used in solving Cubic equations.

Although on moral ground Cardano’s act was wrong, but keeping any Mathematical truth as own asset is also wrong. In this case:
Wrong x Wrong = Right
[(-) × (-)= (+)]

(Note: As a side track, interestingly the same story goes to the Chinese Taiji widespread in the world today. It was in the 19th century stolen from its secretive inventor Master Chen 陈长兴 by Yang Luchan 杨露禅, who disguised himself as his mute servant for 10 years.)





Quartic Equation (degree 4):



Probability by 2 Great Friends

Today Probability is a “money” Math, used in Actuarial Science, Derivatives (Options) in Black-Scholes Formula.

In the beginning it was “A Priori” Probability by Pascal (1623-1662), then Fermat (1601-1665) invented today’s “A Posteriori” Probability.

“A Priori” assumes every thing is naturally “like that”: eg. Each coin has 1/2 chance for head, 1/2 for tail. Each dice has 1/6 equal chance for each face (1-6).

“A Posteriori” by Fermat, then later the exile Protestant French mathematician De Moivre (who discovered Normal Distribution), is based on observation of “already happened” statistic data.

Cardano (1501-1576) born 150 years earlier than Pascal and Fermat, himself a weird genius in Medicine, Math and an addictive gambler, found the rule of + and x for chances (he did not know the name ‘Probability’ then ):

Addition + Rule: throw a dice, chance to get a “1 and 2” faces:
1/6 +1/6 = 2/6 = 1/3
(Correct: 1 & 2 out of other six faces)

Multiplication x Rule: throw two dices, chance to get a “1 followed by a 2 ” faces : 1/6 x 1/6 = 1/36
(Assume 1st throw does not affect or influence the 2nd throw: independent events)

Pascal never met Fermat personally, only through correspondences (like emails today), but Pascal regarded with respect Fermat as superior in Math than himself.

Fermat and Descartes were not so. Descartes openly criticized Fermat as a second class mathematician. Both independently discovered Analytical Geometry, but Descartes scored the credit in ‘Cartesian’ coordinates.

Fermat never published any books in Math. As a successful judge in Toulous, he spent his free time as an amateur mathematician, especially in Number Theory. He showed his Math discoveries to friends in letters but never provided proofs. Hence the Fermat’s Last Theorem made the world mathematicians after him (Gauss, Euler, Kummer, Sophie Germaine, Andrew Wiles…) busy for 380 years until 1994.

Fermat died 3 years after Pascal. Another Modern Mathematics was being born – Calculus – in UK by Newton and Germany by Leibniz. Probability was put in the back seat over-taken by Calculus.