Pacioli concluded that a general solution for cubic equations was impossible.
Ancient Civilizations
Attempts to solve cubic equations by:
Babylonians
Greeks
Chinese
Indians
Egyptians
Persians
All civilizations failed to find a solution despite successfully solving quadratics.
Quadratic Equations
Example: x² + 26x = 27
Ancient mathematicians visualized variables geometrically (e.g., area of squares/rectangles)
Completing the square illustrated how to find solutions visually.
Negative solutions were ignored due to their inconsistency with reality (e.g., length can't be negative).
Development of Cubic Solutions
Omar Khayyam (11th Century)
Identified 19 different cubic equations with positive coefficients.
Sought a general solution but was unsuccessful.
Scipione del Ferro (1510)
Found a method for solving depressed cubics (cubic equations without x² term).
Kept the method secret for job security until his death in 1526 when he revealed it to his student, Antonio Fior.
The Challenge (1535)
Fior challenged Niccolo Fontana Tartaglia to solve depressed cubics.
Tartaglia, who was self-taught and had overcome adversity, solved all problems quickly while Fior couldn't solve any.
Tartaglia developed an algorithm based on geometric interpretations to solve depressed cubics.
Cardano's Breakthrough
Gerolamo Cardano (1539)
Gained Tartaglia's method under oath of secrecy and discovered a way to solve general cubic equations by transforming them into depressed cubics.
Published "Ars Magna" in 1545, which included the general solution to the cubic equation.
Acknowledged contributions of other mathematicians but faced controversy from Tartaglia concerning credit.
Imaginary Numbers and Geometry
While exploring cubic equations, Cardano encountered the square roots of negative numbers, leading to geometric paradoxes.
Rafael Bombelli (16th Century) addressed these negatives by treating them as a new type of number, ultimately leading to the concept of complex numbers.
This was crucial for solving equations like x³ = 15x + 4.
Evolution of Mathematics
17th Century Developments
Francois Viete introduced modern symbolic notation, transitioning from geometric representations to algebraic forms.
Rene Descartes popularized the use of imaginary numbers, coining the term and establishing them within mathematics.
Euler later introduced i to represent the square root of negative one.
Physics and the Schrödinger Equation
Erwin Schrödinger (1925) developed the Schrödinger equation, which incorporates the square root of negative one (i).
This equation describes quantum behavior and integrates complex numbers, highlighting their significance in physical theories.
Schrödinger expressed discomfort with using complex numbers but found they were essential in describing atomic behavior.
Conclusion
Imaginary numbers, initially viewed as abstract, became integral to mathematical and physical frameworks, illustrating how abstraction can lead to significant advancements in understanding reality.
The journey from solving a cubic equation to understanding complex numbers showcases the evolution of mathematics and its profound implications in physics.