Cómo se Inventaron los Números Imaginarios

Summary

This script narrates the intriguing history of mathematics, from its origins in quantifying the physical world to the discovery of 'imaginary' numbers, which unexpectedly became central to modern physics. It recounts the ancient struggle to solve cubic equations, the secretive development of solutions, and the eventual integration of complex numbers into physics, exemplified by Schrödinger's equation. The story illustrates how the abstract nature of math, once disconnected from reality, unveiled profound truths about the universe.

Takeaways

• 📏 Mathematics began as a way to quantify and measure the physical world, including land, planetary movements, and commerce.
• 💡 The separation of algebra from geometry and the invention of 'imaginary' numbers were key to solving previously intractable problems and understanding the nature of reality.
• 📚 In 1494, Luca Pacioli, Leonardo da Vinci's math professor, published 'Summa de Arithmetica', a comprehensive summary of all known mathematics at the time, including a section on cubic equations.
• 🧩 Ancient civilizations struggled to find a general solution to cubic equations for over 4,000 years, with each ending up empty-handed despite solving quadratic equations much earlier.
• 🔍 Negative numbers and solutions were ignored by ancient mathematicians because they dealt with the real world's tangible quantities like length, area, and volume, where negative values didn't make sense.
• 🤓 Omar Khayyám, a Persian mathematician in the 11th century, identified 19 distinct cubic equations and found numerical solutions for some by considering intersections of shapes like hyperbolas and circles, but did not find a general solution.
• 🔑 Scipione del Ferro, a professor from the University of Bologna, found a method to reliably solve reduced cubic equations around 1510 but kept it a secret to secure his job for almost two decades.
• 🤝 Niccolò Fontana Tartaglia, after defeating del Ferro's student, Antonio Fior, in a mathematical duel, developed a method to solve reduced cubic equations by exploring the idea of completing the cube in three dimensions.
• 📖 Gerolamo Cardano, after promising not to reveal Tartaglia's method, eventually found a solution for the complete cubic equation and published it in 'Ars Magna', breaking his promise to Tartaglia.
• 🔄 The concept of square roots of negative numbers, initially seen as a mathematical curiosity, was later embraced by engineers like Rafaelle Bombelli, who found a way to use them to solve cubic equations.
• 🌌 The 'imaginary' numbers, which were once considered a mathematical abstraction, are now central to our best physical theory of the universe, as seen in Schrödinger's equation describing quantum behavior.

Q & A

How did mathematics initially start and what was its purpose?

Mathematics initially started as a way to quantify the world, measure land, predict the movement of planets, and record commerce.

What was the problem in mathematics that was once considered unsolvable?

The problem considered unsolvable was the separation of algebra from geometry, which led to the invention of new numbers called 'imaginary numbers'.

Why were imaginary numbers ironically central to our best physical theory of the universe 400 years later?

Imaginary numbers, once thought to be purely theoretical, were found to be central to our understanding of the universe because they allowed for the abstraction of mathematics from the real world, enabling the discovery of the nature of reality.

What significant work did Luca Pacioli publish in 1494, and what was its importance?

Luca Pacioli published 'Summa de Arithmetica', a comprehensive summary of all known mathematics at that time in Renaissance Italy, which included a section on cubic equations.

Why was finding a general solution to the cubic equation challenging for ancient civilizations?

Finding a general solution to the cubic equation was challenging because each ancient civilization that encountered it ended up with empty hands, unable to solve it despite having solved quadratic equations thousands of years prior.

How did ancient mathematicians approach solving quadratic equations without algebraic notation?

Ancient mathematicians approached solving quadratic equations by visualizing them geometrically, using words and drawings to represent the equations, such as thinking of 'x squared' as an actual square with sides measuring 'x'.

Why were negative numbers and coefficients initially rejected by mathematicians?

Negative numbers and coefficients were initially rejected because they did not correspond to the real-world quantities of length, area, and volume that mathematicians were accustomed to working with.

Who was Omar Khayyám, and what was his contribution to the cubic equation?

Omar Khayyám was a Persian mathematician in the 11th century who identified 19 distinct cubic equations and found numerical solutions for some by considering intersections of shapes like hyperbolas and circles, although he did not find a general solution.

What was the significance of Scipione del Ferro's method for solving reduced cubic equations?

Scipione del Ferro's method for solving reduced cubic equations was significant because it provided a reliable way to solve a type of cubic equation that had stumped mathematicians for millennia.

How did Gerolamo Cardano contribute to the solution of the general cubic equation?

Gerolamo Cardano contributed by finding a solution to the general cubic equation by transforming it into a reduced cubic equation using a substitution method, which could then be solved using Tartaglia's formula.

What was the role of Rafaelle Bombelli in the understanding of complex numbers?

Rafaelle Bombelli played a crucial role in understanding complex numbers by providing a method to solve cubic equations involving the square root of negative numbers, thus demonstrating the practical utility of imaginary and complex numbers.

How did the cubic equation lead to the invention of complex numbers?

The cubic equation led to the invention of complex numbers as mathematicians like Cardano and Bombelli found that introducing the square root of negative one allowed for the solution of equations that were previously intractable.

What is the significance of the Schrödinger equation in physics, and how does it relate to imaginary numbers?

The Schrödinger equation is significant in physics as it governs the behavior of quantum particles and is fundamental to quantum mechanics. It relates to imaginary numbers because the equation uses the imaginary unit 'i', which is the square root of -1, to describe wave functions.

Why did Schrödinger's use of imaginary numbers in his equation initially cause discomfort among physicists?

Schrödinger's use of imaginary numbers initially caused discomfort because they were not accustomed to these numbers appearing in a fundamental physical theory, and they seemed to lack a clear physical interpretation.

How did the understanding of complex numbers evolve from being an intermediate step in solving cubic equations to being essential in describing reality?

The understanding of complex numbers evolved as mathematicians and physicists realized their unique properties, such as rotation in the complex plane, which allowed for a more complete and powerful mathematics that could solve real-world problems and describe the fundamental behavior of the universe.