From: veritasium
Mathematics initially served practical purposes like quantifying the world, measuring land, predicting planetary motions, and tracking commerce [00:00:00]. However, the attempt to solve the cubic equation led to a problem considered impossible [00:00:08]. The key to its solution involved separating mathematics from the real world, splitting algebra from geometry, and inventing “imaginary” numbers [00:00:11]. Ironically, 400 years later, these very numbers became central to our best physical theory of the universe, demonstrating that abandoning math’s direct connection to reality could reveal reality’s true nature [00:00:23], [00:00:30].
The Cubic Equation and its “Impossibility”
In 1494, Luca Pacioli, Leonardo da Vinci’s math teacher, published “Summa de Arithmetica,” a comprehensive summary of all known mathematics in Renaissance Italy [00:00:38], [00:00:44]. In this work, he concluded that a general solution to the cubic equation, written today as ax³ + bx² + cx + d = 0, was impossible [00:00:51], [00:01:15]. This was surprising, as many ancient civilizations had solved quadratic equations (which lack the x³ term) thousands of years earlier [00:01:27].
Ancient mathematicians typically derived solutions geometrically, using words and pictures rather than modern equations [00:01:46], [00:01:54]. For example, x² + 26x = 27 could be visualized as a square of side ‘x’ and a rectangle of sides ‘26’ and ‘x’ [00:01:58]. The “completing the square” method involved cutting the rectangle in half and repositioning the pieces to form a larger, almost complete square, which could then be completed by adding a smaller square (e.g., 13x13) to both sides of the equation [00:02:26].
A significant limitation of this geometric approach was its inability to account for negative solutions [00:03:34]. For millennia, mathematicians disregarded negative numbers because they had no real-world geometric meaning, such as a square with sides of length negative 27 [00:03:42], [00:03:54]. This aversion meant there wasn’t a single quadratic equation, but rather six different versions arranged to ensure all coefficients remained positive [00:04:13]. Similarly, in the 11th century, Persian mathematician Omar Khayyam identified 19 different cubic equations, avoiding negative coefficients and seeking graphical solutions through the intersection of shapes like hyperbolas and circles, but he never found a general algebraic solution [00:04:28].
The Breakthrough: Solving the Cubic
The solution to the cubic equation began to take shape in the 16th century in Italy [00:04:53].
Scipione del Ferro’s Secret
Around 1510, Scipione del Ferro, a mathematics professor at the University of Bologna, discovered a method to reliably solve “depressed cubics,” which are cubic equations without an x² term [00:05:00], [00:05:07]. In the 1500s, mathematicians often guarded their discoveries to secure their positions, participating in public “math duels” where job security depended on solving more problems than opponents [00:05:28]. Del Ferro kept his method secret for nearly two decades, only revealing it to his student, Antonio Fior, on his deathbed in 1526 [00:06:02], [00:06:06].
Tartaglia’s Discovery
Fior, lacking his mentor’s talent, boastfully challenged Niccolo Fontana Tartaglia, a self-taught mathematician known for his stutter, to a math duel in Venice on February 12, 1535 [00:06:13], [00:06:26], [00:06:36]. Fior’s 30 problems were all depressed cubics [00:07:09]. Tartaglia, skeptical of Fior’s boast but aware that a solution was possible, set about solving the depressed cubic himself, and succeeded [00:07:30], [00:07:46]. He solved all 30 problems in just two hours, while Fior failed to solve any [00:07:17].
Tartaglia extended the geometric idea of “completing the square” into three dimensions to solve depressed cubics [00:07:56]. For an equation like x³ + 9x = 26, he visualized x³ as a cube’s volume [00:08:02]. By extending the cube’s sides by a distance ‘y’, he created a larger cube [00:08:22]. The additional volume could be broken into seven shapes, six of which (rectangular prisms) could be rearranged into a block representing the ‘9x’ term [00:08:36]. This geometric manipulation led to a new equation (y⁶ + 26y³ = 27) that, when y³ was treated as a new variable, became a quadratic equation [00:09:47], [00:09:59]. Solving this yielded y=1, leading to x=2, a solution to the original cubic [00:10:09].
To avoid repeating the geometry for each new cubic, Tartaglia summarized his method as an algorithm, written not in equations (as modern algebraic notation wouldn’t exist for another century), but as a poem [00:10:32], [00:10:41].
Cardano’s General Solution and “Ars Magna”
Tartaglia’s victory made him a celebrity, attracting the attention of Gerolamo Cardano, a polymath from Milan [00:10:54], [00:11:00]. After persistent correspondence and the promise of an introduction to a wealthy benefactor, Cardano lured Tartaglia to Milan [00:11:10]. On March 25, 1539, Tartaglia revealed his method to Cardano, but only after making Cardano swear a solemn oath not to publish it, or even to write it down except in cipher, “So that after my death, no one shall be able to understand it” [00:11:25], [00:11:30].
Cardano immediately began experimenting with the algorithm and, amazingly, discovered a way to solve the full cubic equation, including the x² term [00:11:48]. He found that substituting x minus b over three a for x would cancel out all the x² terms, transforming any general cubic into a depressed cubic that could then be solved by Tartaglia’s formula [00:11:59].
Eager to publish, Cardano was bound by his oath. However, in 1542, during a visit to Bologna, he saw del Ferro’s old notebook, which contained the original solution to the depressed cubic, predating Tartaglia’s discovery by decades [00:12:43], [00:13:00]. Feeling released from his oath, Cardano published “Ars Magna” (The Great Art) three years later, an updated compendium of mathematics [00:13:15]. He included a unique geometric proof for each of the 13 arrangements of the cubic equation and acknowledged the contributions of Tartaglia, del Ferro, and Fior [00:13:25]. Despite this, Tartaglia was furious, and to this day, the general solution to the cubic is often called Cardano’s method [00:13:36], [00:13:46].
The Emergence of Imaginary Numbers
“Ars Magna” represented a peak of geometric reasoning, pushing it to its breaking point [00:13:51]. While writing it, Cardano encountered cubic equations, such as x³ = 15x + 4, whose solutions, when plugged into the algorithm, yielded square roots of negative numbers [00:14:01], [00:14:03]. When he asked Tartaglia about this, Tartaglia evaded, implying Cardano’s lack of skill [00:14:17].
Cardano’s geometric derivation of a similar problem revealed a paradox: completing the square required adding “negative area,” leading to the square roots of negatives [00:14:30], [00:15:02]. Prior to this, square roots of negatives, as seen in problems like finding two numbers that add to 10 and multiply to 40, were understood as mathematics indicating “no solution” in the real numbers [00:15:08], [00:15:43].
However, the cubic equation presented a unique challenge: x³ = 15x + 4 clearly has a real solution (x = 4), yet Cardano’s method produced square roots of negatives [00:15:51], [00:15:54]. Unable to reconcile this, Cardano avoided this case in “Ars Magna,” calling the idea of square roots of negatives “as subtle as it is useless” [00:16:06].
Bombelli’s Breakthrough
Around 10 years later, the Italian engineer Rafael Bombelli picked up where Cardano left off [00:16:16]. Undeterred by the impossible geometry, Bombelli sought a way through the mathematical “mess” [00:16:22]. Observing that the square root of a negative “cannot be called either positive or negative,” he allowed it to be its own new type of number [00:16:30], [00:16:35].
Bombelli assumed that the terms in Cardano’s solution could be combinations of ordinary numbers and this new type of number involving the square root of negative one. He then found that the two cube roots in Cardano’s equation were equivalent to 2 plus or minus the square root of negative one. When added together, the square roots cancelled out, yielding the correct answer: 4 [00:16:39], [00:16:49], [00:16:57]. This was seen as miraculous: Cardano’s method worked, but only by abandoning the geometric proof that generated it, requiring “negative areas” to exist as an intermediate step to a real solution [00:17:07], [00:17:13].
Modern Mathematics and the Importance of Imaginary Numbers
Over the next century, modern mathematics took shape [00:17:21]. In the 1600s, Francois Viete introduced modern symbolic notation for algebra, ending the millennia-long tradition of math problems described by drawings and words [00:17:25], [00:17:27]. Geometry was no longer the sole source of truth [00:17:35].
Rene Descartes heavily used the square roots of negatives, popularizing them, though he called them “imaginary numbers”—a name that stuck [00:17:39], [00:17:42], [00:17:46]. Later, Euler introduced the letter ‘i’ to represent the square root of negative one [00:17:50]. When combined with regular numbers, they form complex numbers [00:17:55]. The cubic equation led to the invention of these new numbers and liberated algebra from geometry [00:17:59]. By letting go of what seemed like the best description of reality (visible geometry), a more powerful and complete mathematics emerged, capable of solving real problems [00:18:05].
Impact of Imaginary Numbers on Physics
The cubic was just the beginning. In 1925, Erwin Schrödinger was searching for a wave equation to govern quantum particles [00:18:20]. He developed the Schrödinger equation, one of the most important and famous equations in physics, which prominently features ‘i’, the square root of negative one [00:18:31], [00:18:37]. While mathematicians had grown accustomed to imaginary numbers, physicists were uncomfortable with their appearance in such a fundamental theory. Schrödinger himself wrote, “What is unpleasant here, and indeed directly to be objected to, is the use of complex numbers. The wave function Psi is surely fundamentally a real function” [00:18:43], [00:18:51].
Imaginary numbers exist on a dimension perpendicular to the real number line, forming the complex plane [00:19:18]. Multiplying by ‘i’ repeatedly rotates a point by 90 degrees in this plane [00:19:27]. The function e^(ix) creates a spiral in the complex plane, with its real part being a cosine wave and its imaginary part a sine wave [00:19:56]. These two quintessential functions that describe waves are contained within e^(ix) [00:20:12]. Schrödinger used this exponential formulation for his wave equation because its derivatives are proportional to the original function, a property not shared by sine functions, and its linearity allowed for the creation of arbitrary wave shapes as solutions [00:20:26], [00:20:42].
Physicist Freeman Dyson noted, “Schrödinger put the square root of minus one into the equation, and suddenly it made sense. Suddenly it became a wave equation instead of a heat conduction equation” [00:21:10]. The Schrödinger equation correctly describes atomic behavior and forms the basis of chemistry and most of physics [00:21:31], [00:21:38]. The presence of the square root of minus one indicates that nature operates with complex numbers, not just real numbers, a discovery that surprised everyone [00:21:42], [00:21:50]. Thus, imaginary numbers, initially a quirky intermediate step in solving the cubic, proved to be fundamental to our description of reality [00:21:56].