The Surprising Discovery of Imaginary Numbers
A general solution to the cubic equation was long considered impossible, until we gave up the requirement that math reflect reality.
The Impossible Problem of the Cubic Equation
Mathematics began as a way to quantify our world, to measure land, predict the motions of planets, and keep track of commerce. Then came a problem considered impossible: the cubic equation.
For thousands of years, ancient civilizations from the Babylonians to the Persians tried to find a general solution, but all came up empty-handed.
In 1494, Luca Pacioli, the math teacher of Leonardo da Vinci, published a comprehensive summary of all mathematics known in Renaissance Italy at the time.
In it, he concluded that a solution to the cubic equation was impossible. This should have been at least a little surprising since without the x-cubed term, the equation is simply a quadratic, which many ancient civilizations had solved thousands of years earlier.
The Limitations of Ancient Geometry-Based Mathematics
Back in those days, mathematics wasn’t written down in equations. It was written with words and pictures, using geometric reasoning.
For example, to solve the quadratic equation x² + 26x = 27, ancient mathematicians would think of the x² term as a literal square with sides of length x, and the 26x term as a rectangle with one side of length 26 and the other side of length x.
They could then use the technique of “completing the square” to find the solution.
However, this geometric approach had its limitations. Mathematicians were so averse to negative numbers that there was no single quadratic equation — instead, there were six different versions, all arranged so that the coefficients were always positive.
The same approach was taken with the cubic equation, with Persian mathematician Omar Khayyam identifying 19 different cubic equations, again keeping all coefficients positive.
The Breakthrough: Solving the Depressed Cubic
The solution to the cubic equation began to take shape 400 years later and 4,000 kilometers away, in the 16th century. Scipione del Ferro, a mathematics professor at the University of Bologna, found a method to reliably solve a subset of cubic equations known as “depressed cubics” — those with no x² term.
However, he kept his discovery a secret, fearing that other mathematicians would challenge him for his position.
Del Ferro’s student, Antonio Fior, eventually boasted about his own ability to solve the depressed cubic, leading to a mathematical duel with the self-taught mathematician Niccolo Fontana Tartaglia. Tartaglia solved all 30 of Fior’s problems in just two hours, and then went on to develop his own algorithm for solving the depressed cubic, which he expressed as a poem.
Cardano and the Invention of Imaginary Numbers
Gerolamo Cardano, a polymath based in Milan, was desperate to learn Tartaglia’s method. After much persistence, Tartaglia revealed the algorithm to Cardano, but only after forcing him to swear a solemn oath not to tell anyone the method, not to publish it, and to write it only in cipher.
However, Cardano later discovered that the solution to the full cubic equation, including the x² term, could be derived from Tartaglia’s formula. He was so excited to have solved the problem that had stumped the best mathematicians for thousands of years that he wanted to publish it, even though it would violate his oath to Tartaglia.
In his publication “Ars Magna” (The Great Art), Cardano included a unique geometric proof for each of the 13 arrangements of the cubic equation. But he came across some cubic equations that couldn’t be easily solved using the usual method, like x³ = 15x + 4. Plugging this into the algorithm yielded a solution that contained the square roots of negative numbers, which Cardano called “as subtle as it is useless.”
The Breakthrough of Imaginary Numbers
Around 10 years later, the Italian engineer Rafael Bombelli picked up where Cardano left off. Undeterred by the square roots of negatives and the impossible geometry they implied, Bombelli let them be their own new type of number, which he called “imaginary.” He realized that the two cube roots in Cardano’s equation were equivalent to two plus or minus the square root of negative one, and that when he added them together, the square roots canceled out, leaving the correct answer.
Over the next hundred years, modern mathematics took shape. Geometry was no longer the source of truth, and the square roots of negatives, now called “imaginary numbers,” became an integral part of the new algebraic notation introduced by Francois Viete and Rene Descartes.
Imaginary Numbers in Quantum Mechanics
The discovery of imaginary numbers was just the beginning. In 1925, Erwin Schrödinger was searching for a wave equation that governs the behavior of quantum particles, and he found that the square root of negative one, or the imaginary number i, was a fundamental part of his equation.
While physicists were initially uncomfortable with this, it turned out that the unique properties of imaginary numbers were essential for describing the wave-like behavior of matter at the quantum level.
As physicist Freeman Dyson wrote
“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. And Schrödinger found to his delight that the equation has solutions corresponding to the quantized orbits in the Bohr model of the atom. It turns out that the Schrödinger equation describes correctly everything we know about the behavior of atoms. It is the basis of all of chemistry and most of physics. And that square root of minus one means that nature works with complex numbers and not with real numbers. This discovery came as a complete surprise, to Schrödinger as well as to everybody else.”
Only by giving up mathematics’ connection to reality could it guide us to a deeper truth about the way the universe works. The invention of imaginary numbers, born out of the quest to solve the seemingly impossible cubic equation, ultimately led to our modern understanding of quantum mechanics, the foundation of much of our modern technology and scientific understanding.
