Early Developments

At the core of this story are polynomial equations, which have fascinated scholars since ancient times and continue to be vital in contemporary algebra.

The Babylonians were adept at solving quadratic equations, expressed as ax² + bx + c = 0, employing these techniques for area calculations. Today, these formulas and their proofs are standard curriculum in high schools and even some elementary schools globally.

Over two millennia later, the Persian mathematician Omar Khayyam made the initial strides toward solving the cubic equation, represented as ax³ + bx² + cx + d = 0.

Khayyam was an exceptional mathematician whose contributions included devising a calendar that measured the year to 365.24219858156 days, remarkably close to today's measurement of 365.242190 days.

He approached cubic equations geometrically, leveraging conic sections to identify positive solutions, although he surely yearned for algebraic methods to do so.

Renaissance Europe

It would take nearly 400 years before an algebraic formula for the cubic equation emerged. By the 1400s, the mathematical spotlight had shifted back to Europe, now equipped with Arabic numerals and algebraic strategies.

Previously, Europe favored Roman numerals for calculations, but the practicality of Arabic numbers soon became clear. For example, which is easier: LXX+XXX = C or 70 + 30 = 100?

The Church, however, clung to Roman numerals, and during a dark epoch, the usage of Arabic numerals was banned in Italy. This likely stemmed from cultural tensions; only foreign traders employed this "dark magic" for swift calculations, leading to trust issues with local customers.

Fortunately, through logic and the influence of early pioneers like Fibonacci, Arabic numerals ultimately triumphed. A couple of centuries after Fibonacci popularized this notation in Europe, the mathematical community was ready to tackle the cubic equation algebraically.

It is important to note that while Arabic numerals had reached Europe prior to Fibonacci, they had not gained widespread acceptance until later.

Before proceeding, we must consider how mathematicians of that era engaged with equations. Lacking knowledge of negative numbers, they always represented equations with positive coefficients.

Consequently, they categorized polynomial equations. For instance, the quadratic equation ax² + bx + c = 0 had three associated forms:

• ax² + bx = c
• ax² = bx + c
• ax² + c = bx

Here, a, b, and c are positive values.

Mathematicians at that time were also unaware of the number 0, necessitating special rules for equations with missing terms. Each category had its own method for solutions.

This was also the case for cubic equations, with Khayyam reportedly identifying 14 distinct equation classes, each with unique geometric approaches.

The Quest for the Cubic Formula

The first to derive a formula for a cubic equation class was Scipione del Ferro, a mathematician at the University of Bologna. In the early 1500s, he formulated a solution for cubic equations structured as x³ + bx + c = 0 where b and c are positive.

If del Ferro were to discover his formula today, he would likely publish it immediately. However, during that time, discovering a formula was akin to finding a secret recipe for gold.

This secrecy stemmed from the competitive nature of securing academic positions in Renaissance Europe. Candidates would engage in public mathematical duels, posing problems to one another, and the one solving the most questions would win the position. Candidates also had to demonstrate their ability to solve their own problems, sometimes even challenging the incumbent professor.

Understandably, any new formula could serve as a secret weapon in these duels, ensuring a successful career as a mathematics professor.

Del Ferro never revealed his findings publicly but did share them with his successor Annibale della Nave and his student Antonio Maria Fior before passing away.

Equipped with this knowledge, Fior believed he could easily outmatch any mathematician, and in 1535, he challenged the established mathematician Niccolo Tartaglia. This proved to be an unwise decision.

Tartaglia, quick-witted, discerned the pattern in Fior's questions, which predominantly related to the cubic equation that del Ferro had resolved. He deduced the existence of a general method for solving it.

Working diligently, Tartaglia rediscovered del Ferro's formula just before the duel and emerged victorious.

The duel garnered significant attention, leading Tartaglia to receive a request from the esteemed Gerolamo Cardano to examine the formula. Initially reluctant, Tartaglia eventually conceded, recognizing Cardano's influential status and connections. They agreed on the condition that Cardano would not publish the formula.

Privately, Cardano began exploring solutions for other cubic equation classes and, after several years, had addressed all forms of cubic equations.

Traveling to Bologna, Cardano examined del Ferro's original notes. Rumors suggested that del Ferro had been the true discoverer of the solution, not Tartaglia, and Cardano confirmed this was indeed the case.

Ultimately, Cardano opted to publish despite his prior commitment to Tartaglia. In the meantime, he persuaded his brilliant student, Lodovico Ferrari, to share his newfound formula for solving quartic equations.

In 1545, Cardano's acclaimed work Ars Magna was released, presenting not only Tartaglia's formula and its variations but also Ferrari's quartic solution.

Although Tartaglia's contributions were acknowledged, he was infuriated. Cardano managed to convince Ferrari to support him, and in 1548, Ferrari triumphed over Tartaglia in another mathematical duel.

Ars Magna is often regarded as the first major advancement in mathematics in Europe since ancient times.

The accomplishments of these early mathematicians are remarkable, especially considering their challenging working conditions. They lacked knowledge of negative numbers and zero, and had to express their formulas in words, similar to a recipe. They also did not have coordinate systems to relate to their polynomial equations.

The Emergence of Complex Numbers

Today, we understand solutions to polynomial equations as the points where the graph intersects the x-axis. However, mathematicians of that era lacked this geometric insight.

If we translate their verbal descriptions into formulas, we discover intriguing results. One of Cardano's findings, derived from Ferro's and Tartaglia's work, relates to the positive solutions of cubic equations represented as x³ - ax - b = 0. Such equations invariably have a positive solution, outlined by Cardano’s formula:

Here, a and b are positive constants.

For instance, consider the equation x³ - 15x - 4 = 0. Applying Cardano’s formula yields:

This reveals "forbidden" square roots of negative values. At the time, mathematicians struggled to interpret these results. Cardano recognized that the expression should yield a positive solution, specifically x = 4. So, what role do these peculiar symbols play in his formula?

Rafaello Bombelli demonstrated that:

and

When substituting these expressions back into Cardano's formula, the troublesome square roots cancel out, confirming x = 4 as expected.

Mathematicians of that era quickly recognized that if they could treat these "alien" numbers as real, Cardano's formulas could yield genuine positive solutions for cubic and quartic equations.

Remarkably, these "alien" numbers consistently canceled out, revealing the true solutions. Many mathematicians regarded these unconventional square roots as illegitimate and only employed them when absolutely necessary, often dismissing them as dubious mathematics.

They feared that utilizing these numbers could lead to inconsistencies and contradictions in their calculations.

Despite attempts to suppress their use, these numbers persisted until the 18th century when the brilliant Leonhard Euler linked them to function theory.

Euler, however, lacked a geometric understanding of complex numbers and was uncertain of their significance. He believed in the validity of the fundamental theorem of algebra (which states that n-degree polynomials have n complex roots, counting multiplicities) but failed to provide a convincing proof.

Numerous mathematicians endeavored to prove this theorem unsuccessfully until Carl Friedrich Gauss achieved a formal proof.

By the late 18th century, these "alien" numbers were validated as legitimate entities, thanks to the contributions of the esteemed Norwegian-Danish mathematician Caspar Wessel. He revealed that complex numbers existed in a dimension beyond the comprehension of mathematicians at the time.

These "alien" numbers, now known as complex numbers, are integral to our number system and have been shown to be consistent with real numbers. In fact, real numbers are now regarded as a subset of complex numbers.

Final Reflections

The foundational development of complex numbers was fueled by competition, economics, religion, and culture.

In mathematics, experimentation is sometimes necessary, as noted by mathematician Edward Frenkel. Euler adeptly utilized complex numbers, significantly influencing the field.

Fortunately, mathematicians like Wessel, Gauss, Cauchy, Riemann, Weierstrass, and others established a robust foundation for complex numbers, demonstrating their necessity in solving problems involving real and whole numbers.

Riemann, for example, showcased their utility in studying prime numbers, and today we recognize their importance in quantum physics, radar technology, and beyond.

Thus, complex numbers are just as real as negative numbers, which were also once deemed alien by Cardano and his contemporaries.

This narrative was inspired by Norwegian mathematician Tom Lindstrøm. Thank you, Tom, for breathing life into this subject!

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