Origin of Complex Numbers

 by Rukhsar Naguman

A complex number is any number in the form of a+bi (a being the real part and bi the imaginary part) where a and b are real numbers and i is the square root of -1. When talking about the history of complex numbers, it is impossible to not mention cubics. In 1494, Luca Pacioli published “Summa de Arithmetica” in which there is a section for cubics. For over 4,000 years, mathematicians from several civilisation have been trying to find a general solution to cubics but to no avail. So, Pacioli concluded that a solution for cubics is impossible. To many this was surprising as quadratics had been solved thousands of years earlier. In the early days, mathematics was visual so when solving a quadratic, the x2 term was imagined as a literal square with side lengths of x. Because of this, Mathematicians were oblivious to the negative solutions of quadratics which to them made sense as length cannot be a negative number. Mathematicians were so opposed to negative numbers that rather than having a single quadratic equation, they came up with 6 different versions of the quadratic equations arranged in such a way that the coefficients were positive, and this was the same approach applied to cubics.

Around 1510, Sipione del Ferro found a method to solve depressed cubics (a cubic without the x2 term). Now, because of the secretive nature of mathematicians in the early days, we cannot tell if he was the first person to do so. It could possibly be that a mathematician early on before del Ferro had identified a method to solve depressed cubics but kept it a secret for the sake of keeping their job safe due to threats from other mathematicians. Del Ferro was no different. After his marvellous discovery, he kept the solution to himself for over two decades. However, on his deathbed, he slipped the solution to Antonio Fior – a student of his.

After del Ferro’s death, Fior challenged Mathematician Niccolo Fontana Tartaglia giving him 30 problems all of which were depressed cubics and in return Tartaglia gave him a very discernment of 30 problems. Tartaglia managed to solve all 30 depressed cubics in just under 2 hours. He did this by extending the idea of completing the square into three dimensions and summarised his method as a set of instructions in the form of a poem. Tartaglia’s achievement caught the attention of multiple mathematicians who were desperate to learn as to how he solved the cubic, especially Gerolamo Cardano who lured Tartaglia to finally revealing his method but only after Cardano swore a solemn oath not to tell anyone the method. Using Tartaglia’s method, Cardano discovered a solution to the full cubic equation but was unable to publish it due to the oath he swore to Tartaglia.

In 1542, Cardano visited a mathematician in Bologna who happened to be the son-in-law of Scipione del Ferro. He found the solution in del Ferro’s old notebook which predates Tartaglia’s by decades. So, in Cardano’s eyes, he could finally publish his discoveries without violating his oath to Tartaglia. Three years later, he published “Ars Magna” in which there is a chapter solely focused on the unique geometric proof for each of the 13 arrangements of the cubic equation and in which he acknowledges the contributions of Tartaglia, del Ferro, and Fior. But to this day, the general solution of a cubic is often referred to as Cardano’s method. In the chapter, Cardano comes across some cubic equations that cannot be solved in the usual way using his method, the solutions come out with roots of negative numbers. When looking at the geometric derivation of a similar quadratic equation, he finds that the final quadratic completing the square step leads to a geometric paradox where he finds a part of a square that must have an area of 30 but with side lengths of 5. So a negative area of 5 must be added since the full square has an area of 25. This did not make sense, which leads to Cardano avoiding these cases in “Ars Magna”.

Around 10 years later, Rafael Bombelli – an Italian engineer - picks up where Cardano left off trying to find the solutions despite the mess of square roots of negative numbers. He lets the square root of a negative be its type of number after observing that it “cannot be called either positive or negative”. He assumes the two terms in Cardano’s solutions can be represented as some combination in the form of an ordinary number (the real part) and a new type of number involving the square root of a negative one(now known as the imaginary part of a complex number). And using this idea and Cardano’s method, he finds the solutions to the cases that Cardano avoided. Ironically, the geometric side of Cardano’s method which generated it in the first place needed to be left out when solving this set of equations.

In modern mathematics, the letter “i” is used to represent the square root of a negative one. And combined with real numbers, they form complex numbers. The cubics led to the invention of these new types of numbers (Imaginary and Complex Numbers). Imaginary numbers appear in a lot of real-life problems, one of the most famous being the wave equation in which the square root of negative one “i” is featured suggesting that to understand the reality of nature, we need to push our understanding beyond reality itself.