The Italian Origins of Imaginary Numbers

If you happened to be taking a stroll around Bologna or Milan in the mid-16th century, it’s possible you might have stumbled upon a duel. Not the type that was fought with pistols or swords, but an altogether different type: a duel of minds. It was common practice at the time for mathematicians to challenge each other to solve problems in public contests of how smart they were.

One common type of problem that mathematicians would duel over was to find a solution of low-degree polynomials — often cubic polynomials of the form ax³+bx²+cx+d = 0 for certain (usually integers) a,b,c,d. Akin to choosing the right weapon, many such mathematicians would come prepared with their own (secret) general method that would work for certain groups of such equations, in the hope that one of those equations would come up in the contest. One such method that was developed during this time was Cardano’s formula for the solution of a ‘depressed cubic’ of the form x³+px+q=0. Here it is:

Cardano’s formula

It’s a bit unfair to call this formula Cardano’s, because it was actually discovered by Scipione del Ferro — a professor at the University of Bologna. But Girolamo Cardano was the first to commit it to print alongside other methods by Niccolo Fontana (aka ‘Tartaglia’ or 'The Stammerer') when he realized that a simple substitution meant that the formula could be used to find solutions for any cubic equation, and not just a depressed cubic.

To see this, let’s take a general cubic of the form ax³+bx²+cx+d=0, and we can safely divide through by a because a is clearly non-zero or we would not have a cubic. So effectively we can say that the general form of a cubic is actually x³+ax²+bx+c=0. Now we define a new transformed variable, y = x+a/3, so that x = y-a/3. Substituting this into our equation and doing the relevant expansions delivers y³+py+q=0, where

Now we can calculate p and q from any general cubic and use Cardano’s formula to find a solution.

When Cardano published this formula, it caused a scandal in the Italian scientific community at the time, because the original formula was not his and had been told to him in confidence. See my other article here if you are interested in how this scandal played out.

Weird results from Cardano’s formula

Cardano’s formula only helps find one solution to a cubic. We know today that a cubic can have up to three distinct solutions, but only one of these is guaranteed to be real. Lacking an understanding of complex numbers, the mathematicians at the time were not about to go further than finding the single root that came out of Cardano’s formula. Nevertheless, complex numbers were about to make their first appearance in the minds of mathematicians at this time, even though they were not to know it.

Once scholars started playing around with Cardano’s formula, they found that it worked well in some cases but produced incomprehensible results in others.

First let’s look at a case where Cardano’s formula works nicely. Let’s take a simple depressed cubic like x³+6x-2=0, so p=6 and q=-2. Using Cardano’s formula we have a solution as:

But having results come out so neat is actually quite rare in Cardona’s formula, and this led to its usefulness being called into question at the time. For example, let’s try solve x³+3x-36=0. If we substitute in x = 3, we can see easily that it must be a root. But if we put p=3 and q=-36 in the formula, we get:

which doesn’t really look like it should equate to 3. However, observe the following:

and following the same steps with the conjugate reveals that:

So clearly the formula works, but often delivers a very obtuse answer which needs more work to reveal its underlying truth.

The accidental emergence of the imaginary number

Let’s look at another example coming out of Cardano’s formula that baffled the Italians. The depressed cubic x³-15x-4=0 can be seen to be satisfied by x=4. But when we use Cardano’s formula, we obtain the following expression for x:

Now this really stumped people. There was, at the time, no concept for how to deal with square roots of a negative number, and this led to many people declaring Cardano’s formula as useless. But Rafael Bombelli, an algebraist from Bologna, made an interesting observation. He suggested that if we ignore the meaning of the negative square roots and work with them as if they were just abstract objects whose squares equal a negative number, we can still perform manipulations that lead to the truth. Observe the following:

And using a similar expansion of the conjugate, we can come to the conclusion that

Therefore Bombelli observed that if we release our objection to the existence of square roots of negative numbers and allow them to exist as algebraic objects, we have a technology with which to reveal truths that would otherwise be inaccessible.

Of course all of this technology was yet to be developed in the coming centuries, but effectively Bombelli had introduced i = √-1, the imaginary number, to the world.

What did you think about how imaginary numbers came to exist as algebraic objects? Feel free to comment.