Why Imaginary Numbers Exist
A Brief History of i
Imaginary numbers have long been a source of wonder for me.
Take i, which is defined by the following equation.
This is curious. We know from school that a negative times a negative makes a positive. Also, a positive times a positive makes a positive. Lastly, zero times zero equals zero. So how can you multiply a number by itself and get a negative number? Does this actually serve any purpose?
First, think of negative numbers
At one point in your childhood, you learned about addition and subtraction, perhaps through an example like this:
If Ramona has 5 apples and she gives 2 apples to Sonja, how many apples does Ramona have left?
The answer is 3, and indeed 5 minus 2 equals 3.
Unfortunately, this kind of reasoning only works when you’re subtracting a smaller number from a bigger number.
If Ramona had 5 apples and wished to give Sonja 8, then the logic falls apart.
Taking 8 away from 5 only makes sense once you’ve wrapped your head around negative numbers.
For much of history, mathematicians believed in no such thing. To them, the concept of a negative number was as absurd as the concept of a line with negative length, or a shape with negative area.
Negative numbers are abstract, but they help us solve all kinds of problems that we once thought were unsolvable, like the problem of 5 minus 8.
The story of imaginary numbers is similar.
The History of Imaginary Numbers
In Renaissance Italy, it was common for mathematicians to engage in math duels to show off their intelligence. Kind of like drag queens lipsyncing for our lives. Two mathematicians would give each other a list of equations to solve, like this one:
At the time, such an equation (called a depressed cubic) was thought to be unsolvable. Only a handful of people knew the trick to solving them. Those who figured out the formula kept it close to their chests, keeping it like a trade secret in order to win more duels, only passing it down to their protégés on their deathbeds.
One man named Gerolamo Cardano felt the world deserved to know this formula. He burned some serious bridges by publishing it! But thanks to him, we have a solution:
There’s just one big problem with this formula. In many cases, it can lead to taking the square root of a negative number. Take for instance the following equation:
Here, we take m = -15 and n = -4 and plug that into Cardano’s formula, and we get the solution:
Cardano saw the square root of -121 here and thought that this solution was a dead end. But in fact, this equation does have a solution! One of them is x = 4. Simply plugging it in will confirm this:
One Italian mathematician named Rafael Bombelli demonstrated how this could be the case. Using some simple algebra, he showed that:
And similarly,
Therefore, the solution that Cardano cooked up using his equation actually simplifies to 4:
(The square roots of -1 simply cancel each other out, leaving 2+2 = 4).
Taking the imaginary leap
All of this meant that if you were willing to treat these square roots of negative numbers as regular numbers that you could multiply, add and subtract, then you could use them to solve actual math problems.
The only problem was that you had to leave behind the geometric interpretation of numbers as representing lengths, areas, and volumes and embrace the abstract world of negatives and imaginary numbers, which historical mathematicians were highly hesitant to do.
It is due to the vicissitudes of history that we have given them the name imaginary, while calling other numbers real, but don’t let the name confuse you. Imaginary numbers exist just as much as negative numbers, fractions, and whole numbers do.
I like to describe imaginary numbers as queer, in the sense that they are peculiar, strange, curious, and represent a diversion from the norm. Difficult as they may be to accept, embracing their existence expands our mathematical toolkit, allowing us to solve new kinds of problems, and opening up new branches of scholarship like complex analysis and Fourier analysis. To those struggling to wrap their heads around it, I say, why not simply take the imaginary leap?
(This blogpost was adapted from Chapter 2 of my book, Math In Drag).