Last time, we decided to let the square
root of minus 1 be its own new type of number, hoping it would help us solve Cardan’s problem. Doing so is helpful, but finding a solution requires one more insight from our friend, Rafael Bombelli. Bombelli knew that because of the way cubics are shaped, our function had to have a solution that didn’t involve the square root of minus 1. It had to be a plain old regular positive or negative number as he had seen before. The second big insight here was that for this to be the case, the root of minus 1 parts of each half of the equation must cancel out when added together. Bombelli use this idea to equate the two parts of the equation to a+b√(-1) and a-b√(-1), where a and b are constants that we need to find. To solve, we can first eliminate that annoying cube root by cubing both sides. The result is a particularly tough system of equations. Bombelli was able to get around this through some clever guessing and checking. If we look at our original equation and test a few integers, we eventually see that 4 is a solution. If we substitute 4 into our new equations, we can solve for a and b and obtain a=2 and b=1. These values make the two parts of our expression equal to 2+√(-1) and 2-√(-1). We can cube these and show that they are in fact equivalent to what we started with. And more importantly, when we add the two parts as the formula tells us to do, we just get 4, which we know is a solution to our original equation. We have found the solution to Cardan’s problem. And what’s really interesting is that our problem had nothing to do with the square root of minus 1 and neither did our answer. However, along the way, we found that by extending our number system to include the square root of minus 1, we were able to find a solution. And it turns out that extending our number system in this way is helpful in lots and lots of other problems as well. So what did Bombelli do to celebrate after discovering a number so crucial to the future of science and mathematics? He actually did nothing. He just scattered
his discovery and basically said it was a hack. As ridiculous as that might seem now, Bombelli drew a pretty reasonable conclusion at the time. It just seemed a little too convenient, like a trick devised just to solve problems like this. Squaring numbers had up until that point largely been associated with what the operation is named for: squares. A square’s area is equal to the length of its side squared, so positive areas makes sense. But what could a negative area be? What even is the square root of minus 1? Questions like these slow down the development of imaginary numbers. It turns out there is a much deeper and richer meaning lurking below the surface, but it would take long after Bombelli’s death to be revealed.