Need for imaginary numbers and the role of outsiders
After watching this video by Veritasium. I realized that geometry was the origin of all of algebra, at least a lot of advancements to it. For eg: Let’s take a simple Quadratic Equation x^2 + x = 20, anyone can apply the quadratic formula to solve it but before that, it was Geometry that solved such problems ( I highly recommend you to watch the video to understand how it was done, my notes below might not help you imagine it as clearly as was done in the video)
Let’s try to do it together We will imagine a square of side x and add a rectangle of one side equal to 1 and the other equal to x. Now the total area is equal to x^2 + 1x, we already know that this is equal to 20. Let’s cut the rectangle in half and place the part perpendicular to the square. The side that was 1 is not half. So each side now becomes x+0.5, there is a new square area that we add to this figure, which is of side 0.5. If we add this area to both side of the equation then we get (x+0.5)^2 = 20 + 1/4, which makes is (x+0.5)^2 = (9/2)^2. Which is easy to solve and we get x = 4
For me, the larger point of the video was that Geometry and Classical Mechanics are very intuitive subjects as they are based on reality. But when it comes to imaginary numbers or relativistic mechanics, the lack of examples that you can see with your naked eyes makes them harder to understand. For eg: It is quite easy to imagine matter (anything that occupies space, has mass or volume is matter), but when it comes to dark matter our imagination starts falling apart. On the same lens, it is easy to imagine a square of 16 sq meters, but how do you imagine a square of -16 sq meters? Since mathematicians were using geometry to solve these algebraic equations, they got stuck while solving a cubic equation that required them to imagine a square that had a negative area, which became a hurdle. This is where the concept of imaginary numbers comes to the solution, imagine a number whose square was equal to -1 and denote it with a separate symbol, i. Now you are still able to solve all your equations.
That answers only half of the title, what about outsiders and their role?
So if you have watched the video (which I highly recommended) you would see that Mathematicians in the past had a job that involved status, they used to participate in duels with other mathematicians where the loser of the duel would lose his/her job. Because of this incentive, even when a mathematician found a solution (here the solution to the cubic equation), he would try to keep it a secret, just so that if he is challenged in a duel, he could use it to defend his job.
The person who published or popularised the method of solving cubic equations first, Cardano, was not a mathematician by profession, but a physician. So his incentives (means of earning bread) weren’t impacted negatively by publishing the solution. In a way, he had more room to experiment or play compared to other mathematicians. An outsider published one of the most important methods of solving equations, which opened the doors for other interesting ideas (one among them being imaginary numbers).
The modern equivalent of this would be open source contributors, open source statisticians/scientists, whose incentives do not lie in publishing a paper, but in sharing actionable insights. Outsiders usually have more room to experiment and explore the edges of a domain, as their cost of failure is low and the upside is uncapped.