>One fact we all remember about numbers is that squaring a number gives you something non-negative. 7^2 = 49, (-2)^2 = 4, 0^2 = 0, and so on. But it certainly d

>One fact we all remember about numbers is that squaring a number gives you something non-negative. 7^2 = 49, (-2)^2 = 4, 0^2 = 0, and so on. But it certainly doesn’t have to be this way. What if we got sick of that stupid fact and decided to invent a new number whose square was negative? This presentation of complex numbers makes the whole article flawed and useless in my opinion. When I first learned about complex numbers, they were presented in a very similar fashion: let's just invent a number i such that i^2 = -1. This is inane: we have a multiplication operation that we are all familiar with and we know exactly what it does, and then somebody tells us that we can use it on some "imaginary" thing (??) such that it times itself equals -1? How is anybody supposed to make any sense of it? It's like saying: let's invent a "number" j such that j-j=the letter z. What does it even mean? Nothing! it's gibberish, and a similar definition of "i" is also gibberish. We cannot make sense, under our normal understanding of multiplication and under our normal understanding of numbers as including only the real line, of how can something times itself be -1, and neither should we, because there are much better ways to present the whole thing from the beginning. The correct way to present complex numbers is either in the context of abstract algebra - where there is a very obvious question of whether we can embed the real line as a field inside the real plane, or simply present them geometrically without going into fields. It is simply wrong, in my opinion, to present i as something we "invent" so that i^2=-1 (why would anybody do that??), and then go on and say that after we have invented this, there are ways to imagine this geometrically. No! If you want to talk about geometry, then define i geometrically, then extend the definition of multiplication geometrically to the plane, and then it becomes clear that i^2 = -1, and there is no mystery about anything. Edit: it should also be noted that historically, complex numbers didn't come into existence because somebody decided on a whim to "invent" a number i such that i^2=-1. Rather, it was a result of the fact that cubic equations such as x^3=15x+4 clearly had a solution (for example x=4), but using the cubic formula to solve them resulted in weird terms such as sqrt(-121). Bombelli, in the 16th century, decided to try and compute with those terms anyway, and through this process eventually succeeded in producing the right results (x=4), so it eventually became clear that the roots of negative numbers aren't just gibberish: they interacted somehow with the reals, and there was some way to "make them work" to produce real results, though the full realization of what was happening probably came much later.