Historically, the complex numbers were discovered by accident while solving cubic equations using the general formula. The general formula only works in general

Historically, the complex numbers were discovered by accident while solving cubic equations using the general formula. The general formula only works in general if you can take the square root of a negative number. One of the first attempts to provide a geometric meaning to the complex numbers was by John Wallis, and I haven't been able to make much sense of it. I suspect he didn't see it the way we do. Also, there's no indication that in spite of the work Euler did on the complex numbers, that he knew of their geometric meaning. The mystical sounding name "imaginary number" was coined by Descartes in the 1600s at least partly because he didn't have the modern view of them. Of course, teaching by explaining historical developments is not a common thing in mathematics, and the above facts illustrate why. But you have to be aware that things do start off being mysterious before they're fully understood. The mystery has been somewhat reawakened with the quaternions and octonions, and some other hypercomplex number systems. And mystery gets some people out of bed, so don't be too hard on it. > It turns out that physical problems are much more accurately modeled in 2 dimensions than in 1 dimension, and even more so in higher dimensions. Are you talking about matrices?