One of the ways to avoid imaginary numbers is to use matrices instead. You wouldn't exactly solve sqrt(-1). You'd solve the sqrt of the negative of the 2x2 iden

One of the ways to avoid imaginary numbers is to use matrices instead. You wouldn't exactly solve sqrt(-1). You'd solve the sqrt of the negative of the 2x2 identity matrix (-I) You can't solve sqrt(-1) without a second dimension. So even with imaginary numbers you kind of have an implicit -1+0i when solving sqrt(-1). Complex numbers is just a way of working with two numbers at once, just like 2 dimensional vectors and matrices. There nothing special to it, other than some syntactic shorthands.