My favorite mental model for imaginary numbers is that they are placeholders or stepping stones. A speed of (30 + i*10) miles per hour is nonsense, but sometime

My favorite mental model for imaginary numbers is that they are placeholders or stepping stones. A speed of (30 + i*10) miles per hour is nonsense, but sometimes in the course of calculating a nice real 40 mph, you might take a detour through some steps including complex numbers. Having them handy sure will make the derivation easier. Same as negative numbers. I can’t be negative six feet tall, but negative feet (as a magnitude) make the calculations way easier. You can always reformulate these derivations into derivations that don’t use e.g. imaginary numbers, but there’s no need to. If I were apointed Senior Exectutive Math Concept Namer For Humanity, I’d call them something like “algebraic placeholders” or “algebraic closure stepping stones” or something in that vein instead of “imaginary numbers.”