I think a lot of the problem in mathematics education is motivation. Students (and people learning math in general) encounter two problems: 1. They don’t reali

I think a lot of the problem in mathematics education is motivation. Students (and people learning math in general) encounter two problems: 1. They don’t realize what you’re saying, i.e. number systems and mathematics in general are built from definitions, not observations, and any definitions work as long as they’re compatible with what we’ve already proven, 2. If they realize #1, even implicitly, they still do not know why mathematical definitions are motivated. A student might say, “Okay so I understand what complex numbers are, but why do we have them? What’s the point of all of this?” In my opinion, new mathematical concepts should be taught by placing them in the context of the least complicated thing the student already knows, then motivating them by demonstrating what problem they resolve. In the case of complex numbers, this can be explained to a student via analogue to the irrational numbers, which are the set of all numbers p/q where p and q are positive integers. We “invent” and define irrational numbers to resolve real world problems involving numbers that cannot be in the integral or rational systems. Similarly, we define complex numbers to resolve real world problems involving negative numbers on a plane, and so on...