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The phrase “what IS it” has always bugged me in mathematics. The answer to that is always “what we defined it to be.” The question math answers is “what does it
by gaze 9y ago
The phrase “what IS it” has always bugged me in mathematics. The answer to that is always “what we defined it to be.” The question math answers is “what does it do.”
- dsacco 9y agoI 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...
- roywiggins 9y agoThe definition of complex numbers is not arbitrary, though. It falls out of the way real numbers work, and it's pretty much predetermined once you start investigating "impossible" roots of polynomials. Interesting questions are 1) why do we care about polynomials and 2) how do polynomial roots motivate complex numbers.
- Retra 9y agoWell any reasonable person will know that those are the same question.