I have a claim, and I'm trying to figure out whether I believe it or not.
When learning fractions, it's tempting to treat it as a two-part number, but that's a mistake. Really, a fraction is a single number. It's exactly the opposite case with complex numbers. Really, they're two-part numbers, though it's tempting to see it as a single number.
Is this true? What sort of evidence would support or contradict this claim?
You could treat both fractions and complex numbers as ordered pairs (i.e. two-part numbers). Fractions with the same value but different representations (e.g. 1/2 and 2/4) are then just different points on a line that goes through the origin.
ReplyDeleteI think I prefer the holistic view that fractions and complex numbers (and real numbers) are just "numbers". It's important to note that one cannot just distribute arithmetic operations over individual parts of a fraction or complex number, but depending on what kind of "number" you have, you may need to do things in a slightly different or novel way. I wouldn't imagine this to be any different from the kind of abstraction needed when one first learns algebra and manipulating algebraic expressions.
To me, that correction that you have to make for equivalent fractions points to their fundamental one dimensionality.
DeleteI'm with you on this one Michael. I think of them - along with vectors - as an inherently visual thing. I see direction and length, I see horizontal change and vertical change wrapped up in one number that has two characteristics.
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