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At this level, associating numbers with potentially infinite decimals would not be too bad. The examples of π and √2 are shown as decimals, suggesting that stud
by penteract 5y ago
At this level, associating numbers with potentially infinite decimals would not be too bad. The examples of π and √2 are shown as decimals, suggesting that students are already expected to think something like this.
I think that this pair of definitions for real and irrational numbers are unacceptable - the examples of irrationals mean that they don't mislead anyone (unless you can find someone who thinks of complex numbers as numbers but hasn't encountered the term "real numbers"), but it's wrong to pretend that this is a mathematical definition. Even just removing the words "defined as" would be fine.
- yodsanklai 5y ago> At this level, associating numbers with potentially infinite decimals would not be too bad. I find number with an infinite number of decimals quite an abstract concept. Another issue with this is that it confuses numbers with their representation.
- penteract 5y ago> I find number with an infinite number of decimals quite an abstract concept. If students are familiar with the idea that fractions can be represented as repeating decimals and are introduced to π as a non-repeating decimal, then I think the idea of the set of all possibly repeating decimals should make sense to them. > Another issue with this is that it confuses numbers with their representation. It's probably worth acknowledging that some numbers have multiple representations just to avoid arguments about whether 0.999.. = 1. However, is it much worse to say that a real number is an equivalence class containing 1 or 2 infinite decimals than to say that a real number is a non-empty, downwards closed, proper subset of the rationals? I agree it's nicer to define the reals as a field with some extra properties, but I'm not convinced it's wrong to say that they are their representation.