Comment by nilkn
16 days ago
> I believe real numbers to be completely natural
You can teach middle school children how to define complex numbers, given real numbers as a starting point. You can't necessarily even teach college students or adults how to define real numbers, given rational numbers as a starting point.
well it's hard to formally define them, but it's not hard to say "imagine that all these decimals go on forever" and not worry about the technicalities.
An infinite decimal expansion isn't enough. It has to be an infinite expansion that does not contain a repeating pattern. Naively, this would require an infinite amount of information to specify a single real number in that manner, and so it's not obvious that this is a meaningful or well-founded concept at all.
I don't quite get what you mean here. While you need to allow infinite expansions without repeating patterns, you also need to expansions with these pattern to get all reals. Maybe the most difficult part is to explain why 0.(9) and 1 should be the same, though, while no such identification happens for repeating patterns that are not (9).
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The way I think of it is this:
Imagine you have a ruler. You want to cut it exactly at 10 cm mark.
Maybe you were able to cut at 10.000, but if you go more precise you'll start seeing other digits, and they will not be repeating. You just picked a real number.
Also, my intuition for why almost all numbers are irrational: if you break a ruler at any random part, and then measure it, the probability is zero that as you look at the decimal digits they are all zero or have a repeating pattern. They will basically be random digits.
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What are you talking about? Infinite decimals give reals, do they not? Repeating decimals give rational which are a subset of the reals.
The colloquial phrase 'infinite decimal' is perfectly intelligible without reference to whether it's an infinite amount of data or rigorously defined or whatever else.
There's a lot of trickery involved din dealing with the reals formally but they're still easy to conceptualize intuitively.
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