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I asked my math teacher this when I was a kid. He told me to accept that's the way it is so I did.
Kinda like why 5 rounds up instead of down
https://stats.stackexchange.com/questions/218821/round-to-ev...
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This is not an infinite decimal. The digit 1 is somewhere out there.
Though you could treat it as an infinite series in a similar way: 0.9 -> 0.99 -> 0.999 -> ... -> ? 0.1 -> 0.01 -> 0.001 -> ... -> ?
9/10 + 9/100 + 9/1000 + ... + 9/10^n + ...
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It's more fundamental: People seem to have the intuition that the decimal representation of a number is a number . I don't know if it's because decimals resemble the ntural numbers or what, but decimals seem to have a primacy for people that fractions do not. The idea that there's a gap between the symbol for a thing and the thing itself is the stumbling block.
I think this is a very insightful remark. People think that numerals _are_ numbers, and it's hard to explain why this is not the case, because we have no way to talk about specific numbers _except_ by using numerals. But many frequently-asked questions are based in a confusion between numbers and numerals. For example, many beginner questions on Math SE about irrational numbers are based in the mistaken belief that a…
I think the numerals and numbers issue is more complex because numerals are fundamentally hard to reason about. Even the question "what is a number?" is deceivingly deep.
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I think this is a very insightful remark. People think that numerals _are_ numbers, and it's hard to explain why this is not the case, because we have no way to talk about specific numbers _except_ by using numerals. But many frequently-asked questions are based in a confusion between numbers and numerals. For example, many beginner questions on Math SE about irrational numbers are based in the mistaken belief that a…
>the mistaken belief that an irrational number is one whose decimal representation doesn't repeat ...which is true for any base-n representation where n is a natural (even rational) number. And that's kind of implied most of the time, so it seems like a useful definition. Where would this lead to problems?
Or a rational number whose decimal representation doesn't repeat?
There is no proof that will ever satisfy a person dead-set against this. Ever since I brought this home from school as a child, my whole family ribbed me mercilessly for it. If you tell a person that 3/6 = 1/2, they'll believe you - because they have been taught from an early age that fractions can have multiple "representations" for the same underlying amount. People mistakenly believe that decimal numbers don't hav…
> We cannot imagine infinity Now try imagining that some infinities are bigger than others: https://en.wikipedia.org/wiki/Aleph_number
Although I do understand the concepts presented, the notion of "greater" makes no sense when applied to something without boundaries.
Yet it's used all the time.
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Sure. Instead of "0.99..." please substitute lim n->inf sum(1..n)(9 times 10^-n). The point I'm making is that the "obvious truth" 0.99... = 1 that we're all talking about depends on the assumption that we're working in the real numbers. I claim that the real numbers are not something intuitively obvious to every sufficiently intelligent person; instead they are kind of weird and technical. I go on to claim, though I…
> I go on to claim, though I'm more unsure of this, that the real numbers are not even the only way to make calculus work. That claim is certainly true. The proof is that calculus (Analysis) exists for the complex number system too. Although I don’t think that’s what you meant and I doubt the complex number system is “more intuitive.” Just out of curiosity, have you heard of real analysis? How do you define calculus?
https://en.wikipedia.org/wiki/Nonstandard_calculus
It is based on the hyperreal numbers:
https://en.wikipedia.org/wiki/Hyperreal_number
Practically speaking, I don't think it buys you anything over traditional calculus/analysis. It's just pointing out that there are alternative approaches to formalizing calculus.
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It's more fundamental: People seem to have the intuition that the decimal representation of a number is a number . I don't know if it's because decimals resemble the ntural numbers or what, but decimals seem to have a primacy for people that fractions do not. The idea that there's a gap between the symbol for a thing and the thing itself is the stumbling block.
great point. My thought on (1/3 = 0.3333...) * 3 = 1 = 0.999... was that it is intuitively obvious that the "problem" is that we use base-10 for decimals. There is nothing magic or unknowable about the quantity 1/3. I've often wondered if there is some alternate base or mathematical system entirely that would be "better" in these respects. The thought usually comes up thinking about why pi is such an "ugly" number in…