341–350 of 647 posts
Earlier quoted context omitted.
No, there's no 1. 2OEH8eoCRo0 is exactly right. Subtract 0.999... from 1 and you get 0.000...
0.999... + 0.000...1 = 1 0.000...1 = 1/∞ 0.999... = 1 - 1/∞
It's 0.999... and not 0.999...0
In the same way, it's 0.000... and not 0.000...1.
Earlier quoted context omitted.
That doesn't exist. An open interval doesn't have a smallest number.
Of course it does. It's called the infinitesimal. It's common definition for real number is 1 / infinity: https://en.wikipedia.org/wiki/Infinitesimal If you've taken Calculus, you've already worked with math that requires the infinitesimal to exist. It's not a value you can meaningfully write out, but you can't write out pi, e, phi, root 2, 1 / 3 in base 10, root -1, etc. "I can't write it down" isn't a particularly…
Not at all. Standard calculus uses standard real numbers, for which there is no infinitesimal. One may well speak of infinitesimals as a mental tool when building a mental model for calculus, but those infinitesimals are not actual real numbers (or a well-defined mathematical object at all - in standard calculus).
Earlier quoted context omitted.
The GP's definition is the fundamental definition of the decimal notation. It's exactly what the "0.9..." symbol means. You can redefine the "0.9..." symbol to mean something else as much as you want, you can have it meaning pi if you like, but then you are just changing the subject on the most unhelpful way.
> The GP's definition is the fundamental definition of the decimal notation. It's exactly what the "0.9..." symbol means. > You can redefine the "0.9..." symbol to mean something else as much as you want, you can have it meaning pi if you like, but then you are just changing the subject on the most unhelpful way. I _know_ that. I think I maybe should just bow out of this conversation. I'm apparently incapable of expl…
It’s worth remembering that most people who understand and agree rarely leave a reply.
Earlier quoted context omitted.
>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?
Yeah, can we have an example of an irrational number whose decimal representation repeats or terminates? Or a rational number whose decimal representation doesn't repeat?
Earlier quoted context omitted.
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…
Now sure, make a square using those measures and measure the diagonal, like an awkward mathematician - "see, see, we need irrationals!" - but then we can just cut another measure that's exactly that length ... stupid mathematicians!
Yeah representation is not reality.
Now, says the ratio of those two measuring sticks ...
Earlier quoted context omitted.
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 -> ... -> ?
The first is an infinite series: 9/10 + 9/100 + 9/1000 + ... + 9/10^n + ... The second is not?!
1/10 + -9/100 + -9/1000 + -9/10000 + ...
But I just meant them as series, not necessarily as sums.
Earlier quoted context omitted.
When I was a child I was convinced by a pretty simple conversation with my father: Me: 0.9999... is not the same as 1 Him: Well if it's not the same is it more than 1 or less than 1? Me: Less Him: Okay then how much less is it? At this point I started trying to do 1 - 0.999..., using the methods I'd been taught, and after a few iterations of "borrowing" the 1 I realized the answer was 0.000... which I was pretty conv…
Hehe... smart man. Another one is that 1 / 3 * 3 = 1 0.333... * 3 = 1 0.999... = 1
1/10*10=1, 0.1*10=1