Earlier quoted context omitted.
Why not 0.00...1?
What is 0.00...1 times 34?
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0.999...= 1
331–340 of 647 posts
Re: 0.999...= 1
#332Earlier quoted context omitted.
I find the algebraic way convinces most people: x = 0.9999.. 10x = 9.9999... 10x - x = 9 9x = 9 x = 1
black magic! I wonder if there are any programming languages that are able to handle this properly?
There are programming languages with rational number types, but none that I know of that represent numbers as repeating decimals.
Re: 0.999...= 1
#333Earlier 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…
Base 12 is a better base overall. It is divisible by more numbers. That is why it is used in various monetary, time and measurement systems.
It's the one issue I have with the metric system... But that ship has sailed :) look up the dozenal society if you're curious how fervent some supporters might be.
Re: 0.999...= 1
#334My 5 year old stumped me with this, and I had to look it up. He asked me why 1/3 + 1/3 + 1/3 = 1, since it's equal to 0.333... + 0.333... + 0.333... which is 0.999... How can that possibly equal 1.000...? And is 0.66... equal to 0.67000...? I didn't have a good enough answer for him, so I had to look it up and found this page. I tried to explain it to him but since I'm a terrible teacher and he's only 5, it was hard…
> He asked me why 1/3 + 1/3 + 1/3 = 1, since it's equal to 0.333... + 0.333... + 0.333... which is 0.999... How can that possibly equal 1.000...? And is 0.66... equal to 0.67000...? This would make me very proud.
Yes, it's quite clever. An equivalent proof is dividing 0.999... by 9 using long division, which comes out to 0.111... which is equal to 1/9. Now use fraction notation and it simplifies to 9/9 = 1. Not quite as robust as the limit-based proofs but it's a quick answer and gets to the heart of the issue of repeating notation not capturing the whole picture.
Re: 0.999...= 1
#335There 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…
I didn't grok infinity until I started thinking in terms of verbs rather than nouns. As a static number, the concept of infinity makes no sense; but once reimagined as a process (start counting up from 1, and never stop), all apparent paradoxes disappear. This is the inverse problem: it could just as easily be reframed as 0.000...0001 = 0. Defined as static nouns (does such a thing exist in nature?), it's seemingly p…
> it could just as easily be reframed as 0.000...0001 = 0
But it can't be, because there's nothing after "0.000..."; that ... goes on infinitely. It's literally "0s forever, never stopping". It's not a process of "keep adding 0s", it's the end result of never adding 0s. It's not a process, it is a noun.
Re: 0.999...= 1
#336Earlier quoted context omitted.
No .666666 is not equal to .6700000 0.666... is equal to 0.666...7
Not sure if you’re joking, but 0.666...7 is not a real number. Can you define it?
I don't think it's a real number, but if it was this would be yet another intuitive proof that 0.999... = 1.
Re: 0.999...= 1
#337Earlier quoted context omitted.
> which is indeed false, you can also interpret the expression as: > lim[eps->0] ((1-eps) Assuming you don't mean some special notion of limit, I would guess that by `(1-eps) If so, `f` does indeed have a limit from above that is `false` and a limit from below that is `true`. Where do you wanna go from here? Edit: Corrected stupid wrong assertion about limit from below, d'oh.
> If so, `f` does indeed have a limit from above that is `false` and a limit from below that is `true`. Where do you wanna go from here? That's the other way around. From above you get true, and from below you get false. Note that 0.999... represents the limit from above for 1-eps. Hence the result is true.
> That's the other way around. From above you get true, and from below you get false.
Yeah, my bad.
> Note that 0.999... represents the limit from above for 1-eps. Hence the result is true.
I mean you can define 0.999… like that if you want. You'll get 1. So what? Your detour via `f` provided nothing.
Re: 0.999...= 1
#338Earlier quoted context omitted.
> you’re basically just a priori defining 0.9... to be 1. I think the point is not defining 0.9... to be 1, the point is that “...” means an infinite number of 9s. If you shift the decimal point by 1, then nothing changes, there are still an infinite number of 9s. If you shift the decimal point by 5 places, there are still an infinite number of 9s to the right. And here is the logical (induction) step: if you shift t…
> I think the point is not defining 0.9... to be 1, the point is that “...” means an infinite number of 9s. If you shift the decimal point by 1, then nothing changes, there are still an infinite number of 9s. If you shift the decimal point by 5 places, there are still an infinite number of 9s to the right. And here is the logical (induction) step: if you shift the decimal point by an infinite number of places, then t…
Don't conflate "I don't understand it" with "hand-wavy". The 2 are different things.
Another proof by contradiction I've heard of this is that if 0.9... != 1.0... then there exists a number in between the 2. What is it?
Re: 0.999...= 1
#339Earlier quoted context omitted.
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…
Maybe I'm misunderstanding, but I think the issue with dates is strictly different. Dates are hard not because time is fundamentally hard, but because there is lots of complexity in human representation of time (different places at different times have had similar but different representations of time). But that's not inherent to time. Ignoring relativity, if everyone throughout time used something like seconds since…
No, dates are harder than that. Humans use time to coordinate; the representation of time is fundamentally about communication. Only timestamps of events in the physical world are easy (ish). But that's not always, or perhaps mostly, what people are interested in.
When people receive a time, they may (usually) want it in their own time zone, but they might instead want it in the time zone of the entity they're getting the time from, if they're subsequently going to talk to that entity about the time. When they talk about meeting someone else, when they convey the time of the meeting, they usually mean whatever that time means in the place where they meet, which might be different from the current location of either. It might even be different due to political changes around time zones and daylight saving if the meeting is far enough in the future.