My apologies. I thought I had a point to make but it’s no longer apparent. Can I just plead a head cold?
Well, it does give us a way to express real numbers, although maybe not the best way in almost all situations. There are exceptions though:
Expressing common angles in radians … a right angle is π/2, sin π/6 = 1/2, etc etc etc,
Using determinants as coefficients of polynomial equations, as in Cramer’s Rule (which is sometimes presented in high school Algebra).
Not that I’m trying to excuse my piss poor writing, I’ve only claimed I was presented with instruction on correct and proper Englishing, I’ve never claimed to have actually learned such.
All this out of one little comment. 
> When the OP says “For practical purposes you can treat .333~ as the limit of the sequence (.3, .33, .333, …) which is equal to 1/3. Consequently you can say that .333~ is equal to 1/3 because it is defined as equal to it,” I have to wonder what other definition of .333` he’s thinking of where it’s not equal.
The point of infinity is that it is not a number, it is a concept. It means “keeps going.” The quoted infinite sequence never contains the value of 1/3 because you never get to the end of the sequence; you can always add another term that is closer. Even if you take it to infinity you never get to the point where the term is exactly 1/3. It is always an infinitesimal short. The point of taking the limit is that the limit is exactly 1/3.
In situations where you have to do arithmetic with infinity the result you get is always going to be defined rather than calculated because you cannot do arithmetic with a concept.
I find it fascinating that when you divide a finite interval by infinity, the result is usually taken as zero, but is in fact an infinitesimal. It has to be, otherwise integration wouldn’t work! You gotta love maths!
This is incorrect and the source of all your confusion.
Of course you can do arithmetic with a concept. How could it be otherwise? All numbers are concepts.
I don’t think “getting to the end of a sequence” has any bearing on the existence of such a sequence. This sequence does exist and it is equal to exactly 1/3.
Perhaps think of it this way: (0.333…) - (0.333…) = (0.000…) which is zero no matter how many zeroes we add.
OK, even if I just provide a target for the mathematicians…
Isn’t the problem for 1/3 just a function of the number base we use?
If I worked in ternary/trinary 1/3 would be 0.1
Or in nonary, 0.3
So long as we state that 0.333 recurring IS 1/3 when using decimal, then all is cool.
This was one advantage of unit systems that had different number bases for the divisions (e.g. pound-shilling-pence, yard-foot-inch); you could always try a different unit!
No, infinity divided by infinity is undefined, not zero. Don’t take my word: here’s a cite. Some good lines on that page.
That’s not what engjs said: He said a finite interval divided by infinity. What that means depends on just what you mean by infinity, but there are many contexts in which it’s valid to call that zero, and a few in which it’s valid to call it an infinitesimal. I don’t think there’s any context at all where infinity is treated as a number but where a finite number divided by infinity is undefined.
I’m just going to go pull the blankets back up over my head.
I completely agree with the first paragraph, that ‘infinity’ can mean many different things, so you can’t lay out a single set rules for arithmetic with infinity involved.
Which is why I don’t understand why the second paragraph states, as a ‘fact’ what something divided by infinity is. My understanding – and I’ll defer to any real mathematicians – is that infinitesimals are these days pretty much seen as an element of an alternative formulation of numbers, and unnecessary for most mathematical work. Certainly, even <mumble> decades ago when I learned calculus, the rigorous derivation was presented using limits, not infinitesimals-- integration works perfectly fine based on standard delta-epsilon limits.
And even in those formulations of math which do use infinitesimals, they’re still not usually applied to mean things like “1/3 - 0.3”.
> I don’t think “getting to the end of a sequence” has any bearing on the existence of such a sequence. This sequence does exist and it is equal to exactly 1/3.
If the infinite sequence were to reach its limit and stop, it wouldn’t be an infinite sequence. The fact that it is infinite means it can never reach its limit. It doesn’t matter how many 3s you put on the end it will never be exactly equal to 1/3. Thinking that the limit of a sequence is the last term in it is a tyro’s mistake.
> I don’t understand why the second paragraph states, as a ‘fact’ what something divided by infinity is.
You can’t divide by infinity, but you can split an interval into an infinite number of segments. Infinity means ‘keeps going without ever coming to and end’, so the result is not a process that terminates :-).
I never trust things like 9.99~ minus 0.999~, where there is an assumption that it doesn’t matter how you do the sum because “infinity so it’s okay”.
Take the sum (1 - 2 + 3 - 4 + 5 …). The correct way to sum it is to form the sequence of partial sums, which gives the result (1, -1, 2, -2, 3, -3, …), and take its limit. There is no limit here; the sum flip flops between infinities.
You can sum it by adding consecutive terms: (1-2) + (3-4) + (5-6) … = negative infinity. Or 1 + (-2+3) + (-4+5) … = positive infinity. Neither answer is correct, but at least you can see them coming. A third way is to add the sequence to itself but shift it by one place: 1 + (-2+1) + (3-2) + (-4+3)… = (1-1) + (1-1) + (1-1) … = 0. Which is not only WRONG, but is also hiding the fact that at the end of the sequence the bit being dropped is infinite! And, yes, I’ve seen that used in a real, live proof. And been ridiculed for pointing it out.
You’re right, it’s obviously 1/2.
See that’s exactly what you are doing when you say stuff like
My edit
You want to stop after a finite number of terms, but you can’t because the term you stop at wouldn’t be the infinite decimal expansion indicated by the notation 0.333…
I think you get all this. And if you don’t, please answer the question posed by watchwolf:
Behind every simple notion in Math lies an incredibly complex world.
For infinite series, it does get weird if you don’t do things exactly right. The proof by multiplying by ten and subtracting the original series does work … for certain infinite series. There are a lot of different types of convergence. Absolute convergence works here.
Technically, one would have to prove .333… converges absolutely first. Obviously it does so the rest is fine.
As for odd ducks like 1-1+1-1+1-1 …, if you want to make this “converge” to some value, you need something like Cesàro summation. But that’s waaay off topic and is significantly unrelated to the matter at hand.
> You want to stop after a finite number of terms, but you can’t because the term you stop at wouldn’t be the infinite decimal expansion indicated by the notation 0.333…
Okay, let’s be rigorous.
Consider the sequence (0.3, 0.33, 0.333, … ).
Assume (a) the sequence is infinite; (b) each term in the sequence is followed by another that is strictly larger; (c) the limit of the sequence is 1/3; and (d) there is a term in the sequence whose value is exactly 1/3. By (b) and (c) and the definition of limit, no term in the sequence can have a value greater than 1/3. By (b) and (d) the term whose value is 1/3 must be followed by another whose value is greater than 1/3. These deductions are contradictory, so at least one of the assumptions must be wrong. My money’s on (d). Which one do you think it is? 
>> 1 + (-2+1) + (3-2) + (-4+3)… = (1-1) + (1-1) + (1-1) … = 0. Which is not only WRONG
> You’re right, it’s obviously 1/2.
I’ve been mulling this over for a while and today I decided to write it up for my website. You might enjoy it. http://home.pacific.net.au/~jimnsharon/mot.html Note that this thread is not the place to discuss it.
There is no term in the sequence that has a value of exactly 1/3 - the sequence has a value of 1/3. You get hung up looking for a specific term in an infinite sequence; that’s nonsensical.
d) is nonsense and no one is making that assumption, so what is your attempted point? For the rest of us, 0.333… is considered to be the decimal expansion of 1/3, and can also be described as the limit of the sequence 0.3, 0.33, 0.333.
It’s “actually equal to -1/12” is shorthand for:
Yes, it’s infinite/diverges, but a specific set of rules can be chosen that give it a value of -1/12, and that specific set and that value are mathematically interesting in a way that your write up isn’t.
Or to put it another way, 0.3333… isn’t a sequence. It is very closely related to a sequence, specifically the sequence ( 0, 0.3, 0.33, 0.333, 0.3333 …) which everyone in this thread has been talking about. But 0.3333… isn’t that sequence; it’s defined to be the limit of that sequence.
In technical terms, when we say that 0.3333… = 1/3, what we mean is that, for any nonzero epsilon you care to name, there is some number of digits such that a zero and a decimal point followed by that many threes differs from 1/3 by an amount less than that epsilon. If you wish to assert that 0.3333… ≠ 1/3, then you need to falsify that statement.