Does 0.333333... equal 1/3 ?

This thread is pedantic and insufferable.

it has devolved into a sophomoric who-knows-more contest

the original question was does an informally expressed infinite sum "equal" something

The answer depends on what formally did the whole expression mean.

Maybe for scalars that can be pulled out of every term, you can manipulate and produce series whose sums are coherent: +infinity + +infinity = +infinity.

But is it valid to subtract one divergent series from a different one, or from an integer: e.g. S3 - S2; or 1 - S1?

Normally, addition and subtraction of integers are closed operations. If we are adding and subtracting a finite number of integers, the results will always be in the set of integers. (It is also closed for rational and real numbers, but all your series were summing integers, so I will stick with integers).

But if we add an infinite series of integers, and the sum diverges, the sum of the infinite series is no longer an integer. I don't know what it means to add and subtract infinite sums which are not integers. Many of the properties taken for granted may have to be re-evaluated for these sums that don't converge to some finite value.

For example, if the series has alternating positive and negative terms, the associative property looks like it is broken:

Let's take your sum S2, which "equals" 1/4, and group the terms in pairs:

S2 = (1-2) + (3-4) + (5-6) + (7-8) + ... = -1 -1 -1 -1 ... "=" 1/2

Or alternatively:

S2 = 1 + (-2+3) + (-4+5) + (-6+7) + = 1+1+1+1 ... "=" -1/2

Or, since you want to split and join infinite sums, so we can add +infinity to -infinity (please check my math):

S2 = (1+3+5+7...) + (-2-4-6-8...) "=" (1/3) + (1/6) = 1/2

I haven't worked a whole lot with infinite sums, or these special physics problems which can use these non-intuitive results, but it sure seems like you have to be careful how you manipulate the terms, and how you interpret addition, subtraction, and equals.

I’ve taken real analysis, functional analysis and basically any math course related to this sort of question. But how about I take a one meter step, and you take a .9 meter step then .09 meter step, then .009 meters, etc. Will you ever get to the one meter mark? Nope. The proof basically works because of how mathematicians define infinity, which makes math work but is also a concept of philosophical debate for millennia.

This argument (Zeno's dichotomy whatever) also seems to fall apart to me when you consider that the time to travel successively smaller units scales linearly with the distance - I get the point is to question the infinite divisibility of matter/the universe but the problem as stated is too contrived.

Let's look at it from another angle.

The hypotenuse of a 45-45-90 triangle, with legs equal 1 unit, is an irrational number sqrt(2). The length is a prefectly, simply defined quantity. It also takes an infinite number of decimal places, all added up to sqrt(2), approximated by 1.41412. There really is no debate in the math community over these quantities in analysis, as to whether they exist and can be defined with an infinite process. Although some of the admissible methods in analysis have been debated.

Leibniz invented a new number called an infinitesimal, that Cauchy and Weierstrass dispensed with in the 19th century, but Abraham Robinson resurrected them in 1960 with a consistent model that included infinitesimals, which were small numbers that were not zero, but less than any non-zero real number.

Just my 2 cents.

Just because some people say "ain't never" when they mean "never" doesn't change the truth of what the words mean. It's just that in most cases being imprecise with language is OK. That doesn't mean it's some different set of truths. The truth is the truth, what varies is the level of precision in relation to that truth that different situations and disciplines call for.

Is the speed constant?

The practical implications to this question are well described through error or tolerance or measurement analysis. These approaches are taught in any rigorous stem program.

Yes, we do have to be careful with how we manipulate the terms. As the user 10KSU85:[CR]18.32s demonstrated, if you manipulate the terms in certain ways aligning with the basic properties of math, you can get different results. I have known that with imaginary numbers (and by extent quaternions and beyond [if imaginary numbers can be seen as a 2d number (ie. complex plane), then quaternions can be seen as a 4d number, and (if I remember correctly) there are infinitely many number groups, each a 2^n dimensional number, and each lose some properties of the previous]), you lose some properties, and easy way to see this is the following (false) proof that i=1.

i

i^1

i^(4/4)

(i^4)^(1/4)

1^(1/4)

1

This proof also is completely correct assuming that the rules that govern the real numbers also govern the imaginary numbers, however, this proof is false because not all exponent (or logarithm) rules apply to imaginary numbers.

One could make an argument saying that 'infinity' could be seen as its own number group that might have different rules. For example indeterminate forms. inf/inf is not always 1. inf/0 is not always inf. 0/0 is not always 0 (or 1). inf-inf is not always 0. etc etc etc. Despite these things seeming to be pretty common sense, and they seem to align with the basic mathematical properties, they are not always true.

I wouldn't say that a divergent series would follow the same rules as infinity, and I don't know if divergent series that diverge to infinity would even follow the rules of infinity. Heck I don't even fully understand the different rules that govern infinity in the first place! But I do think its safe to say that they (divergent series) don't (always) follow the same rules as the real numbers.

If the time it takes to travel each distance is constant, then you will reach 1 meter after an infinite amount of time, which practically means you will never reach one. However, here is another version of the question.

Again, say you take a 0.9 step, and a 0.09 and a 0.009... But say that the 0.9 step takes 1 second, the 0.09 takes 0.1, the 0.009 takes 0.01... Then will you reach 1 meter?

Yes. You would take an infinite amount of steps and reach 1 meter after a finite amount of time. The first step would take 1, then 0.1, then 0.01 etc. Adding all the times together gives you 1.111111111111111111111... seconds (10/9 seconds)

Woudln't "1^(1/4)" have four roots? Namely 1, -1, i, and -i?

I would also say that common sense says "inf/0 is always inf" (if we suspend for discussion that /0 is not defined).