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        riddles >> general problem-solving / chatting / whatever >> Revisiting 0.99999.....
        
(Message started by: Mickey1 on Nov 13^th, 2014, 6:55am)

Title: Revisiting 0.99999.....
Post by Mickey1 on Nov 13^th, 2014, 6:55am

Many questions – including one I was given it on this site - have been raised about the result of this process, i.e. adding nines after “zero point”. In this respect the question is not unlike a riddle. This is my attempt of a complete, if not the only possible, answer.

The first issue is about interpretation. My answer relates to the interpretation that the symbol is a reference to an infinite series S(n), where S(n) equals the sum of 9/(10)^n from 1 to n. This is how I interpret the question, an infinite series of numbers, not necessarily a limit.

It is tempting to equate (NB equate, not interpret) the series with a limit of S(n) as x goes to infinity (I have some problems with the English term “approaches infinity”), but there is a problem with such an “equation”.

The problem lies in that addition is understood to allow for an infinite number of terms, and it can be illuminated in the following way: Addition is defined (in the case of numbers) for two terms A+B where (A+B) is assumed to be a new number of the same type as A and B to which we can add another number (A+B)+C etc. The number of terms can therefore be extended, or similarly, reduced from any number to the first addition of two terms. There is no way we can do that for an infinite number of terms.

It might seem to be a formal objection I raise with no real consequence, since the series look very much like the limit. The series S(n) is absolutely converging and therefore the commutativity of addition runs parallel to the property that the rearrangement of terms in the series have no impact on its limit.

The problem lies in that definition of addition includes also adding negative numbers.

For (-1)^(n)/n, rearrangement implies that the corresponding series can have any value including causing the series to diverge, all according to Riemann’s rearrangement theorem. The simple approach equating a series with its limit is therefore problematic if it is to be done in a consistent manner.

Since the sum of an infinite series has no meaning, we can now move forward and formulate an axiom about such a sum.

In an earlier comment (on Math.SE) this year Professor Christian Blatter (from Switzerland) made the comment to me: “Mathematics does not have "an actual sum of an infinity of numbers" in store”.

I take it to mean that such an axiom (or definition?) has not (yet) been proposed and I therefore formulate it now (under the name Mickey1’s axiom):
The sum of an infinite number of terms is

1 if the series is absolutely converging - the limit of the series

2 if the series is conditionally converging – the limit of the series p1+n1+p2+n2... where p1, p2, p3.. etc and n1, n2, n3 ... are the positive and negative terms in decreasing order of the term’s absolute values. (Gerry Myerson member of Math.SE pointed out to me that a conditional converging series not initially presented with alternating sign of its terms can nevertheless be so rearranged). The interpretation can therefore be made in a consistent way.

This concludes my view of 0.99999... interpreted as 1. (I must admit that the new definition or axiom is not part of a great scheme where new results are presented. It is simply something can be done without contradictions, like having a good time without breaking the law).

Title: Re: Revisiting 0.99999.....
Post by 0.999... on Jul 20^th, 2015, 4:54pm

Given my username, I kind of feel obligated to respond to this one [no pun intended].

Let's work backward.
The axiom: I guess I will simply say that you haven't written it as a formula (or "schema" of formulas) over a certain language. The reason for this setup is that you create axioms as a sort of description of a class of structures that you feel are worth distinguishing (e.g. Cayley's axioms of group theory). Thus, having the structure in place (real numbers) and then specifying an axiom that is to apply to the structure is useless.

As for Christian Blatter's quote. Out of context, it can be interpreted in multiple ways, but a natural way of reading it is that if you have a theory like that of abelian groups (a good theory of additive structures) and you want to incorporate infinite sums with a new symbol and reasonable looking axioms, (unlike your own since you implicitly rely on a topological structure!), then you will fail in the sense that no structures actually exist which satisfy those criteria.

Quote:

Since the sum of an infinite series has no meaning, we can now move forward and formulate an axiom about such a sum.

I will just go ahead and call this sentiment ridiculous. As a rule, axioms formalize meaningful concepts.

Besides the possibility of diverging (which does not have to do with the possibility of some terms being negative), I do not understand your objection with equating a series with its limit. The part about rearrangements is even more mysterious. The series is the sequence of partial sums, and automatically the corresponding arrangement of terms is specified.

The notion that commutativity of addition should apply to the infinitary version as well is unfounded. The finite initial pieces can be rearranged however you want, the infinite tails cannot be rearranged. The fact that it does not completely generalize the finite case is not at all a contradiction.

Quote:

The problem lies in that addition is understood to allow for an infinite number of terms, and it can be illuminated in the following way: Addition is defined (in the case of numbers) for two terms A+B where (A+B) is assumed to be a new number of the same type as A and B to which we can add another number (A+B)+C etc. The number of terms can therefore be extended, or similarly, reduced from any number to the first addition of two terms. There is no way we can do that for an infinite number of terms.

I don't understand your objection. If we equate an infinite sum with the limit of the partial sums, then equate the sum of a_n with a and consider (a)+b. In fact, why don't we regard b as b+0+0+..., and note that a+b=(a₁+...)+(b+0+0+...). I can't help but wonder what subtlety I am missing here.

Title: Re: Revisiting 0.99999.....
Post by Mickey1 on Sep 5^th, 2015, 3:01pm

Perhaps I should be more specific on the philosophical issues

Blatter´s comment was precisely on the same issue I mentioned here, not out of context but quite to the point. Some older posts on Math.se can only be seen by member with 10 k + reputation so I can´t prove it (but I could I mail him of course).

The sum of an infinite amount of numbers has no meaning in mathematics. It can be given a meaning by the definition I called an axiom, which you find ridiculous, but still continue to use (equating a series with its limit). We can call it a new definition rather than an axiom if you prefer that, but is has nothing to do with classical addition.

It is related to an expansion of a real number. There is a one-to-one correspondence at least for non-rational real numbers e.g. between 0 and 1 and an infinite series of numbers (rational numbers include an infinite number of doubles such as 0.099999… and 0.10000…. ), but that is set theory and not addition. “Equating a series with its limit” still requires something else than addition.

A conditional convergent series – rearranged - can be given any limit, but the sum of an infinite number of terms (if it existed) cannot change since “all” terms are added. Therefore it cannot exist, which is just an example of the difference of adding finite vs infinite number of terms.

PS I don´t understand your statement which seems to imply that there are conditional converging series which do not have infinite number of both negative and positive numbers.

Title: Re: Revisiting 0.99999.....
Post by rmsgrey on Sep 7^th, 2015, 6:46am

My problem with rearranging conditionally convergent series into the proposed canonical form is that it makes them less useful/applicable.

Consider a toy robot, which takes a step forward, stops, takes a step forward, stops, takes a step backwards, stops, and then repeats that cycle. If you have two such toy robots starting at 0, and facing in opposite directions, then their position at any finite time, t, can be worked out by summing either the finite series 1+1-1+1+1-1+... or the finite series -1-1+1-1-1+1+... each truncated after t terms. The behaviour of both robots is well-defined as t gets arbitrarily large - one heads toward positive infinity; the other toward negative infinity. But when you use the proposed rule on the infinite versions of the series, suddenly the robots' predicted positions after an infinite length of time has passed are both within one step of their starting points.

Okay, you might quibble about whether that example is conditionally convergent or not, but the point holds even if you make each step half the length of the previous one - rearranging a conditionally convergent series changes the limit of the finite truncations of the series, meaning that defining the limit as the limit of the finite truncations of a rearrangement of the original series means the limit of the series is no longer directly related to the behaviour of the process that the original series described.

I'm willing to take Mickey1's word for it that defining the limit of a conditionally convergent series this way doesn't cause any contradictions, but, aside from letting you assign a canonical number to any arbitrary infinite set of numbers and calling that its sum, I don't see any benefits, and the cost is that you lose applicability.