Are the surreal numbers at all relevant to this discussion? That is, the surreal numbers include numbers that are, for all finite n, greater than sum_{i=0}^{i=n}(9 * (10)^(-i). They are "infinitesimally less" than 1. But that doesn't mean that .99999.... is infinitesimally less than 1! I don't know if there is a standard or useful definition of the sum of an infinite series in the surreal numbers. But if there is, then every step in the chain of equalities (letting S denote the sum of the series in question, if it exists) 9 S = 10 S - S = 9.9999999 - .99999999 = 9 is still valid in the surreal numbers, as is the deduction that if 9S= 9, S = 1. The bottom line is that this proof that if .999999... has a meaning at all, it means 1, works not only in the standard real numbers, but in many other systems. If you want a system where .999999.... is a well-defined number less than 1, you need a system where the axioms are weakened in some significant way so that one of the equalities above isn't true. They are all true in the surreal numbers, so if .999... has a value at all, it's still 1, not one of the numbers infinitesimally less than 1. Andy On Fri, Nov 16, 2012 at 4:58 PM, Dan Asimov <dasimov@earthlink.net> wrote:
=
1. The symbol = for equals means that the left side and the right side represent the same element of the set under consideration. In this discussion, that set has usually been the real numbers -- and a few times, the surreal numbers. (Though technically speaking, the surreals are not a set but a class.)
2. If the mathematical expression is all about real numbers, then yes: the two sides of = are interchangeable (assuming the equation is a true one!).
(This doesn't really pertain to this discussion, but of course if the mathematical expression is about the set of mathematical expressions, then the two sides of = will be interchangeable only if they are the same expression.) -----
converge
To say that a sequence x: Z+ -> R converges to a real number c means that for any real number eps > 0, there is an integer N (depending on eps) such that, for all k >= N, it holds that |x_k - c| < eps.
I.e., for any desired closeness eps > 0 to c, then all the terms of the sequence after a certain point are that close or closer to the number c.
This does not require the sequence to always get closer to c, as long as it eventually does. Nor does it require the terms of the sequence to be unequal to c.
E.g., the sequence 0, 1, 2, 0, ½, 1, 0, ¼, ½, 0, ⅛, ¼, . . . (where each successive three terms are exactly half of the last one) exemplifies these things, since by the above definition it converges to 0.
...
The ellipsis does not have a formal definition, but instead is an informal way of saying "continue in the same pattern", with the assumption that readers will know which pattern is referred to.
But after a few terms of a sequence -- when the pattern is clear -- the ellipsis just (generally) signifies only the rest of the terms of that sequence. Not its limit (if any).
--Dan
On 2012-11-16, at 10:17 AM, Gary Antonick wrote:
My question appears to be partly about notation. Am wondering what the following symbols mean.
"=" Does x=y mean 1. "x" and "y" are different symbols or expressions for the same thing 2. "x" and "y" are interchangeable in any mathematical expression
"converge" Does converge mean gets close but never touches?
"limit" Could it be said that something that converges always converges toward what could be called a limit? Is the x axis, therefore, a limit for y=1/x?
"..." Does ... mean, for a series that converges, infinite steps in that series, and, by definition, the limit of that series?
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