If you're having math problems, I feel bad for you, son...

[Joseph8th]

22. Had to jump in on this thread...

Last edited Thu Apr 5, 2012, 11:53 AM - Edit history (1)

... as a mathematician. Napoleon_in_rags wants to have a spiritual quality, and that's fine, but it's wrong, Napoleon, to resent eggheads for spending the time, hard work and money to tackle the difficult subjects mathematical. If there is one field of study where the spirit of creativity still reigns supremum, it's pure mathematics. But what isn't acceptable... or rather it's asking way too much... is redefining basic math concepts and objects, then expecting everybody to toss everything they've learned and start over from scratch with someone's half-baked idea. If that person's goal is to somehow justify the existence of their deity by redefining mathematical concepts, then, yeah... most of us mathematicians are going to slam our minds closed in your face.

On to the actual math: the limit of a sequence of real numbers, if it exists, is always a real number. That said, the limit of a sequence of numbers, each of which is in a subset A of R, might not be in the subset A, but will always be in R. An example is the limit of 1/k as k --> infty for k a natural number (positive integer not zero). Every term is 1/k in (0,1], an interval that does not include zero. But the lim(1/k) = 0 as k --> infty, since as k grows infinitely large, 1/k grows approaches zero.

The expansion of Euler's number (e) that you mentioned actually extends this idea to the concept of a series, which the sum of an infinite sequence of numbers. For instance, we might sum 1/k^2 for k=1 to infty. If we fix k, then we call one such sum a partial sum of the series, denoted S_k, and we can write it using an ellipsis (rather than sigma notation) using the familiar S_n = 1/1^2 + 1/2^2 + ... + 1/k^2. Then we can look at whether or not S_k < S_k+1 or not, for each k, and use some theorems, to discover whether or not the sequence of S_n's converges in the reals to some some finite number. If so (roughly), then we can look for the limit of the sequence of sums as k --> infty. This turns out to be profoundly powerful, especially when working with power series, which generalize the notion of polynomials.

Well... this is how e was discovered by Bernoulli as he tried to evaluate the limit of the series (1 + 1/n)^n as n --> infty. Then given this series definition of e, we can define the function log as the inverse of the series that defines e, such that e^log(x) = x, x>0. Or logarithm can be defined 1st as an integral (infinitesimal Riemann sums, related concept for increasingly fine partitions of an interval), and then e can be got from that. The point here is that even though e is irrational (and transcendental), it can still have a series expansion (of infinitely-many terms) anyway. Sorry there's no LaTeX at DU, but...

pi = 4 * \sum_{k=0}^{\infty} (-1)^k / (2k + 1)

is the same number as

pi = circumference/diameter, for any circle.

Both of these equations are precise. That is, they express pi completely as transcendental irrationals, the way that 1/3 is a precise rational number, but .3333... is a decimal expansion is a floating point number with variable precision. A computer can't deal with 1/3, and always does math with rationals using FP arithmetic, and so has limited precision (depending on the language and computer, of about 10^30). This is always true of decimal expansions: the ellipsis at the end indicate that the representation is imprecise. If they follow a repeating pattern, convention says repeat the pattern. If they do not follow a repeating pattern, convention says do not repeat the pattern. That said, in pure math we rarely ever use ellipsis notation and in analysis don't truck in decimal expansions, at all. Decimal expansions are the difference between applied and pure math, in many ways. Even in number theory (source of elliptical curve theory) we are rarely ever interested in dealing with numerals. Letters and symbols keep things nice and general.

In short: arithmetic is not mathematics, and vice versa. So when we write 0.999... it's understood that this is not a precise expression of any number. But it's a number we can write precisely by using our heads to expand decimals to infinity to find the limit of the decimal expansion (sim. as we did with partial sums for e and pi above) as the number of digits --> infty. The precise representation of this number is 1. Really the point is that there's a number of different ways to represent numbers in general, and some numbers in particular. Right? Especially on the real line.

A better way (more 'natural') to real numbers is by continued fractions:

pi = [3;7,15,1,292,1,1,1,2,1,3,1,…] = 3 + 1/(7 + 1/(15 + 1/(292 + 1/(...)))). The sequence of integers in the C.F. representation are apparently random, but nobody's proved it yet.

To get really trippy, yet another way to extend the reals to completion is with the field of p-adic numbers (https://en.wikipedia.org/wiki/P-adic_number). In particular, I draw your attention to the "Constructions" section at Wikpedia, which states:

The real numbers can be defined as equivalence classes of Cauchy sequences of rational numbers; this allows us to, for example, write 1 as 1.000… = 0.999… . The definition of a Cauchy sequence relies on the metric chosen, though, so if we choose a different one, we can construct numbers other than the real numbers.

Ain't math cool?