Tropers that do NOT hate math

There only is one infinity if you're using the reimann sphere.

edited 1st May '13 5:36:08 PM by Jate88

@Jate: Just point out that a decimal representation is a sequence of digits, so a string of symbols like "0.00...1" is complete gibberish from the standpoint of real numbers. It has as much meaning as "0.00potato01".

Also, the proof using infinite series (or Cauchy sequences, basically the same argument) is the best proof, since it exposes the real reason for non-uniqueness of decimal representation: namely, that the sequence 1/10^n becomes arbitrarily small for sufficiently large n.

@Raichu: Actually, it sounds like you might have mixed up several different concepts of infinity. The collection of all cardinal numbers (i.e., sizes of sets) is a proper class, which intuitively means that it's "too large" to even be a set. This means that, in a certain sense, there are "more than infinitely many" infinite cardinal numbers. Another way of seeing this is that the class of all cardinal numbers has subclasses of every cardinality.

Furthermore, "negative infinity" isn't meaningful for cardinal numbers. After all, you can't have a set with less than zero elements — that just doesn't make sense. (You could probably play around with nonstandard set theories or something to define a concept of "set with less than zero elements", but that certainly doesn't coincide with the familiar notions of "set" and "cardinality".)

There are certain contexts where "negative infinity" makes sense, but those shouldn't be conflated with cardinal numbers. I can think of two such contexts at the moment:

1. The affinely extended real number line is obtained by adjoining elements +∞ and -∞ to the real line. These are basically just formal symbols that we define to behave in a certain way, and they definitely aren't cardinal numbers. Also, there are just the two "infinities" in this system, because we just define those two symbols.
2. The surreal and hyperreal numbers, along with a few other systems used in non-standard analysis. These are particular ordered fields*
   A field is basically just a number system (such as the rational, real, or complex numbers) where you can do addition, subtraction, multiplication, and division more or less as usual.
   that fail to satisfy the Archimedean property, and thus have "elements at infinity" (in both directions), as well as "infinitesimals". Once again, these shouldn't be confused with cardinal numbers; however, in some cases — e.g., the surreal numbers — we end up with a very large collection of "infinite" elements.

edited 1st May '13 5:44:19 PM by Enthryn

Oh, I understand that what infinity is and the amount of infinities there are change depending on what they are; not really what said contexts are, but I'll get into that eventually. Negative infinity wasn't one of the "infinite infinities" bit; he said there was only one and I tried to get him to accept there being at least two. And I did intend the affinely extended number line's negative infinity.

edited 1st May '13 6:00:01 PM by RaichuKFM

@Enthryn: I don't get the argument but he says the sequence doesn't converge for the same reason y=1/x doesn't touch the y-axis, so the calculus/analysis is being misapplied in this case. I asked him if he even knows how to tell if a series convergences or not but haven't gotten a reply yet.

He uses inductive reasoning to form those strange numbers. I commented on how to use inductive reasoning on arithmetic involving repeating decimals to come away with the right answer but no reply back yet.

I've usually just used inductive reasoning myself to do the arithmetic but I'm starting to see how calculus/analysis can be used to give an answer to at least some arithmetic problems involving repeating decimals.

Where did this idea originate that all or a portion of the fractions don't have an exact decimal representation? It seems to destroy some useful properties of the rational numbers and real numbers.

edited 1st May '13 7:17:41 PM by Jate88

@Jate: Sounds like he's making a common error: thinking that an infinite decimal representation is some sort of ongoing process that "gets closer and closer" to something, but is never completed. In fact, it refers to a specific, well-defined, completely determined, actual number.

If you cut off the decimal expansion after a certain number of digits — something like 0.9999999 — then you get something that approximates the value in the limit; however, the full decimal expansion (e.g., 0.999...) is this limit, not a "process" or an approximation.

The (incorrect) idea that some fractions don't have an exact decimal representation is probably an extension of this misconception, since many rational numbers' decimal representations are infinite in length (though always infinitely repeating — a characteristic property of the rational numbers).

I'm not sure what you mean by "inductive reasoning", by the way. Whatever it is he's doing, if it results in a nonsensical expression like "0.000...1", clearly something's gone wrong along the way.

edited 1st May '13 7:38:52 PM by Enthryn

1/243 has the best decimal expansion.

@Enthryn: This is inductive reasoning.

http://en.wikipedia.org/wiki/Inductive_reasoning

This is why mathematicians don't use it.

http://en.wikipedia.org/wiki/Problem_of_induction

Not to be confused with this.

http://en.wikipedia.org/wiki/Mathematical_induction

edit: If you don't feel like reading all that, inductive reasoning looks for patterns and theorizes they hold everywhere and deductive reasoning constructs rules, sees if they accurately model reality, and tries to see if the rules can be used to expand our knowledge in other areas.

Sorry. I didn't fully understand it myself :(

edit: Scratch the above edit. In deductive arguments if the premises are true then the conclusion must also be true, and they're judged on whether the premises are true and if the conclusion follows from the premises. In inductive arguments there's anywhere from 0 up to but not including 100 percent chance that the conclusion is true. Inductive arguments are judged on whether the premises are true and how likely the conclusion is true. The more likely the more the conclusion can be justifiably believed.

@Noaqiyeum: I got confused for a bit when trying to find the fraction form for 1.0999180999180999...

edited 8th May '13 6:55:29 PM by Jate88

Mathematical induction actually is a form of inductive reasoning. The reason it works for maths when general inductive reasoning doesn't is because mathematical properties are defined, whereas scientific properties are observed. Mathematicians don't guess. :P

The other wiki says differently :(. I got confused for a bit because both types of reasoning use probabilistic arguments to prove their premises.

edited 8th May '13 3:27:19 PM by Jate88

Well... maybe Wikipedia knows something I don't, but... Induction as a process basically just means generalising from a finite number of samples of a given type to all samples of that type. Mathematical induction can do this by establishing a fact for the first element in a sequence, and then showing that the sequence is defined in such a way that this property is not lost by moving to the next element. This works only because the properties of numbers are defined rather than observed; it's a much more rigourous form of induction than the usual methods, basically.

Where did you learn this at? Maybe math being defined instead of observed is what causes the process of mathematical induction to be deduction.

...logic class, I think. :P

I came here out of curiosity. I'm not really good at math. I don't understand what u people are saying :/

How does contraposition still work when you're not talking about a set and its subset? Such as innocent/guilty or true/false.

Contraposition? As in the contrapositive of a statement? I'm not sure what you mean; that usually just refers to the fact that "P implies Q" is logically equivalent to "(not Q) implies (not P)", where P and Q are any statements.

Yeah I it get now. This stuff is all new to me.

It is just like science. If A is true I should see B. I don't see B. Therefore A isn't true.

edited 28th May '13 10:00:05 AM by jate88

So, what's the glaring error I'm missing here?

It's the part where you remove the square.

a²=b² does NOT imply a=b, since it is also satisfied when a=-b.

In fact, try substituting x²-2πx+π² in the right hand side of the fourth-from-the-bottom equation (you're just rearranging the terms here), and see where that gets you.

edited 29th Jul '13 12:10:11 AM by KylerThatch

There's no error. There's no indication that the pi in those equations is the same 3.14 pi that comes from circles. It's just another symbol standing in for an unknown number. pi=3 and x=3 is one solution to the equation. However, since there are two unknowns—pi and x—and one equation, there are an infinite number of answers. As Kyler Thatch pointed out, the second to last step where the squares are removed should actually be written as

|3-x| = |pi-x| (where |z| is the absolute value of z)

We then write

3-x = pi-x

or

3-x = -(pi-x)

The first works out as in the picture. The other

3-x = x-pi

3+pi = 2x

(3+pi)/2 = x

which you'll notice is the equation we started with. Pick any value for pi, and you'll find a corresponding value for x. pi=11 means that x is 7. Notice also that, if pi is 3, then the third line reduces to 0 = 0. True, but rather trivial.

edited 31st Jul '13 9:57:18 PM by JohnnyAdroit

When has pi ever been used like a variable and not a constant?

Well, whoever made that did so, presumably accidentally. You could treat anything as a variable, but using a symbol like pi that way is just confusing. If they treated pi like the constant it was, x would equal 3.0707... and you wouldn't achieve anything.

Actually, the letter π (pi) is sometimes used as a variable in certain contexts; the most common one I've seen is that a uniformizing parameter of a discrete valuation ring is often denoted by π.

Of course, that's clearly not what was intended in this case, as that's a fairly technical context, and you also don't see π used as a variable in places where it could be confused with the number π. Plus, in cases where π is used as a variable, it's always explicitly mentioned (usually with phrasing like "let π ∈ R be a uniformizer")

I don't even know how they got from line two to line three in those equations. :/

So when it comes to infinity there are three ways to think about it?

1) A process that goes on forever and never stops at any amount.

2) A process that stops at some amount which is always larger than any natural number.

3) A process that goes on forever but this itself could be described as a kind of amount.

They just multiplied each side by (pi-3) to get where they got.

@jate: All of those ways of thinking about infinity are vague and inaccurate. In mathematics, it's important to precisely define all terms. There are actually several completely different things that are called "infinity", and they often all get conflated in plain English. However, none of them are really a "process", and that way of thinking will just lead to confusion.

The common theme for all these: The word "infinite" is used to describe anything that's "larger" than any positive integer, in some sense that depends on which type of "infinite" you're talking about.

First, there are infinite cardinal numbers, basically, sizes of things. If you're talking about the size of some collection, or answering the question "how many", you're dealing with cardinal numbers. There are many different infinite cardinal numbers; some infinite sets are larger than others. An infinite cardinal number is not a "process" or anything like that — they're sizes, so the cardinal associated to an infinite set is infinite in size. (Example: the cardinal number associated to the set of integers, usually denoted ℵ_0, is infinite because there are infinitely many integers.)

Closely related are the ordinal numbers. These are like cardinals, except they measure the size of certain types of ordered collections, rather than unordered collections. There are lots of infinite ordinal numbers, too — they're basically just cardinal numbers that also care about ordering.

Then, there is infinity as a limit. We say that a sequence "goes to infinity" if the terms of the sequence become arbitrarily large, i.e., increase without bound. Infinity in this sense isn't a number, but a description of the behavior of a function or sequence. As the name "limit" suggests, this can be thought of as the "limit of an infinite process", but note that the limit is not the process itself. For example, you can think of the sequence 1, 1/2, 1/3, 1/4, 1/5, 1/6, 1/7, ..., which has n-th term 1/n in general. The "limit at infinity" of this sequence is 0. Similarly, the "limit at infinity" of the sequence 1, 2, 4, 8, 16, 32, ... is often said to be "infinity", but this is just shorthand for saying that the sequence increases without bound.

Another meaning is infinity as a number on the extended real line. This is a number system that extends the real numbers by including infinity. The "infinity" here is really just a formal symbol that we define to behave like we'd expect a "point at infinity" to behave; it's closely related to limits.

Finally, in measure theory, the area of mathematics dealing with concepts intuitively based in length, area, and volume, infinity is often treated as a value. For example, the real line with the standard Lebesgue measure has measure (or "length") infinity.

There might be some other notions of infinity I've forgotten about, but those are all the main ones. To sum it up, the word "infinity" is horribly vague, so you need to be clear about which one you mean. This is basically a matter of figuring out what you're using it for: a size? an ordering? a length, area, volume? the limit of a process? etc.