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Infinity is not a number

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Re: Infinity is not a number

#21

Post by granite_scones »

jpallen wrote: post #14
impendia wrote: post #6

The problem is that "a number" is meaningless. Is i (square root of -1) a number? Is x? Is 3/2? Depends on your perspective. You can define the integers, the rationals, the real numbers, the complex numbers, and plenty of more exotic systems. What about C[x], the ring of polynomials in one variable. Are these "numbers"? There is no a priori reason to say no. The integers mod 7? Quaternions? Etc. etc. etc. And, yes, y…

The problem is that "a number" is meaningless. No it isn't. It's fairly clear from the article that in this context a number is something which you can have that many of . You can have 3 dollars. It even makes sense to have -2.5 dollars. Ok, an irrational number of dollars is pushing it a bit, but you don't need to go that far to see that infinity doesn't work when trying to count things consistently. A lot of the co…

You clearly can't have 2i + 3 dollars, so by that logic, complex numbers aren't real numbers.

Re: Infinity is not a number

#22

Post by lmkg »

It's worse than that. "Infinity" isn't even a single concept.

There are at least two distinguishable uses of infinity (there may be more, but I haven't figured them out yet, not that my opinion counts for much). There's the adjective "infinite" that refers to a property of sets. This is the type that Cantor studied, and it turns out to have many different types, which are pretty strictly ordered into layers. Then there's the noun "infinity" which is either a point or a location that points can exist at, and while it's usually possible for there to be several such infinities in several different directions, they don't come in layers. I believe Rider of Giraffes refers to these as "set-theoretic" and "geometric" infinities, respectively.

For the first definition, it's easy to distinguish between it and traditional numbers: an infinite set is bijective to a proper subset of itself. However, it's also useful to consider it a generalization of numbers, so you can count forever.

The second definition is more problematic. Here, infinity is just a point, just like all your other points. You can choose to add it to your set, or not. It usually behaves a little funny (like it makes certain operators not invertible), but you may have to get subtle to define it, or a set that contains it. Sometimes, it's not different than normal points at all, and sometimes it depends on the context. For example, if you take the real line and add +/-infinity, in topology you just get a closed interval like [0,1], whereas in analysis based on metric spaces you get something outright broken (fails to satisfy the axioms of the objects being studied).

Re: Infinity is not a number

#23

Post by clebio »

"Most people seem to struggle with this fact when first introduced to calculus..."

When I took calculus in college, I _did not_ struggle with this idea, despite at that time not having had any deeper background in advanced mathematics. Intuitively, the idea that you _approach_ some absolute as you edge the denominator ever larger made perfect sense to me.

Formally, my instructor made it clear that the _limit_ as you approach something was, in nature, different from any particular fixed value (of x). So, in a clearly defined manner, as you _apply the limit operator_ to the left side of the equation, the right side correspondingly behaves differently.

This concept never troubled me. As other comments here imply, this is an idiom specific to (differential) calculus. The only caveat might be in the use of strict equality, since limit operations by definition indicate asymptotic behavior. One could argue that a different type of relation is described (such as 'approximately equal': ≈). But then it's not infinity itself which is at issue.

Re: Infinity is not a number

#25

Post by [deleted] »

mcpherrinm wrote: post #17
jpallen wrote: post #14

Earlier quoted context omitted.

The problem is that "a number" is meaningless. No it isn't. It's fairly clear from the article that in this context a number is something which you can have that many of . You can have 3 dollars. It even makes sense to have -2.5 dollars. Ok, an irrational number of dollars is pushing it a bit, but you don't need to go that far to see that infinity doesn't work when trying to count things consistently. A lot of the co…

In this context, infinite limits, I think the distinction is beginning to become important. I agree that colloquially it often doesn't matter, but I think it does here. "Something which you can have that many of" is a terrible definition: You can have sets of infinite size. Do you mean physical things? What, then, is a "thing"? I can have infinity intervals of different length on my arm. So are intervals not a "thing…

[deleted]

Re: Infinity is not a number

#26

Post by tokenadult »

The discussion that has begun in this thread suggests that the Wikibooks chapter submitted here could use some more work. (Its last revision was a 9 October 2011‎ reversion of I.P. edits to restore a version from 22 May 2011.) Evidently, not every reader of Hacker News is convinced that infinity is not a number, despite several websites by mathematically learned people who say exactly that,

http://scienceblogs.com/goodmath/2008/10/infinity_is_not_a_n...

http://nrich.maths.org/2756

so equally evidently, some readers here are not convinced that there is a rationale for drawing a distinction between infinity and numbers. Does it help to take a look at a discussion of "not a number" concepts

http://scienceblogs.com/goodmath/2006/12/nullity_the_nonsens...

as they are implemented in computer science? What I see here, from my view as an educator in primary mathematics (in a program in which I can define "primary" to include topics like Hilbert's Hotel), is that some readers here have had educational experiences in which they "remember" seeing infinity treated as a number. The classic case, which prompts the Wikibooks chapter, is taking the limit of a rational quantity as the denominator approaches zero. This appears (based on previous HN discussion

http://news.ycombinator.com/item?id=728026

from more than two years ago) to suggest that physical quantities can be divided by zero with a quotient that becomes infinity. Perhaps this is an example of how an engineering calculus course isn't always interpreted by learners quite the way it was presented by teachers. (I presume all but the tiniest number of teachers of engineering calculus would agree that infinity is not a number, and that no one can divide any number by zero.)

What would be a good way to clarify the point so that people are communicating with one another well as they speak about infinity and about what numbers are?

AFTER EDIT: impendia's kind top-level comment here

http://news.ycombinator.com/item?id=3592101

has sent me looking at a Wikipedia article, the talk page of which leads to a WolframMathWorld article,

http://mathworld.wolfram.com/AffinelyExtendedRealNumbers.htm...

and there is a discussion of the affinely extended real numbers. Does the limit example in the submitted Wikibooks chapter fit the characteristics of that number system fully?

Re: Infinity is not a number

#27

Post by nandemo »

jpallen wrote: post #14
impendia wrote: post #6

The problem is that "a number" is meaningless. Is i (square root of -1) a number? Is x? Is 3/2? Depends on your perspective. You can define the integers, the rationals, the real numbers, the complex numbers, and plenty of more exotic systems. What about C[x], the ring of polynomials in one variable. Are these "numbers"? There is no a priori reason to say no. The integers mod 7? Quaternions? Etc. etc. etc. And, yes, y…

The problem is that "a number" is meaningless. No it isn't. It's fairly clear from the article that in this context a number is something which you can have that many of . You can have 3 dollars. It even makes sense to have -2.5 dollars. Ok, an irrational number of dollars is pushing it a bit, but you don't need to go that far to see that infinity doesn't work when trying to count things consistently. A lot of the co…

> Ok, an irrational number of dollars is pushing it a bit, but you don't need to go that far to see that infinity doesn't work when trying to count things consistently.

Well, if your definition doesn't work for irrational numbers, then your argument isn't good enough: you exclude infinity on criteria that also exclude "good" numbers. In the context of the article, π is a number. And you cannot count things with an irrational number,　by definition.

Re: Infinity is not a number

#28

Post by drt1245 »

thaumasiotes wrote: post #12

The argument "Addition breaks" proves just as well that zero "isn't a number", since it breaks division rather badly. (Yes, division is a badly-behaved version of the more upstanding multiplication. The article uses subtraction in what is nominally a complaint about addition). It doesn't address ordinal numbers at all, since addition works just fine with infinite ordinals, exactly the way the article claims you'd exp…

> The argument "Addition breaks" proves just as well that zero "isn't a number", since it breaks division rather badly.

Mathematically, numbers (be it natural, rational, real, or complex) are defined as a field. Fields (or, more accurately, rings, which all fields are) are defined by addition and multiplication, not both. [1]

[1] https://en.wikipedia.org/wiki/Ring_(mathematics)

Re: Infinity is not a number

#29

Post by powertower »

fferen wrote: post #18
powertower wrote: post #15

Infinity can be relative and can be defined unlike a division by zero which in undefined (as in ... it can't be relative to anything else, and can't be used in a formula). And hence infinity can be used in a formula and can cancel out with another relative infinity... Example: 1) There are an infinite amount of real numbers between 1 and 2. 2) The amount of real numbers between 2 and 4 is twice the amount of real num…

Correct me if I'm wrong, but I believe the cardinality of the two sets you describe are equal, as there exists a bijective mapping between them, meaning there are an equal "number" of real numbers in both.

For every real number between 1 and 2...

1) I can find that number also in the set between 1 and 4.

2) But I can't find that number in the set between 2 and 4.

To myself numbers are always relative to one another, don't exist outside the mind, and the set between 2 and 4 is twice the set between 1 and 2.

But like I said, it's a personal spin on it.

Re: Infinity is not a number

#30

Post by drt1245 »

powertower wrote: post #29
fferen wrote: post #18

Earlier quoted context omitted.

Correct me if I'm wrong, but I believe the cardinality of the two sets you describe are equal, as there exists a bijective mapping between them, meaning there are an equal "number" of real numbers in both.

For every real number between 1 and 2... 1) I can find that number also in the set between 1 and 4. 2) But I can't find that number in the set between 2 and 4. To myself numbers are always relative to one another, don't exist outside the mind, and the set between 2 and 4 is twice the set between 1 and 2. But like I said, it's a personal spin on it.

Where 1 f(x) = (x-1)/9 + 1, maps bijectively from [1,4] to [1,4/3]

g(x) = (x-1)/9 + 4/3, maps bijectively from [1,4] to [4/3,5/3]

h(x) = (x-1)/9 + 5/3 maps bijectively from [1,4] to [5/3,2]

Therefore, [1,2] must contain three times as many numbers as [1,4], right?

It doesn't work like that.

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