
Infinity is not a number - tokenadult
http://en.wikibooks.org/wiki/Calculus/Infinite_Limits/Infinity_is_not_a_number
======
impendia
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,
you can define the "extended real number line" (cf. the Wikipedia article of
that title) which includes infinity and satisfies a list of axioms which you
can write down.

This article is dangerously misleading. "Numbers" is not a well-defined set
and there is no way to say that infinity does or does not belong to it. As is
often the case in mathematics, you need to make the discussion more precise
before you can reasonably answer questions.

~~~
jpallen
_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 comments here are getting hung up on trying to pin down the
mathematics of what you can and can't do with infinity. That's fine, it's been
keeping mathematicians busy for centuries, but this article is for the layman
who doesn't know about rings, groups, algebras or any other mathematical
structure which you might call 'numbers'. It's for someone who thinks the
obvious when someone says 'number'.

~~~
mcpherrinm
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"? You appear to think irrationals are numbers, so what
physical thing can you have an irrational number of?

This is a dangerous path to walk down, but that doesn't make it an unimportant
one. And that's why this article is ultimately flawed. The idea is important
though. We need to understand how to operate formally on mathematical objects.
If there's one thing I learned in my Philosophy of Mathematics class as a math
undergrad, it is that we shouldn't struggle over defining what it means to be
a number. We can just use them. I'll leave the philosophy to the philosophers.

~~~
jpallen
I pretty much agree with you. I'm just worried that with all the comments
about 'what is a number?', and 'sure, we can make infinity a well defined
number', it will dilute the message of the article. Which is that infinity is
not a 'number' (in the sense of real numbers). It's important to understand
why it's not a number and what it means when we write it on one side of an
equality.

------
yequalsx
The article deals with the infinity symbol as it is used in calculus courses.
With such a usage in mind the infinity symbol is not a number and does not
represent a number. The usage of the infinity symbol in calculus is merely a
shorthand notation for a more complicated statement.

Suppose that we have

Lim x->5 f(x) = infinity

What is meant by the use of the infinity symbol is that the limit is not
bounded in the real number system. More specifically that given any large real
number I can find a number d such that whenever |x - 5| < d then f(x) > M.

Here the use of infinity is not meant to be as a number though making such an
association is helpful to beginners in terms of visualizing what is going on.
Students have trouble with the precise definition of being unbounded and so
it's convenient to say "it's infinity" and treat the symbol as a number.

------
thaumasiotes
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 expect (what have I missed?).

The immediately obvious uses of infinity in calculus (the topic of the
wikibooks article) are, according to wikipedia, termed the "affinely extended
real number system" (I learned to just refer to the "extended reals"), which
heavily implies that the points within are considered extended real _numbers_.
<http://en.wikipedia.org/wiki/Extended_reals>

The terms "cardinal number" and "ordinal number" both definitively include
infinite quantities -- infinitely many, even.

The IEEE standard for floating point defines two infinite numbers.

Essentially, I'm in full agreement with tokenadult; the only relevant question
is "what do you mean by number?". But we can easily observe that varying
infinities, including the calculus uses of infinity, are referred to as
numbers all up and down the chain, including in the most unimpeachably correct
sources, and that it walks and quacks like a duck, even if it may not quack in
the precise manner of Anas Platyrhynchos.

~~~
drt1245
> 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)>

~~~
mturmon
Sometimes you may use a "number", but you just want ordering properties from
it.

You may not want addition and multiplication in such a "number".

This may be the case in numbers used for ranking outcomes, or counting, or
optimization. Using infinity in this context does not cause problems.

------
T_S_
It turns out that treating infinity like it _is_ a number is very convenient
(e.g. in convex optimization).

I think these black and white statements are not a good way to explain things.
Just start with the set of real numbers. Point out infinity is outside the
set. Then introduce the extended reals (R + infinity), with special rules for
arithmetic involving +/-infinity. This theoretical convenience extends to the
computer when the rules for Inf are implemented correctly. Follow the rules
and you might sensibly get an Inf result, break them and you should get a NaN.
It's all implemented in IEEE 754. In fact Inf is in your computer but not all
the reals are. So there.

------
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).

------
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.

------
powertower
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
numbers between 1 and 2.

I would guess that if numbers are defined in terms of relativity/relationship,
then infinity is a number.

But it seems that people wrongly define numbers in absolute terms, as if they
exist outside the mind, and are separate from one another. Like the Universe
cares about 1.24545434 and 7656.45433477.

But that's just my guess.

~~~
fferen
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.

~~~
powertower
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.

~~~
drt1245
Where 1 <= x <= 4:

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.

~~~
powertower
I'm not sure what you are doing with the above.

------
firefoxman1
Javascript and I both disagree. Type the following into your JS console:

    
    
      typeof Infinity
    

(Obviously I'm joking. I'm not going to argue with the mathematics on that
page)

------
sixbrx
The article only shows it is not an ordinal number.

~~~
tokenadult
_The article only shows it is not an ordinal number._

Because I'm a Wikipedian, I've learned that whenever I visit an article on
Wikipedia, Wikibooks, etc. I can visit the article talk page too. Many of the
usual misconceptions about infinity can be found in the talk page of that
article. Another article showing that infinity is not a number

[http://scienceblogs.com/goodmath/2008/10/infinity_is_not_a_n...](http://scienceblogs.com/goodmath/2008/10/infinity_is_not_a_number.php)

which has previously been submitted to HN

<http://news.ycombinator.com/item?id=331581>

evoked discussion that illustrated confusion about what a number is, as did a
different HN submission

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

of a very interesting article

<http://nrich.maths.org/2756>

by a young mathematician with some demonstrated chops in mathematics.

The last discussion of this issue on HN, which appears to have been from about
two years ago, was interesting, so when I saw the article submitted here today
(while looking up sources for the teaching I do), I thought I'd invite HN
participants to discuss the issue again.

Two follow-up questions:

1) What do you mean by number?

2) Supposing the claim is that infinity is a number, how would that claim be
verified by accepted principles of mathematics?

~~~
yequalsx
I'll discuss various aspects of the issue.

I think it is fair to say that a number should be an object of a ring. I'm an
algebraist though. That is, there should be an addition operation and a
multiplication operation that satisfy certain conditions.

1\. Calculus. One sees in calculus things like

Lim x->5 f(x) = infinity

In this case infinity is not a number but rather a shorthand notation for a
more complicated statement. What is meant by the use of the infinity symbol is
that the limit is not bounded in the real number system. More specifically
that given any large real number I can find a number d such that whenever |x -
5| < d then f(x) > M. Here the use of infinity is not meant to be as a number
though making such an association is helpful to beginners in terms of
visualizing what is going on.

2\. There are ordinal numbers that are infinite. That is, that represent the
order type of an infinite well ordered set. Ordinal numbers do not for a ring
but they do have an arithmetic defined. They do form a semi-ring though. If
one wants to say that objects of a semi-ring are numbers then there are
infinite numbers. This also applies to cardinal numbers.

3\. There is the extended real number system which has the symbols -infinity
and positive infinity attached to the real number system to form a compact
set. Think of the compact closure of the reals. Again not a ring though.

4\. That said, there are infinite sets. An infinite set is one that is not
finite. Or, in more precise terms, and infinite set is one that can be put
into 1-1 correspondence with a proper subset of itself. Cardinal numbers and
cardinal number arithmetic deal with comparing sizes of sets. Mostly useful
for dealing with infinite sets. Some infinite sets are bigger, in a meaningful
way, than other infinite sets. The set of reals is much bigger than the set of
integers.

------
ec429
What about nonstandard analysis?

Or, to put it another way, Infinity is a hyperreal number.

------
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://scienceblogs.com/goodmath/2008/10/infinity_is_not_a_number.php)

<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...](http://scienceblogs.com/goodmath/2006/12/nullity_the_nonsense_number_1.php)

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...](http://mathworld.wolfram.com/AffinelyExtendedRealNumbers.html)

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?

~~~
nandemo
(Check your link to the WolframMathWorld article)

> _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_

Like impendia said, it depends on the context.

To give a simpler example: at elementary school I was told that I couldn't
write "2 - 3". Essentially my teacher was saying that numbers less than zero
did not exist! She wasn't lying to me. She was telling me that we were working
with natural numbers only, and that our operation "-" was supposed to be
closed on the natural numbers.

In the context of calculus, infinity is not a number. lim(n->c) foo = ∞ is
just another way of saying that "foo does not converge as n goes to c".

But in realm of surreal numbers (or combinatorial game theory), infinities are
first-class numbers. You can add them and subtract them alright! Besides, in
that realm ω + 1 is not the same as ω. It is greater than ω, as expected. I
can even show you a children's game -- that is, we can teach kids how to play
it without ever uttering the words _sets_ and _cardinals_ \-- that has
positions with such values.

