/sci/ - Science & Math

I would say the second one.

To tell you the truth, I definitely haven't seen infinity or differential variables used in equalities, having gone through Calc III. I would generally say that infinity is greater than 1, and differentials are less than one (and greater than zero).
Now I'm wondering the point of the question. Are these for some reason incorrect or not formal?

id say the first one, i feel the same way but when i m doing physics id say the second one cause using those ideas makes physics a little easier to understand.

TL;DR: if math then the first one, if physics then the second one, if you chose the third one your a fuckhead

>>5198742
infinity is not a real number, its a concept that makes calculus work. differentials make things become infinitly close to make a tangent line, however in the real world this is impossible. Also if you take the case where 0.999...=1 you assume that the 0.999.. goes to infinity. now is that infinity greater than 1 smaller or equal to one? Even in differentials you are using infinity to become very small. Infinity can be smaller than one or larger or approach a single value.

TL;DR
>>5198782

Infinity > 1 is true, because by definition, an infinite number is one which is greater than any natural (or real or rational, depending on construction) number.

Differential variables are not comparable with integers, in part because they are variables, not numbers. It just doesn't make sense because they don't have a defined size. Infinitesimals are, however, smaller in magnitude than any real.

>>5198812
> infinity is not a real number
That is true, in the sense that no infinite number is an element of R. However, there are classes of numbers containing infinities, and they are just as valid as the reals.

>>5198742
>Now I'm wondering the point of the question. Are these for some reason incorrect or not formal?

My question is whether your instructor views them as correct/incorrect or formal/informal.

There can be differences in approach among different instructors and textbooks.

Some feel that a statement such as ∞ > 1 is unnecessary to develop calculus, and so they may choose to say it's undefined. Others don't see the harm of saying ∞ > 1, because it doesn't yield any faulty application results, and it may help the student understand it better.

Not everyone teaches calculus the same way. Some of the more radical teachers even use what's called "nonstandard analysis", which develops calculus using an alternative number system that promotes differentials to be full-fledged numbers. That's an extreme example, of course, but there are other gray areas like whether ∞ > 1 is "true" or "undefined".

(I'm an aficionado of mathematical gray areas, like 0^0 and such. Many people claim to have a "simple" explanation of 0^0, but when you put various peoples' "simple" explanations together it's interesting to see how they disagree.)

>>5198837
>Infinity > 1 is true, because by definition, an infinite number is one which is greater than any natural (or real or rational, depending on construction) number.

Did you hear that from your instructor, or from a textbook?

Or is that just your own personal assessment?

(I'm just curious -- I have nothing against the claim that ∞ > 1 is true.)

In high school my teacher didn't really make note of it, but seemed to say it was valid. My current professor would be with the second option of it just helping understanding, but it not being formally defined, and I happen to agree with him there.

I essentially agree with
>>5198837

One should state that both symbols <span class="math">\infty[/spoiler] and <span class="math">dx[/spoiler] can be made sense of via various definitions.

The concept of infinity is well captured in the two (essentially algebraic) constructions of ordinal and cardinal numbers

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

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

The one orders numbers <span class="math">0,1,2,...[/spoiler] and eventually ending up at infinite numbers <span class="math">\omega[/spoiler] and so <span class="math">\infty[/spoiler] will be interpreted as some infinite number and then yes, the is an odering for which <span class="math">1<\infty[/spoiler].

Cardinal numbers represent the sizes of sets and if you define < to mean "this set or one with a bijection to it smaller than the other" then the equation is certainly true to, because an infinite set contains one set.

there is a wiki, which I consider funny:
it's basically a list of the constructed infinities
http://cantorsattic.info/Upper_attic

for differentials, there are a basillion definitions.
I usually like to think of them as functionals of a tangent space. Then they are maps and have no "value" as such. They need to be feeded a vector/direction to produce some number and the notion of infinitesimal smallness only comes from the interpretation of the Riemann integral (which some purists dismiss).

Taking the last perspecive more seriously, peolpe have formalized the notion of an infinitesimal

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

but as it doesn't seem particularly necessariy (for a physicist like me), I don't know if the relation <span class="math">dx<1[/spoiler] is part of the theory - intuitively, it will be considered correct there.

So again, it will depend on your definitions/formalizations of <span class="math">dx[/spoiler] and <span class="math">\infty[/spoiler]. I see no use in <span class="math">dx<1[/spoiler].

>>5198837
>Infinity > 1 is true, ...
>Differential variables are not comparable with integers, in part because they are variables, not numbers.

Infinity is not a number either. So your second argument seems to suggest that ∞ > 1 should be undefined.

Do you have an explanation for why infinity (a non-number) can be compared with a number, but yet a differential variable (also a non-number) cannot be?

(I'm not trying to pick a fight. I just want to better understand the nuances of what you were taught.)

>>5198888
>I see no use in dx<1.

Do you also see no use in ∞ > 1 ?

Or, more precisely, do you consider ∞ > 1 to be "true" or to be "undefined"?

>>5198837
>>>5198812
>> infinity is not a real number
>That is true, in the sense that no infinite number is an element of R. However, there are classes of numbers containing infinities, and they are just as valid as the reals.

I agree. But the question I'm asking is whether it's appropriate to extend the definition of > so that the expression (∞ > 1) is defined as being true. Is that something you would approve of or not? And did you have a teacher or textbook that had an opinion on that?

And the same question for differential variables, too.

I think it's okay to regard these as vaguely true, but not in any strict sense. Infinity and dx aren't quite numbers unless you treat them very carefully. See: non standard analysis and transfinite numbers.

>>5198742
>To tell you the truth, I definitely haven't seen infinity or differential variables used in equalities, having gone through Calc III

Well, infinity is definitely used as an equality in certain special constructs, such as:

lim (x -> 0+) = ∞

My question stems from this. If it's ok to say that ∞ can participate in this equation, then is it ok to say something like (∞ > 1)?

Do you recall if your instructor or textbook had anything to say about that?

>>5198929
>Infinity and dx aren't quite numbers unless you treat them very carefully.

Once you veer into that territory, then a lot of things change.

I'm more interested in the standard, conservative approach that's most common in teaching calculus.

The standard approach says -- unequivocally -- that:

"inifinty is not a real number, nor does it represent a real number"

and

"a differential variable is not a real number, nor does it represent a real number".

My question pertains to a more subtle point. I'm asking if your teacher implied that it's ok to extend the definition of the > operator by allowing it to compare a non-real with a real -- for example: ∞ > 1.

>>5198933

Boy did I screw that up.

I meant:

lim (x -> 0+) (1/x) = ∞

>>5198933
But that is not using the equality sign in a strict, meaningful way. It's just shorthand for "it diverges." Otherwise:
lim as x-> infinity (x) = infinity
lim as x-> infinity (2x) = infinity
lim as x-> infinity (x) = lim as x-> infinity (2x)
lim as x-> infinity (x) = 2*lim as x-> infinity (x)
1=2
Clearly invalid.

>>5198905
the dominant framework "people work in" (in principle) is Zermeno Freankel Set Theory as the foundation of mathematics. (I say in principle because you don't need to care about the foundations usually, but you'll not be able to name me something which doesn't fit in this theory.)
This is a particular theory in the language of (first order) logic with the symbol <span class="math">=[/spoiler] and the symbol <span class="math">\in[/spoiler] followed by about 15 axioms.
On of them is the axiom of infinity

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

which basically says there is a set of the integers. This is done via considering the empty set {} and then apply a function S to it n times.
so
1:=S({})
2:=S(S({}))
3:=S(S(S({})))
...
see
http://en.wikipedia.org/wiki/Successor_function
the numbers of S's correspond to the symbol. This works because we write down symbols to do math and people have an intuitive understanding for "the number of symbols I wrote down."
See the construction here

http://en.wikipedia.org/wiki/Peano_axioms#Set-theoretic_models

Notice what they do:
They define the function S to bracket the argument such that if you write 0:={}
then
1:=S(0)={0}
2:=S(1)=...={0,1}
5:=...={0,1,2,3,4}
This process gives a so called model of the numbers (this post is really about the model of numbers in set theory). And each number contains the previous numbers.

In a way, this is also how we teach machines how to think of numbers, when we program. Eighter we define the number of steps, of we definie objects as in
http://en.wikipedia.org/wiki/Lambda_calculus#Encoding_datatypes
to get a grasp of numbers.

Now there is a philosophical problem, namely that you can't in fact write down arbitrary long symbols, so formally, induction must be axiomatized

http://en.wikipedia.org/wiki/Axiom_of_induction#Axiom_of_induction

Moreover, there are some delicate logical barriers to induction too, see

http://en.wikipedia.org/wiki/Peano_axioms#Models

and so not everybody agrees on going to infinity, see

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

or even

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

This is why I demand to speak about something specific (i.e. ordinals, which can be formalized in Zermeno Freankel Choice set theory) and then you have no problem

As far as analysis goes, <span class="math">\infty[/spoiler] is basically used in contexts like <span class="math">\infty[/spoiler]=lim of n to arbitarty big values, so when you ask
>Do you also see no use in ∞ > 1 ?
my answer is a weak no, as I want this to deliver the message "I don't have to name the number of S's I use in S...S(S(S(1)))>1".
Notice here that I use the cardinal understanding of numbers: Every number contains the previous number (like 5:={1,2,3,4}) and the order > is modeled via set inclusion. Hence <span class="math">\infty[/spoiler] is interpreted to mean some infinite number (see link in the previous post) and so <span class="math">\infty>1[/spoiler] amounts to saying "I wanna work in a framework with infinite sets."

I'm not a finitist, I'm probably a modern formalist
http://en.wikipedia.org/wiki/Formalism_%28mathematics%29

I can take a sum from n to infinity, like 1 to infinity, therefore there are numbers between n and infinity and each next n is greater than n therefore infinity is greater than n so whatever i don't really care.

>>5198899
you might be interested in the answer to this question:

http://math.stackexchange.com/questions/36289/is-infinity-a-number

>>5198988

Thanks.

I think most people know by now that if there's any question AT ALL about what you mean when you use the word "number", then you need to qualify it.

If I think there's even the tiniest chance that the listener might be some kind of pedantic freak who loves to pick arguments and brag about his extensive knowledge by whipping out the hyperreals and the surreals and the Cantorian cardinals, then I'll make it a point to say: "infinity is not a real number".

But if I'm talking to a normal ordinary person who has just had some high school math and maybe a bit of college calculus, then I'll typically say "infinity is not a number" so that he won't look at me funny and ask me why I'm making such a big deal about the number being "real" -- and wonder if by emphasizing "real" I am somehow implying that infinity could be a complex number or something.

>>5198976
>>5198980

Thanks. That's all good stuff.

In this thread, my interest is more narrow. I am interested in what first-year calculus students are actually being exposed to in class, and in their textbooks. I am interested in how first-year calculus teachers are typically addressing (or not addressing) some of these finer points.

There's no way that an introductory calculus instructor has the bandwidth to talk about all the stuff that you did in your post. As a practical matter, the instructor must distill all of this down into an easily-teachable, easily-digestible summary that allows the students to understand the ideas correctly in a quick and efficient manner.

GIVEN THAT -- what would you recommend we say to a Calc 1 student who asks if ∞ > 1 ?

>>5199128
I'd say "We decide not to use it, but if you ever come across a problem where you think you actually need that relation, come back (and I'll probably work out why you don't - after all, \infty only shows up because you took a limit somewhere, and why would there be a limit in an equation with one side already evaluated?)"

>>5198980
>>Do you also see no use in ∞ > 1 ?
>my answer is a weak no, as I want this to deliver the message "I don't have to name the number of S's I use in S...S(S(S(1)))>1".

What about the idea of extending the > operator so that it can take non-real operands, in certain limited cases such as ∞ > 1 ?

That wouldn't require you to call infinity a "number". It would just require you to extend operators into new domains, in the same spirit that we extent real multiplication into complex multiplication.

{2 (n - 1) \choose n - 1}

>>5199150
>I'd say "We decide not to use it,....

Good answer.

The reason I started this thread is because I'm writing introductory calculus material.

I spend countless hours fine-tuning my explanations, making them both technically correct, and yet level-appropriate for a student with only 2 years of high-school algebra.

Here's my current draft language on this topic:

> "It's common to wonder if ∞ can directly participate in mathematical expressions.
> A statement such as '∞ > 10^100' seems, intuitively, like it should be a true.
> A statement like this can help students to achieve an
> understanding of infinity — but we actually do not need to use ∞ this way in calculus.
> Infinity has a role in calculus, but it's used only in special restricted
> circumstances, and it's used only when necessary. It's important to keep in mind that
> ∞ is not a number, nor does it represent a number, nor can it be used like a number."

It's probably got a few more rewrites to go, but I think it's getting mostly there.

>>5199150
>and why would there be a limit in an equation with one side already evaluated?

Maybe the limit isn't evaluated yet. Example:

(1): lim (x->0+) (1/x) > 1

(2): lim (x->0+) (1/x) = ∞

(3): Combining (1) and (2) together, we get: ∞ > 1