Reasons why infinity hasn't been implemented into modern math

[epkid08]

I don't really understand why it hasn't been numerically added into modern math. I mean, we make all kinds of properties for zero, but we can't make properties for infinity? It gets messy from time to time, but we could define everything strictly so that it still works algebraically.

Example: (quoted from http://mathforum.org/dr.math/faq/faq.divideby0.html)

5/0 = \infty
5 = 0*\infty
Multiplicative property of 0.
5=0
WRONG!

If we defined \infty numerically:
5/0 = 1/0
5 = 0/0
We then, could define 0/0 as truly undefined, or all real numbers, or some other name. I don't think we need to classify something as undefined, or anything for that matter, unless it is 0/0, 0^0, \infty^\infty etc.

As for infinity, it should be implemented carefully into our modern math.

Now it's time for you to post "Reasons why infinity hasn't been implemented into modern math."

[slider142]

Infinity is present everywhere in modern math. From simple implementations such as the extended real numbers (http://en.wikipedia.org/wiki/Extended_real_numbers) to the more abstract cardinal (http://en.wikipedia.org/wiki/Cardinal_numbers) and exotic ordinal numbers (http://en.wikipedia.org/wiki/Ordinal_numbers).

[CRGreathouse]

Don't forget the projective reals!

[matt grime]

Or indeed the projective anything, the compactifications of spaces, the Riemann sphere, the theory of poles and singularities going back hundred+ years, Laurent series, Mobius transformations,....

[epkid08]

I'm talking numerically, 1/0. You won't ever see its notation in a function etc...

[Hurkyl]

I'm talking numerically, 1/0. You won't ever see its notation in a function etc...
Unless you're working with extended real numbers, cardinal numbers, projective reals, projective anything, compactified spaces, the Riemann sphere, meromorphic functions, Laurent series, Möbius transformations....

You don't see an infinite integer simply because the ring of integers doesn't contain such a thing. The ring of integers doesn't contain 1/2 either.

[kts123]

I'm talking numerically, 1/0. You won't ever see its notation in a function etc...

You won't usually see it in a function over the real numbers, in the same sense you won't see 1/2 in functions over the integers. Infinity is a special class of number, just like rational numbers, irrational numbers, transcendental numbers, imaginary numbers, complex numbers, etc. If you're working with functions that have a domain and range over the real numbers you won't see infinity (OR imaginary numbers or complex numbers.)

[CRGreathouse]

Actually, I'm pretty tired of the word "infinity". A good tenth of the posts on these math forums could be avoided if posters instead named the sort of infinite number they were thinking about. The one-point compactification of the reals? Aleph-2? Epsilon-naught? The IEEE +Infinity?

[kts123]

location.reimannsphere(santa) ???

[epkid08]

You won't usually see it in a function over the real numbers, in the same sense you won't see 1/2 in functions over the integers. Infinity is a special class of number, just like rational numbers, irrational numbers, transcendental numbers, imaginary numbers, complex numbers, etc. If you're working with functions that have a domain and range over the real numbers you won't see infinity (OR imaginary numbers or complex numbers.)

My main problem is that low-level algebra defines infinity as undefined, when in reality we have no reason to classify it so. Zero is also a special class of number, which has its limited uses, but we find the time to define that.

[matt grime]

It doesn't 'define infinity as undefined' (which is a contradiction in terms). It merely, and correctly, states that you can't cancel off zeroes in multiplicative expressions.

[arildno]

I don't really understand why it hasn't been numerically added into modern math.
My main problem is that low-level algebra defines infinity as undefined, when in reality we have no reason to classify it so.

So, according to you, low-level algebra is the heart of modern maths?

[Hurkyl]

Well, epkid08, your post has been answered: mathematics does 'implement infinity'. Allow me to suggest that, before you continue ranting, you spend some time studying mathematics so as to gain an understanding both of how mathematics works and how the notion of infinite is treated and used in mathematics.

Incidentally, your comment on zero is unfounded. It is so incredibly useful that natural languages have evolved to give it special treatment -- the difference between 'zero' and 'not zero' is quite deeply ingrained in grammar. (Apparently so deeply that some people have difficulty comprehending how they could be united into a notion of 'quantity') Grammar can also split the latter case into 'one' and 'more than one', but not in such an essential way. (There is some limited direct linguistic support for larger quantities, but going beyond one usually entails using ordinal or cardinal numbers)

[arildno]

. Zero is also a special class of number, which has its limited uses, but we find the time to define that.

I hardly ever use 2.17, but I am really glad it has already been defined! :smile:

[D H]

, your comment on zero is unfounded. It is so incredibly useful that natural languages have evolved to give it special treatment -- the difference between 'zero' and 'not zero' is quite deeply ingrained in grammar. (Apparently so deeply that some people have difficulty comprehending how they could be united into a notion of 'quantity') Grammar can also split the latter case into 'one' and 'more than one', but not in such an essential way. (There is some limited direct linguistic support for larger quantities, but going beyond one usually entails using ordinal or cardinal numbers)
As an aside, one of my jobs is writing software standards. One such is "All magic numbers shall be named and referenced by that name. For example 'circumference=2*3.14159*radius' violates this rule and is wrong to boot. Pi is a magic number. Whether two is a magic number is debatable. A good starting point is that the only non-magical numbers are zero and one. You can use an unnamed small integer if the usage is well-commented." I never mentioned that zero and one, being the root of almost all mathematics, are actually the most magical numbers of all.

[D H]

My main problem is that low-level algebra defines infinity as undefined, when in reality we have no reason to classify it so.

No. It does define infinity in terms of limits. It does not define operations with infinity as undefined. It simply does not define such operations, period, and for good reason. They are a superfluous and confusing distraction in the topic at hand, which is getting students to grasp the main concepts of elementary algebra.

It doesn't 'define infinity as undefined' (which is a contradiction in terms).
That's going a bit too far. Computer science can be viewed as an offshoot of applied logic / applied mathematics. Several computer programming languages explicitly define "undefined behaviors". Defining what is undefined is not inherently a contradiction in terms. It is just a warning about the potential for nasal demons.

[kts123]

My main problem is that low-level algebra defines infinity as undefined, when in reality we have no reason to classify it so. Zero is also a special class of number, which has its limited uses, but we find the time to define that.

We call it undefined for the same reason we tell grade-schoolers they can't sqaure root negative numbers -- it'd confuse the hell out of them and it's off-topic. When their minds are ripe, and the contexts of our discussion is proper, these concepts are introduced. Crawl before you can walk.

Oh, and, surprise, some infinities are bigger than others and we can divide by zero. Hehehe.

[matt grime]

Computer science can be viewed as an offshoot of applied logic / applied mathematics. Several computer programming languages explicitly define "undefined behaviors". Defining what is undefined is not inherently a contradiction in terms. It is just a warning about the potential for nasal demons.

Declaring something to be undefined is not the same as *defining* it to be undefined. It might be a useful bastardisation of the language, but it is technically cobblers.... "coming up at ten: what is a pin head and how many angels can we make dance on it."

[kts123]

To help explain what Mr. Grime has said, contemplate the following:

"My talent is not having any talents." as opposed to "I have no talents." One makes a declaration, wheras the other implies a sentiment (and in such a way that in contradicts itself.)

[D H]

Well OK then. Is it permissible to say that some operations in mathematics are declared to be undefined, or one must simply say that some operations in mathematics are not defined?

To the OP: This is true even for the extended real number line. For example, the values of 0/0 and \infty/\infty are not defined.

[LukeD]

Well OK then. Is it permissible to say that some operations in mathematics are declared to be undefined, or one must simply say that some operations in mathematics are not defined?

What he means is that when something is undefined, there's no object "undefined" or even 0/0 or infinity/infinity or whatever. They simply don't exist. When you write them in a formula, then you don't have a grammatically well formed formula. It's as meaningless as "5x^2*1 + / = 2x +", just a random string of math symbols. What you've written doesn't have any meaning until you actually do define it.

There are a few engineering and computer science things that are similar to "undefined", but I don't know anything in math like that.

[epkid08]

I remember back in the fifth grade (or whenever), when I was first being taught (x,y) coordinates, and linear functions, any vertical line had a slop that was 'undefined'. Is this only because 'their minds are too ripe'? I mean, I remember back then, I questioned the this, as I thought it should have a slope of infinity. I can definitely see how it would be confusing, depending on how you teach it, but you wouldn't have to get so specific as to confuse the children. For example, if you define the equation, x=5, as having a slope of infinity, the kids wouldn't understand why you can't put it into slope-intercept form.

But my point is that, with the right properties, infinity can be implemented into algebra.

[Take_it_Easy]

But my point is that, with the right properties, infinity can be implemented into algebra.

I agree but some problem would occasionally show up.
How much is \frac{\infty}{\infty}? Or {\infty}-{\infty}??????

[epkid08]

I was only talking about infinity. We don't have uses for those, and other, indeterminates anyways. They lie off of any number line.

Edit: If your post was directed at the fact that kids might get into trouble by trying to divide infinity by infinity etc., we can make properties so that kids don't get mixed up and do that.

[Hurkyl]

any vertical line had a slop that was 'undefined'. ... I mean, I remember back then, I questioned the this, as I thought it should have a slope of infinity.
If you generalize the notion of slope, it turns out the projective real numbers are the 'right' number system to use to measure slopes. And then, a vertical line does indeed have generalized slope equal to projective infinity.

Just for emphasis, this is only true for this generalized notion of slope -- it is perfectly correct to say that the ordinary notion of slope is inapplicable to a vertical line.


But my point is that, with the right properties, infinity can be implemented into algebra.
Have you not been listening? Not only can it be 'implemented' in algebra, it has.

I was only talking about infinity.
Which 'infinity' or otherwise infinite number are you talking about? Or do you even know what you're talking about?

We don't have uses for those, and other, indeterminates anyways. They lie off of any number line.
They can't lie off any number line -- they don't even exist. :tongue: (Assuming Take_it_Easy was referring to projective infinity, or the positive extended real infinity)

[Take_it_Easy]

I was only talking about infinity. We don't have uses for those, and other, indeterminates anyways. They lie off of any number line.

Edit: If your post was directed at the fact that kids might get into trouble by trying to divide infinity by infinity etc., we can make properties so that kids don't get mixed up and do that.


I see.
In fact it is a possible and logic approach.

We learn to be careful (that the divisor is 0) when a division occur, so we can just introduce the propertyies for infinity and warn to be careful when doing such an operation.

By the way, we have to be always careful in every elementary operation.
But that's not bad after all!

[epkid08]

Have you not been listening? Not only can it be 'implemented' in algebra, it has.

Please post an example of usage of numerically defined infinity in algebra.

[D H]

There are several examples throughout this thread, starting from the very first two responses.

[LukeD]

Please post an example of usage of numerically defined infinity in algebra.

Extended reals, projective reals, projective complex numbers, ordinal numbers, cardinal numbers, hyperreal numbers, surreal numbers

need any more examples?

[matt grime]

YOu should't leap in with such counter examples before the OP has defined what he thinks 'numerically defined' means. I have no idea what that phrase indicates.

[epkid08]

al·ge·bra [al-juh-bruh]
–noun
1.the branch of mathematics that deals with general statements of relations, utilizing letters and other symbols to represent specific sets of numbers, values, vectors, etc., in the description of such relations.

This is the definition I used as algebra. Infinity numerically defined would be n/0.

[n_bourbaki]

In that case, the extended complex plane would be the (or an) example you seek. As has been pointed out several times in this thread.

[epkid08]

In that case, the extended complex plane would be the (or an) example you seek. As has been pointed out several times in this thread.

part of the definition of your suggestion-
On a purely algebraic level, the complex numbers with an extra infinity element constitute a number system known as the extended complex numbers. Arithmetic with infinity does not obey all of the usual rules of algebra, and so the extended complex numbers do not form a field.

If it doesn't fallow the rules of algebra, then it's not true algebra.

[n_bourbaki]

Oh, what are the precise definitions of "algebra", whatever that nebulous concept is supposed to be? Not that it matters - it is clear that nothing anyone will say will dislodge your preconceptions.

[D H]

al·ge·bra [al-juh-bruh]
–noun
1.the branch of mathematics that deals with general statements of relations, utilizing letters and other symbols to represent specific sets of numbers, values, vectors, etc., in the description of such relations.

This is the definition I used as algebra.
That is a dictionary definition. It has little meaning to a mathematician.

Infinity numerically defined would be n/0.
No. There are many ways in which division by zero will get you in a lot of trouble, which is why division by zero is not defined in any of the systems described many times in this thread that incorporate the concept of "infinity" as a number.

If it doesn't fallow the rules of algebra, then it's not true algebra.
What, pray tell, are "the rules of algebra"?

One big problem with making "infinity" a number is that the result is an algebraic structure with a lot less to it than the reals. The reals form a field. The extended real number line doesn't even form a ring.

[Hurkyl]

There are many ways in which division by zero will get you in a lot of trouble, which is why division by zero is not defined in any of the systems described many times in this thread that incorporate the concept of "infinity" as a number.
Correction -- dividing a nonzero number by zero is defined in the projective complex numbers (and projective 'anything'), with value projective infinity. Wheels even allow 0/0.

Of course, this is only if you interpret the '/' symbol as denoting that structure's division operation. If we instead interpret '/' as being integer division, then 1/0 is still undefined, even if we want to work in the projective complexes.

[okkvlt]

Something ive been thinking is

if 0^-1 is infinity, then infinity^-1=0, right?

It actually makes alot of sense, both intuitively and algebraicly

[HallsofIvy]

No, they don't make sense for exactly the reasons everyone has been giving, didn't you notice?

[throng]

infinity is strange. it destroys alot of simple algebra if given real qualities. Would i be wrong to say infinity can be infinitely large or infinitely small but doesn't take any other value.

[karthikgnv]

hi dude...if you figure out how a division works..u can always sort the mess you have about the infinity
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.lets analyse how a division works.....

for eg: take 10/2.....the answer is 5
now...what the number 5 tells you about???
it tells you that..."If you repeated ly substract number 2 from number 10 until zero comes, you should do it five times"...after all division is a repeated substraction
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applying the same for your doubt......
if you take 5/0 the answer is obviously INFINITY
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because you can remove the number 0 from the number 5 countless times and still you cant do that until your number 5 becomes zero...
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thats why you cant say that 5/0 is equals to 1/0
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because removing 0 from 5 is different from removing 0 from 1
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is that clear???
is my explanation convincing???
let me know plz...
Thankyou

[Hurkyl]

Would i be wrong to say
You would be better off to read the comments that mathematicians have made in this thread (and in other threads you could find by searching for 'infinity').

[karthikgnv]

hello....Hurky..is that correct..which I wrote abt how we should understand the notation of infinity????

[throng]

I'm toying with the total quantity of energy. it wasn't created (always was) can't be destroyed (always will be) so seems infinite in terms of "eternity". All things are "made of" energy. if energy is everything is its total joules infinite?

I would really appreciate any advise in this regard.

[HallsofIvy]

1. You are assuming a lot of things that are not obvious. We say that something is "infinite" if it is unbounded right now. That has nothing to do with "eternity". The fact that the total amount of (mass-) energy in the universe doesn't change does not imply that it is "infinite".

2. How does this have anything to do with the topic of this thread?

[Hurkyl]

hello....Hurky..is that correct..which I wrote abt how we should understand the notation of infinity????
No, it is not.