1 / 0 = infinity?

As it stands 1/0 is "undefined".

"Approach" is used when talking about 1/x, as x -> 0. However strict pedantry requires one to say "becomes infinite" rather than "approaches infinity".

 

What 1/0 is -- or if the expression even makes sense at all -- depends on what you mean by 1, by 0, and by /.

In the real number system and the extended real number system, it's nonsense.

In the projective real number system, it's equal to [itex]\infty[/itex].

 

The trick is to realize that infinity is not a single object like, say, the number 27. It is certainly not a real number.

Zero is a slightly unusual real number.

It was Cantor who first proved that there are many infinities. Some are 'greater' than others.

The knowledge he derived allow us to evaluate such expressions as

[tex]\frac{\infty }{\infty }[/tex] and [tex]\frac{0}{0}[/tex]

By L'Hopitals rule or other methods and come up with real number answers.

 

Is this equation always true?

5 * (x/x) = 5

 

1/0 by itself is undefined. You only deal with infinities if you set a variable to approach something, such as 1/x where x approaches 0.

@Algr: Because there is an x as the denominator, you must set a restriction on what x can be. In other words, there is a domain where x is all real numbers except 0.

 

If we are talking about real numbers, then the answer is either:

• The equation is true -- because x is a variable restricted to some domain of nonzero reals
• The equation is nonsense -- because the domain x varies over includes zero

(Note that "The equation is false" is not the same thing as "this equation is nonsense")



Another possibility opens up if you are working with partial functions, which works out to that equation being synonymous with the statement "x is nonzero".

 

What number system do you want to use?

In the system called "Riemann sphere" it is indeed true that [tex]1/0=\infty[/tex]. In the real number system this is not true, since there is no real number [tex]\infty[/tex] and operation [tex]1/0[/tex] is not defined there.

And, of course, there are many more numbers systems, too.

So ... Which number system are you using?

 

Another problem with 1/0 is the fact that it could be [itex]\pm \infty[/itex] in a sense, depending on which end you approach zero from.

Also, 1/x where x is infinitely small (but not zero) would equal infinite too. So, is 1/0 infinite or undefined?

 

In the extended reals, 1/0 is undefined (not +∞ or -∞).

In the hyperreals, if e is infinitesimal but nonzero, 1/e is infinite (the exact value depends, naturally, on e). But even there 1/0 is undefined.

In the projective line, just as in the Riemann sphere, 1/0 = ∞.

 

Yes, but I was hoping the OP would come to that conclusion for himself.

And of course the OP is talking about the real number system, else the entire group discussion would have been futile since they would have already learnt the difference of 1/0 in each number system.

 

(emphasis mine)

I'm not so sure. If someone posted "What is 3 divided by 2?", I wouldn't say, "3 can't be divided by 2 if you're working in the integers; 1 and -1 are the only units in Z". I would assume that the OP was working in a system where 3/2 made sense.

Similarly, the question "Is foo infinite?" doesn't make sense in a system, like the real numbers, without a concept of "infinite". I would sooner assume that they're working in a system cobbled together out of the real numbers and a misunderstanding of the calculus limits +∞ and -∞. This system seems to resemble the extended reals, so usually this is my first guess as to the best way to formalize such questions.

If that's how the question is understood ("In the extended reals, is 1/0 infinite?"), then the answer is no: it is undefined.

 

the simplest way to see why 1/0 is an undefinable fraction in the real numbers system is to assume the contrary: let's say
[tex]\frac{1}{0}=a[/tex]
then
[tex]0a=1[/tex].

how many [tex]a\in\mathbb{R}[/tex] do you know that satisfy that relation?
in general actually, division by 0 is undefined in any field, the argument for that is similar.

 

The thing is, if the OP was working in a system where 3/2 made sense then the question wouldn't have been asked in the first place. By the sounds of it, he was disputing between peers whether 1/0 is infinite or undefined. This is obviously indicating that they're using a number system where the norm was to simply state 1/0 is undefined, but 1/0 can also be seen naturally as infinite by the sloppy use of limits in calculus as you've said.

But yes, I see the necessity to ask which number system one is intending to work with.

How about [itex]a=\infty[/itex]? The OP has already said this:

so you haven't really swayed the discussion in favour of 1/0 being undefined.

There are even examples in limits with calculus that show that a is undefined (1/0). Take the functions [itex]f(x)=x[/itex] and [itex]g(x)=csc(x)[/itex]. f(0)=0, g(0) is undefined. But f(0)g(0)=1 (well, as x tends to 0 of course). Now, if you let g(0) be the value a you mentioned then there you go, the answer is 1/0.
Now, is 1/0 infinite or undefined? :tongue:

 

Er, "your question is nonsense" is a perfectly reasonable answer to a nonsensical question. :tongue: In real number arithmetic, "1/0" is the semantic equivalent of an English phrase like "the taste of ideas" -- those English words just don't combine in that fashion, nor do those mathematical symbols.

(In a variant syntax based on partial functions instead of functions, 1/0 does make sense, but it doesn't have any value at all. e.g. the equation 1/0=x is identically false)

Where you get that idea?

Process? What process?

If your intuition tells you that, in the real numbers, 1/0 is infinite, then there is an error in your intuition, plain and simple. We have given you keywords that would let you go searching for further information on other number systems. (Another keyword: wheel, although AFAIK those are more of a curiousity than something people actually use)

However, if you're going to insist that all of mathematics must conform to your intuition the way it is right now, then you aren't going to learn anything.

 

Why do they all have to be in the same number system?

There are many processes, both finite and infinite, in mathematics where the result of a process leads outside the origin set.

So why can the process of taking a reciprocal not lead outside the origin set in the case of zero?

Edit

The statement 'the result of this process is undefined' is really a short way of saying 'the result of this process is not a member of the origin set.' In some instances it has been worthwhile establishing what the result of the process is - even if it has meant creating a new mathematical object.

 

Process? What process?

Zero is not in the domain of the real number reciprocal, so obviously one cannot take the real number reciprocal of zero to get something that isn't real.

 

The domain or origin set are whatever the originator choses it to be. Mathematics does not enfoce any particular choice, it (merely/) describes the consequences of that choice.

 

Sure -- you can work with whatever functions you like. But if 0 is in the domain of that function, it's not the real number reciprocal function, and theorems about the real number reciprocal function will typically not apply to the function you want to work with.

 

Hey, I'm not trying to start an argument here - I'm trying to offer a slightly different viewpoint of the same thing. Some may find the alternative viewpoint easier to swallow.

For instance I have started with the domain and asked what functions are of interest?

Rather than starting with the function and asking what results are of interest?

 

Your argument rests on the assumptions:
1. 1 / ∞ = 0
2. a/b = c implies a/c = b

where #2 probably comes from
3. a/b = c if and only if a = bc
4. a = b if and only if a/c = b/c
5. ab/b = a

#3 through #5 (and thus #2) are valid rules for nonzero real numbers, but you have ∞ so you're not working with real numbers. But then, why assume that #3 through #5 still hold?


There are ways to extend the real numbers to cover division by zero. Define "/" as the preimage of multiplication: that is, a "/" b is the set {c: b * c = a}. Then for a,b in R and b not zero, a "/" b = {a/b}; for a in R and a ≠ 0 = b, a "/" b is the empty set {}; for a = b = 0, a "/" b is the set of real numbers R. If you'd like you can try to extend this to include ∞, however you choose to define that.

 

May I ask a stupid question? (yes, that was typed tongue-in-cheek)

Are the theorems for non real numbers invalid for real numbers (and vice-versa)?

 

A theorem that says, "If x is a real number, ..." does not let you conclude "If x is not a real number, ...". Similarly, a theorem that says, "If x is not a real number, ..." does not let you conclude "If x is a real number, ...".

But you might have a theorem of the form, "If x is an extended real number, ..." which does let you conclude "If x is a real number, ..." since all real numbers are extended real numbers.