Is ##\frac{0}{\infty}## equal to 0 or infinity in mathematics?

In summary: Yes, defining these operations can lead to inconsistencies and contradictions in mathematics. It is important to set clear and consistent rules for operations, and including infinity as a normal number can lead to issues. Therefore, mathematicians have developed specific rules and contexts in which these operations can be used, but they are not considered standard or general-use operations.
  • #1
Rectifier
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Is ## "\frac{0}{\infty}"=0 ## ?
 
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  • #2
Rectifier said:
Is ## "\frac{0}{\infty}"=0 ## ?
No, it is undefined because you are treating infinity as though it was a number. It isn't. You can't, in most circumstances, use infinity in normal math and expect meaningful results.
 
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  • #3
Rectifier said:
Is ## "\frac{0}{\infty}"=0 ## ?
## \infty ## isn't a number, so you shouldn't be dividing by it.
That said, in most contexts, yes, ## \frac{0}{\infty}=0 ##
 
  • #4
.Scott said:
## \infty ## isn't a number, so you shouldn't be dividing by it.
That said, in most contexts, yes, ## \frac{0}{\infty}=0 ##
No, it's still undefined in most contexts, I think. What IS true is that as n approaches infinity, 0/n approaches 0.

EDIT: I hasten to add, I don't do any math where there IS any meaningful context for 0/infinity, so I could be wrong. What contexts did you have in mind?
 
  • #5
phinds said:
No, it's still undefined in most contexts, I think. What IS true is that as n approaches infinity, 0/n approaches 0.
OK, in most meaningful contexts.
As a matter of practicality, if someone ended up with infinity in the denominator, they were probably using shorthand in dealing with a limit. The type of nomenclature is commonly used to explain limits - with the caveat that it is not proper general-use nomenclature. Since the OP put the fraction in quotes, I believe the he already knows about this.

The symbol ## \infty ## can be used in other more proper contexts, none of which would allow it to be in a denominator. One is to indicate the result of a limit. Another is to indicate the cardinality of a set, although Aleph is better at that. In cases where the order of Alephs is important, the omega symbol is used. Division can be defined for the omegas, but significant caveats are needed.
 
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  • #6
phinds said:
What contexts did you have in mind?

I was trying to calculate following limit:
## \lim_{x\rightarrow \infty} \frac{x^4 + x \ln x}{x + \left( \frac{2}{3} \right)^x} = \lim_{x\rightarrow \infty} \frac{x^4 \left( 1 + \frac{\ln x}{x^3}\right) }{ x \left( 1 + \frac{ \left( \frac{2}{3} \right)^x }{x} \right) } = \lim_{x\rightarrow \infty} x^3\frac{1 + \frac{\ln x}{x^3} }{ 1 + \frac{ \left( \frac{2}{3} \right)^x }{x} } ##

## \lim_{x\rightarrow \infty} x^3 = \infty ##

## \lim_{x\rightarrow \infty}\frac{\ln x}{x^3}=0 ##

## \lim_{x\rightarrow \infty} \left( \frac{2}{3} \right)^x = 0 ##

## \lim_{x\rightarrow \infty} \frac{ \left( \frac{2}{3} \right)^x }{x}
= "\frac{0}{\infty}"##

HERE-----------------^
 
  • #7
Rectifier said:
I was trying to calculate following limit:
## \lim_{x\rightarrow \infty} \frac{x^4 + x \ln x}{x + \left( \frac{2}{3} \right)^x} = \lim_{x\rightarrow \infty} \frac{x^4 \left( 1 + \frac{\ln x}{x^3}\right) }{ x \left( 1 + \frac{ \left( \frac{2}{3} \right)^x }{x} \right) } = \lim_{x\rightarrow \infty} x^3\frac{1 + \frac{\ln x}{x^3} }{ 1 + \frac{ \left( \frac{2}{3} \right)^x }{x} } ##

## \lim_{x\rightarrow \infty} x^3 = \infty ##

## \lim_{x\rightarrow \infty}\frac{\ln x}{x^3}=0 ##

## \lim_{x\rightarrow \infty} \left( \frac{2}{3} \right)^x = 0 ##

## \lim_{x\rightarrow \infty} \frac{ \left( \frac{2}{3} \right)^x }{x}
= "\frac{0}{\infty}"##
That last limit is 0. Don't write it as ##\frac 0 {\infty}##, either with or without quotes.

BTW, I am moving this thread to the technical math section, as it is more of a conceptual question than an actual homework-type question.
 
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  • #8
I don't understand why the last limit is 0. :,(
 
  • #9
I think I got it.## \lim_{x\rightarrow \infty} \frac{ \left( \frac{2}{3} \right)^x }{x}
= \lim_{x\rightarrow \infty} \frac{ 1 }{x} \left( \frac{2}{3} \right)^x ##

## \lim_{x\rightarrow \infty} \frac{ 1 }{x} = 0 ##

## \lim_{x\rightarrow \infty} \left( \frac{2}{3} \right)^x = 0 ##
 
  • #10
Rectifier said:
Is ## "\frac{0}{\infty}"=0 ## ?

as for me, this notation is completely clear. If we have two functions ##f(x)\to 0## and ##g(x)\to \infty## as ##x\to a## then ##f/g\to 0##
 
  • #11
Rectifier said:
I think I got it.## \lim_{x\rightarrow \infty} \frac{ \left( \frac{2}{3} \right)^x }{x}
= \lim_{x\rightarrow \infty} \frac{ 1 }{x} \left( \frac{2}{3} \right)^x ##

## \lim_{x\rightarrow \infty} \frac{ 1 }{x} = 0 ##

## \lim_{x\rightarrow \infty} \left( \frac{2}{3} \right)^x = 0 ##
That approach is much better, because 0*0 is defined in the real numbers.
 
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  • #12
phinds said:
No, it's still undefined in most contexts, I think. What IS true is that as n approaches infinity, 0/n approaches 0.

EDIT: I hasten to add, I don't do any math where there IS any meaningful context for 0/infinity, so I could be wrong. What contexts did you have in mind?

You are wrong. In most contexts where the OP makes sense, it is indeed true that ##0/\infty = 0##.
 
  • #13
mfb said:
That approach is much better, because 0*0 is defined in the real numbers.

0*0 is undefined?
 
  • #14
micromass said:
0*0 is undefined?
Why do you think so? 0 is a number, and multiplying any number by 0 is 0.
 
  • #15
micromass said:
0*0 is undefined?
Huh? I said it is defined.
 
  • #16
micromass said:
You are wrong. In most contexts where the OP makes sense, it is indeed true that ##0/\infty = 0##.
micromass, than you for that correction, but I wonder if you could expand on it for me a bit? I'm confused as to how it is that "0/infinity = 0" is not treating infinity like a normal number
 
  • #17
If you treat infinity like a normal number, you should be allowed to multiply by it, but then you get ##0=0\cdot \infty## which is not well-defined any more.
 
  • #18
mfb said:
If you treat infinity like a normal number, you should be allowed to multiply by it, but then you get ##0=0\cdot \infty## which is not well-defined any more.
But that's my point. I've always been told that you CAN'T normally treat infinity like a normal number. Is this wrong?
 
  • #19
phinds said:
I've always been told that you CAN'T normally treat infinity like a normal number.
Correct. You can still perform some limited operations in most contexts, and 0/infinity=0 is one of them (e.g. for limits). That is not treating it as normal number, that is a specific rule for this operation.
 
  • #20
mfb said:
Correct. You can still perform some limited operations in most contexts, and 0/infinity=0 is one of them (e.g. for limits). That is not treating it as normal number, that is a specific rule for this operation.
OK, thanks.
 
  • #21
mfb said:
Huh? I said it is defined.

I'm sorry, I must have misread undefined instead of defined.
 
  • #22
Why is it that we don't define operations like 0*infinity and 0/infinity etc.They do occur in maths.Is it because defining these things causes inconsistencies?
 
  • #23
Daymare said:
Why is it that we don't define operations like 0*infinity and 0/infinity etc.They do occur in maths.Is it because defining these things causes inconsistencies?

Like I said, ##0/\infty## is defined in math to be ##0##. As for ##0\cdot \infty##, how would you define it?
 
  • #24
You could define it as 0
 
  • #25
Daymare said:
You could define it as 0

Right, and this is done in measure theory, where you rigorously define what an area is. The logic is that a straight line has length ##\infty## and width ##0## and has an area of ##0##. So if you define ##0\cdot \infty = 0## then this is consistent.

But why is this not done in the rest of mathematics. Consider ##f(n) = 1/n## and ##g(n) = n^2##. Then ##f(n)\rightarrow 0## and ##g(n)\rightarrow +\infty##. But ##f(n)g(n)\rightarrow +\infty##. So in this sense at least, ##0\cdot \infty = \infty## also makes sense.

And if ##f(n) = 1/n## and ##g(n) = n##, then ##f(n)g(n)\rightarrow 1##. So ##0\cdot \infty = 1## also makes sense in this sense.
 
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  • #26
micromass said:
Right, and this is done in measure theory, where you rigorously define what an area is. The logic is that a straight line has length ##\infty## and width ##0## and has an area of ##0##. So if you define ##0\cdot \infty = 0## then this is consistent.

But why is this not done in the rest of mathematics. Consider ##f(n) = 1/n## and ##g(n) = n^2##. Then ##f(n)\rightarrow 0## and ##g(n)\rightarrow +\infty##. But ##f(n)g(n)\rightarrow +\infty##. So in this sense at least, ##0\cdot \infty = \infty## also makes sense.

And if ##f(n) = 1/n## and ##g(n) = n##, then ##f(n)g(n)\rightarrow 1##. So ##0\cdot \infty = 1## also makes sense in this sense.
In the first example, the limit is that of a sequence of finite length lines becoming longer without any limit so the result is a limit on 0.n=0 as n becomes infinite. In cases like this you must always consider the underlying sequence that leads to the expression.
 

1. What is infinity?

Infinity is a concept in mathematics that represents a quantity or value that is unbounded, limitless, or never-ending. It is often symbolized by the infinity symbol (∞) and is used in various mathematical operations and equations.

2. Can infinity be reached or achieved?

No, infinity is not a tangible or attainable value. It is a theoretical concept used in mathematics to represent a concept of boundlessness or endlessness.

3. What is the concept of infinity in limits?

The concept of infinity in limits is used to describe the behavior of a function or sequence as its input or values approach infinity. It helps determine the ultimate or maximum value of a function or sequence as its inputs become larger and larger.

4. Are there different types of infinity?

Yes, there are different types of infinity in mathematics. For example, there is countable infinity, which is used to describe the number of counting numbers (1, 2, 3, ...). There is also uncountable infinity, which is used to describe the number of real numbers on a number line.

5. What is the relationship between infinity and limits?

The relationship between infinity and limits is that infinity is often used in limits to describe the behavior of a function or sequence as its inputs approach infinity. In other words, infinity is used to represent the ultimate or maximum value of a function or sequence as its inputs become larger and larger.

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