A math Problem

[Greg the Grouper]

So I have a question concerning what hackenslash said earlier in this thread. Not sure if I missed the explanation, or if I'm even on the right track here given my lack of education on the subject.

Anyway.

Hackenslash stated that events with zero probability happen, and the example given was that a specific number divided by infinity is equal to zero.

Now in my mind, the following statements all express the same relationship:

6/3=2
6/2=3
2*3=6

It seems to me like Hackenslash's statement implies a similar relationship:

N/Infinity=0
N/0=Infinity
0*Infinity=N

Is this accurate, or is there something I'm missing?

 

[Master_Ghost_Knight]

Well that is a very good question.
And the answer depends on the properties you are looking for.

To address the simplest answer N/Inf is undefined, it actually doesn't equal anything. Inf is not even a number.

Depending on the mathematical foundation you are working from, division is not really an operation, there is no such thing as division.
X / Y is actually a multiplication, a multiplication of X by another number K, but this K has the special property that if multiplied by Y it equals 1.
i.e. K.Y = 1.
For simplicity sake we just decide to write K by using the squiggly lines that look like 1/Y.
Remember there is no division, there is only a very funny way of representing this place holder value.

Given this, it becomes easy to understand why 1 / 0 is undefined. Not because division by 0 is fundamentally evil, but rather because division does not it exists, what 1/0 represents is an undefined place holder value that multiplied by 0 is 1.
I.e. a K such that K.0 = 1
But there is no number (K) such that multiplied by 0 produces anything other than zero, thus K (=1/0) is not a number that exists.

Conversely 1/inf is a place holder value Z such that Z.inf = 1
inf is not a number, multiplication by inf is not defined either, so asking what Z is such that Z.inf = 1, is a nonsensical statement.


But let's address another point.
Which is or generalization of this triplet:
1) X.Y=Z
2) Z/X=Y
3) Z/Y=X

This are 3 independent statements, the applicability or not applicability of one does not imply the other.
For example if X = 0 and Y not = 0, then
1) Defined
2) Undefined
3) Defined

On the other hand if Y = 0 and X not = 0, then
1) Defined
2) Defined
3) Undefined

So these 3 statements are not equivalent.


I could have also interpreted it some other way, for example, in terms of limits of functions.

Let's say I define f(x) / g(x) = h(x)
If f(x) = 1, and g(x) approaches infinity as X approaches Y, then h(x) approaches 0 as x approaches Y. We call this the limit of H(x) as X approaches Y.

If instead of writing all that laborious stuff with limits and f(x), and I replace the representation of f(x) by the the value of the limit it approaches to as X approaches Y, then you can get a representation like this:
1/inf = 0
That although technically true, it's confusing and ambiguous as to what exactly do you mean or you are trying to represent.

Hope that answers your question.

 

[Greg the Grouper]

Depending on the mathematical foundation you are working from, division is not really an operation, there is no such thing as division.
X / Y is actually a multiplication, a multiplication of X by another number K, but this K has the special property that if multiplied by Y it equals 1.
i.e. K.Y = 1.
For simplicity sake we just decide to write K by using the squiggly lines that look like 1/Y.
Remember there is no division, there is only a very funny way of representing this place holder value.

Given this, it becomes easy to understand why 1 / 0 is undefined. Not because division by 0 is fundamentally evil, but rather because division does not it exists, what 1/0 represents is an undefined place holder value that multiplied by 0 is 1.
I.e. a K such that K.0 = 1
But there is no number (K) such that multiplied by 0 produces anything other than zero, thus K (=1/0) is not a number that exists.

Conversely 1/inf is a place holder value Z such that Z.inf = 1
inf is not a number, multiplication by inf is not defined either, so asking what Z is such that Z.inf = 1, is a nonsensical statement.


But let's address another point.
Which is or generalization of this triplet:
1) X.Y=Z
2) Z/X=Y
3) Z/Y=X

This are 3 independent statements, the applicability or not applicability of one does not imply the other.
For example if X = 0 and Y not = 0, then
1) Defined
2) Undefined
3) Defined

On the other hand if Y = 0 and X not = 0, then
1) Defined
2) Defined
3) Undefined

So these 3 statements are not equivalent.


I could have also interpreted it some other way, for example, in terms of limits of functions.

Let's say I define f(x) / g(x) = h(x)
If f(x) = 1, and g(x) approaches infinity as X approaches Y, then h(x) approaches 0 as x approaches Y. We call this the limit of H(x) as X approaches Y.

If instead of writing all that laborious stuff with limits and f(x), and I replace the representation of f(x) by the the value of the limit it approaches to as X approaches Y, then you can get a representation like this:
1/inf = 0
That although technically true, it's confusing and ambiguous as to what exactly do you mean or you are trying to represent.

Hope that answers your question.

So to be clear, 1/inf=0 isn't a proper calculation in and of itself, but rather an inferred representation of a relationship that certain functions seem to exhibit?

Also, Inf would be a set of all numbers, correct? Does Inf as a concept vary based on your mathematical foundation?

 

[Master_Ghost_Knight]

So to be clear, 1/inf=0 isn't a proper calculation in and of itself, but rather an inferred representation of a relationship that certain functions seem to exhibit?

Correct.

Also, Inf would be a set of all numbers, correct?

No. That's just a set, not Inf.

Does Inf as a concept vary based on your mathematical foundation?

Inf is not even a unique concept on the same framework. We often use it to describe different things, with different properties/meanings. It's context sensitive. It has a particular meaning when applied, we can demonstrate that certain meanings are interchangeable with others (or not).
Maybe the confusion comes from the fact that humans are rubish at naming things.

 

[Greg the Grouper]

So to be clear, 1/inf=0 isn't a proper calculation in and of itself, but rather an inferred representation of a relationship that certain functions seem to exhibit?

Correct.

Also, Inf would be a set of all numbers, correct?

No. That's just a set, not Inf.

Does Inf as a concept vary based on your mathematical foundation?

Inf is not even a unique concept on the same framework. We often use it to describe different things, with different properties/meanings. It's context sensitive. It has a particular meaning when applied, we can demonstrate that certain meanings are interchangeable with others (or not).
Maybe the confusion comes from the fact that humans are rubish at naming things.

So would we have to start discussing specific scenarios in order to explain what Inf is in those specific scenarios?