Finite number divided by infinity

[Rodney182]

TL;DR

How does this make sense - ''Any finite number divided by infinity is to near as to nothing as makes no odds, so the average population of the Universe can be said to be zero.''

Beginner math person here. I was reading Hitchhiker's Guide to the Galaxy when I ran into this:

''It is known that there are an infinite number of worlds, simply because there is an infinite amount of space for them to be in. However, not every one of them is inhabited. Therefore, there must be a finite number of inhabited worlds. Any finite number divided by infinity is to near as to nothing as makes no odds, so the average population of the Universe can be said to be zero.''

The bold part is where the narrator lost me. I have to wonder why that would be zero. And have to wonder why would you need to divide the numbers to get to this answer at all?

''As makes no odds?'' This part confuses me completely.

I tried to reason my way through, but I can't. How do you divide a finite number by an infinite one? And if you can't, why would the answer be zero? And why can't we?

Is there an answer to this? Or is this just a thing Douglas Adam made up?

I'm really enjoying these books, they spawn lots of questions I'd never wondered before.

 

[PeroK]

Technically, infinity is not a number. The rules of arithmetic (addition/subtraction and multiplication/division) apply to numbers. You can't add, subtract, multiply or divide using infinity.

Douglas Adams is playing around with some apparent paradoxes that arise when you imagine that you can divide by infinity - or treat it like a number.

PS The concept of infinity plays an important role in mathematics, but you have to be rigorous and define carefully what you mean by "infinity" or "infinite".

 

[Rodney182]

Technically, infinity is not a number. The rules of arithmetic (addition/subtraction and multiplication/division) apply to numbers. You can't add, subtract, multiply or divide using infinity.

Douglas Adams is playing around with some apparent paradoxes that arise when you imagine that you can divide by infinity - or treat it like a number.

PS The concept of infinity plays an important role in mathematics, but you have to be rigorous and define carefully what you mean by "infinity" or "infinite".

Can't believe that didn't occur to me. Thanks!

 

[A.T.]

''It is known that there are an infinite number of worlds, simply because there is an infinite amount of space for them to be in. However, not every one of them is inhabited. Therefore, there must be a finite number of inhabited worlds. Any finite number divided by infinity is to near as to nothing as makes no odds, so the average population of the Universe can be said to be zero.''

The bold part is where the narrator lost me.

What about the 2 sentences before the part you bolded:

"However, not every one of them is inhabited. Therefore, there must be a finite number of inhabited worlds."

Does this really follow?

 

[DaveC426913]

It doesn't require infinities for the average population the universe to be zero. It just depends on now much precision you want.

Similarly, there is logic out there somewhere (*citation needed) that lays out how an event with zero probability can still occur:

eg.: Choosing a particular real number at random from the interval 0 to 1: The probability of selecting any one exact number, such as 0.5, is zero. Yet you must select some number in that interval, and 0.5 is one of the possibilities.

 

[A.T.]

Similarly, there is a statement out there somewhere (*citation needed) that lays out how an event with zero probability can still occur.

Yes, for example when you have infinitely many equally probable outcomes. Like choosing a random real number between 0 and 1.

 

[Matterwave]

The take-away is probably don't try to learn math from a science fiction book.

''It is known that there are an infinite number of worlds, simply because there is an infinite amount of space for them to be in.

The former is not a consequent (deduced from) of the latter. One could imagine an infinite space with only a finite number of worlds.

However, not every one of them is inhabited.

Probably true.

Therefore, there must be a finite number of inhabited worlds.

This again does not follow from logic/math. For example, the natural numbers are infinite and not every one of them are odd, and yet there are an infinite number of odd numbers.

Any finite number divided by infinity is to near as to nothing as makes no odds, so the average population of the Universe can be said to be zero.''

Division by infinity is not defined in our usual arithmetic because our usual arithmetic works on fields (often the real numbers) that don't include infinity as a number, as @PeroK already mentioned.

However, it is true that if you take the limit of a finite constant term over a variable that grows to infinity, the limit is zero. But this is not the same as "division by infinity".

I tried to reason my way through, but I can't. How do you divide a finite number by an infinite one? And if you can't, why would the answer be zero? And why can't we?

There are number systems in which you can, that's why I had to caveat with "our usual arithmetic" earlier. The hyper reals and the extended reals are two examples.

But this gets into topics that are beyond high school level. For "ordinary math" as we simply know it, you can't divide by infinity.

 

[Matterwave]

Yes, for example when you have infinitely many equally probable outcomes. Like choosing a random real number between 0 and 1.

A more brain bending one might be that the probability of choosing a rational number from a uniform distribution over [0,1] is 0.

 

[Rodney182]

The take-away is probably don't try to learn math from a science fiction book.

I'm not. I just enjoy trying to answer questions from a science-fiction book, that's all.

The former is not a consequent (deduced from) of the latter. One could imagine an infinite space with only a finite number of worlds.

That's a good point.

Probably true.

That would be the lore of the book, I suppose. The worlds are inhabited because that's what they are in the lore and the author of the Hitchhiker's Guide presumably knows this.

This again does not follow from logic/math. For example, the natural numbers are infinite and not every one of them are odd, and yet there are an infinite number of odd numbers.

Could you explain this to someone who doesn't know the terms 'natural numbers' or why they're infinite, and why there's an infinite number of odd numbers?

Division by infinity is not defined in our usual arithmetic because our usual arithmetic works on fields (often the real numbers) that don't include infinity as a number, as @PeroK already mentioned.

However, it is true that if you take the limit of a finite constant term over a variable that grows to infinity, the limit is zero. But this is not the same as "division by infinity".

Just checking if I follow - the limit is zero because it's impossible to put a limit on infinity?

There are number systems in which you can, that's why I had to caveat with "our usual arithmetic" earlier. The hyper reals and the extended reals are two examples.

But this gets into topics that are beyond high school level. For "ordinary math" as we simply know it, you can't divide by infinity.

Yep, that sounds more advanced, haha.

 

[Rodney182]

What about the 2 sentences before the part you bolded:

"However, not every one of them is inhabited. Therefore, there must be a finite number of inhabited worlds."

Does this really follow?

I was going over this a few times as well. If the amount of worlds is infinite, but not every one of them is inhabited, then the number of inhabited worlds is finite - my guess would be because it falls 'below' infinity. But then again what does that even mean? I can't tell you.

 

[Rodney182]

Similarly, there is logic out there somewhere (*citation needed) that lays out how an event with zero probability can still occur:

eg.: Choosing a particular real number at random from the interval 0 to 1: The probability of selecting any one exact number, such as 0.5, is zero. Yet you must select some number in that interval, and 0.5 is one of the possibilities.

Oh, I see. That's actually fascinating. But I'm failing to understand how that connects to this:

It doesn't require infinities for the average population the universe to be zero. It just depends on now much precision you want.

 

[Matterwave]

I'm not. I just enjoy trying to answer questions from a science-fiction book, that's all.

It's good to enjoy the book! I found the movie very enjoyable myself (I never read the book haha). Just don't assume the author is making valid mathematical arguments. The author is writing for entertainment.

Could you explain this to someone who doesn't know the terms 'natural numbers' or why they're infinite, and why there's an infinite number of odd numbers?

Sure. The natural numbers are the counting numbers (no fractions, no irrational numbers) we are familiar with from grade school. Nothing fancy: $\{0,1,2,3...\}$. This set is infinite (it doesn't stop at some point, that's what the $...$ means). The set of odd numbers $\{1,3,5,...\}$ also doesn't stop, it's also infinite. But obviously not every natural number is odd.

The problem with Adam's statement, from a mathematical point of view, is he assumes if you take some (strict) subset of an infinite set you end up with a finite set. This is not true.

You could end up with a finite set, but you don't have to.

Just checking if I follow - the limit is zero because it's impossible to put a limit on infinity?

The limit is zero because of the properties/definition of limit. Very roughly speaking, if you look at a function like $1/x$ on a graph and you make x larger and larger, the function $1/x$ gets closer and closer to 0. There is a rigorous (consistent) way to define that "closer and closer" (with some epsilons and deltas) so that you can say "the limit is 0". It's a concept in calculus.

For the question in this thread, you don't need it. However, a lot of people make the mistake of confusing the limit for the answer so I flagged it.

In symbols (you can ignore this part):

$$
\text{lim}_{x\rightarrow\infty} \frac{1}{x}=0
$$

The above eqn is well defined and the limit is 0.

$\frac{1}{\infty}$ this expression not defined in our usual arithmetic.

 

[Rodney182]

It's good to enjoy the book! I found the movie very enjoyable myself (I never read the book haha). Just don't assume the author is making valid mathematical arguments. The author is writing for entertainment.

I heard he did read some physics to prepare for the writing, then basically threw the concepts together in a way that doesn't make any sense. But it's an absurdist type of media. I think that's rather the point and that's why I enjoy it so far haha.

Sure. The natural numbers are the counting numbers (no fractions, no irrational numbers) we are familiar with from grade school. Nothing fancy: $\{0,1,2,3...\}$. This set is infinite (it doesn't stop at some point, that's what the $...$ means). The set of odd numbers $\{1,3,5,...\}$ also doesn't stop, it's also infinite. But obviously not every natural number is odd.

The problem with Adam's statement, from a mathematical point of view, is he assumes if you take some (strict) subset of an infinite set you end up with a finite set. This is not true.

You could end up with a finite set, but you don't have to.

Oh, I see.

The limit is zero because of the properties/definition of limit. Very roughly speaking, if you look at a function like $1/x$ on a graph and you make x larger and larger, the function $1/x$ gets closer and closer to 0. There is a rigorous (consistent) way to define that "closer and closer" (with some epsilons and deltas) so that you can say "the limit is 0". It's a concept in calculus.

For the question in this thread, you don't need it. However, a lot of people make the mistake of confusing the limit for the answer so I flagged it.

In symbols (you can ignore this part):

$$
\text{lim}_{x\rightarrow\infty} \frac{1}{x}=0
$$

The above eqn is well defined and the limit is 0.

$\frac{1}{\infty}$ this expression not defined in our usual arithmetic.

Yep this is where you lost me. I haven't touched calculus yet. If you feel like explaining, feel free, otherwise ignore and I'll leave it for people here to understand it.

 

[Dale]

why there's an infinite number of odd numbers?

You can always take any odd number and add 2 to get an even bigger odd number. So there is no largest odd number, and therefore the set of all odd numbers is infinite.

In the hyperreals infinity is not a number, but there are infinite numbers. If you want to get a good understanding of infinite numbers, I would recommend the hyperreals

 

[Rodney182]

You can always take any odd number and add 2 to get an even bigger odd number. So there is no largest odd number, and therefore the set of all odd numbers is infinite

Doesn't that go for even numbers as well? Or any number?

 

[Matterwave]

Oh, I see.

Yep this is where you lost me. I haven't touched calculus yet. If you feel like explaining, feel free, otherwise ignore and I'll leave it for people here to understand it.

No worries, you will learn limits in an introduction to calculus class (or pre-calculus). It's not essential here.

If you understand my example with natural numbers and odd numbers, you should be able to answer your own question here:

I was going over this a few times as well. If the amount of worlds is infinite, but not every one of them is inhabited, then the number of inhabited worlds is finite - my guess would be because it falls 'below' infinity. But then again what does that even mean? I can't tell you.

(Hint: Adams is not a mathematician)

 

[Dale]

Doesn't that go for even numbers as well? Or any number?

Even numbers, yes. There are an infinite number of even numbers too. There are also an infinite number of prime numbers.

 

[Matterwave]

Doesn't that go for even numbers as well?

There are also an infinite number of even numbers.

Or any number?

There are indeed an infinite number of numbers.

 

[Rodney182]

No worries, you will learn limits in an introduction to calculus class (or pre-calculus). It's not essential here.

If you understand my example with natural numbers and odd numbers, you should be able to answer your own question here:

(Hint: Adams is not a mathematician)

Basically what you said - Adam's statement is based on the assumption that if you take a set of an infinite set of numbers, you end up with a finite set.

Which is false - because the set of natural numbers are infinite. So the whole thing doesn't follow.

Hm, but you also said:

You could end up with a finite set, but you don't have to.

How could you end up with a finite set?

That kind of violates my previous reasoning. What am I missing?

 

[Dale]

How could you end up with a finite set?

The set of all natural numbers less than 42 is a finite subset of the natural numbers.

 

[DaveC426913]

I was going over this a few times as well. If the amount of worlds is infinite, but not every one of them is inhabited, then the number of inhabited worlds is finite - my guess would be because it falls 'below' infinity.

I think someone else answered this more eloquently but this is easy to show how this is false.

The set of integers is infinite, but not every integer is even. It does not follow that the number of even numbers is finite. The number of even integers is still infinite.

And now you get into degrees of infinities...

 

[Matterwave]

Basically what you said - Adam's statement is based on the assumption that if you take a set of an infinite set of numbers, you end up with a finite set.

First, a wording correction -- Subset. You take subsets of sets. You don't "take a set of a set", you "take a subset of a set".

Second, right, Adams is (roughly, with some translation) making the if-then statement:

If you take the subset of an infinite set you end up with a finite set.

Which is my translation of:

It is known that there are an infinite number of worlds... However, not every one of them is inhabited. Therefore, there must be a finite number of inhabited worlds.

Notice the "must be".

This argument from Adams is invalid because you could still get an infinite set back (my example with odd numbers). Of course, you could get a finite set back, e.g. if you take the subset as Dale suggested, or the subset $\{1,2,3\}$ (notice no $...$ this time).

Which is false - because the set of natural numbers are infinite.

In my analogy, it would be because the set of odd numbers is infinite.

That kind of violates my previous reasoning. What am I missing?

Just the difference between a statement that could be true vs deducing the truth of a statement from an argument.

 

[PeroK]

eg.: Choosing a particular real number at random from the interval 0 to 1: The probability of selecting any one exact number, such as 0.5, is zero. Yet you must select some number in that interval, and 0.5 is one of the possibilities.

Mathematically, we can have a uniform distribution on the interval $[0, 1]$, where the probability density is uniform. In reality, however, it's not possible to select from more than a finite number of outcomes with equal probability.

The paradox "something with zero probability can happen" occurs by misapplying mathematics. Or, at least, imagining that there is a scenario where you can apply that mathematics. It's impossible to select real numbers uniformly. More explicitly, the positive integers cannot be selected uniformly.

Another example is Hilbert's Hotel. That's valid mathematics and a good example of the behaviour of infinite sets; but, we cannot build a hotel with an infinite number of rooms.

The Internet is awash with the fallacy that these things could be physically realised, rather than being purely mathematical constructions.

 

[PeroK]

A more brain bending one might be that the probability of choosing a rational number from a uniform distribution over [*0,1] is 0.

It's only mind bending is you mistake the mathematics of infinite sets with physically realisable scenarios. Lebesque integration is one of the greatest pieces of mathematics, IMO. But, it is abstract mathematics. It's not something you can do (with compass and ruler, or whatever).

With Riemann integration, you cannot integrate a function that is non-zero only on the rationals. Even that quite sophisticated mathematics is not sufficient to come up with a probability for choosing a rational from a uniform distribution on $[0,1]$.

To take a simpler example: Just because it looks like you can draw a continuous number line (with a finite number of ink molecules on a finite lattice of paper fibres) does not mean that a continuous number line (and all the numbers on it) exist other than as mathematical abstractions.

 

[Matterwave]

The Internet is awash with the fallacy that these things could be physically realised, rather than being purely mathematical constructions.

Yeah, it can be liberating when one realizes that math is just a set of consistent rules (axioms) and their logical implications. It is not necessarily, at least not a priori, reflective of the real world. I agree with you that it can be easy to make our heads spin in circles for no reason by taking math too seriously as "reflecting reality".

However, it's not coincidence that math is one of the foundations of physics. Even pure mathematicians are often drawn to problems that intuitively reflect the world in some way. And often, things which we regarded as pure mathematics can find their way into surprising physical applications.

An example I might give in this thread is measure theory actually. One might think "rationals are a set of measure zero in the reals" is just a mathematical curiosity, but this concept of "set of measure zero" actually shows up in ergodic theory in statistical mechanics. The original Birkhoff Ergodic theorem says that the (infinite) time average of a phase function converges almost everywhere in the phase space. It does mean there could be a set of measure zero where it doesn't converge.

 

[PeroK]

Yeah, it can be liberating when one realizes that math is just a set of consistent rules (axioms) and their logical implications. It is not necessarily, at least not a priori, reflective of the real world. I agree with you that it can be easy to make our heads spin in circles for no reason by taking math too seriously as "reflecting reality".

However, it's not coincidence that math is one of the foundations of physics. Even pure mathematicians are often drawn to problems that intuitively reflect the world in some way. And often, things which we regarded as pure mathematics can find their way into surprising physical applications.

An example I might give in this thread is measure theory actually. One might think "rationals are a set of measure zero in the reals" is just a mathematical curiosity, but this concept of "set of measure zero" actually shows up in ergodic theory in statistical mechanics. The original Birkhoff Ergodic theorem says that the (infinite) time average of a phase function converges almost everywhere in the phase space. It does mean there could be a set of measure zero where it doesn't converge.

Of course mathematics is useful. The issue I highlighted is that infinite sets carry paradoxes that do not translate to "real" paradoxes. Hilbert's Hotel, an infinite number of monkeys with typewriters, the slot machine from the gambler's hell puzzle etc.

In fact, there was a zoo that gave its monkeys typewriters to see how close they could get to the complete works of Shakespeare. The monkeys soon got bored, damaged the typewriters or defecated on them:

https://en.wikipedia.org/wiki/Infinite_monkey_theorem#Actual_monkeys

 

[PeroK]

@Matterwave the challenge is to devise an algorithm that will select a real number on the interval $[0,1]$, where every real number has an equal probability of being selected. You can assume that a computer, say, can generate a finite random distribution for you.

 

[A.T.]

It's only mind bending is you mistake the mathematics of infinite sets with physically realisable scenarios.

Do we know for sure that infinite sets (e.g. all planets) are not physically realizable?

 

[Matterwave]

@Matterwave the challenge is to devise an algorithm that will select a real number on the interval $[0,1]$, where every real number has an equal probability of being selected. You can assume that a computer, say, can generate a finite random distribution for you.

The computable numbers are countable. A logical expression is, by definition, a finite set of symbols which means (assuming a countable alphabet) numbers expressed that way are also countable. So... trying to express even a single "some arbitrary but specific real number" is non trivial.

 

[PeroK]

The computable numbers are countable. A logical expression is, by definition, a finite set of symbols which means (assuming a countable alphabet) numbers expressed that way are also countable. So... trying to express even a single "some arbitrary but specific real number" is non trivial.

Exactly. You can't "choose" one in the sense that the zero probability of doing so has any real meaning.