h2g2 The Hitchhiker's Guide to the Galaxy: Earth Edition

Infinity has to be the greatest farce the mathematics community has ever developed. Not are they happy with just infinity, but they have multiples of infinities such as 2*infinity.
Inf we look at what we get when we divide inifinity by infinity, we could get any number. This is where mathematicans decide that taking a limit of and equation that will cause us to get the above stated problem, only to find we get any number (and sometimes infinity) back out of the result.
As if 2*infinity is not enough, we soon move into the powers of infinity. Infinity^2 is a hell of a lot more than infinity, but we don't stop here. You can go to the point to infinity^inifinity. But wait there is more... You guessed it inifinity is not a number, but just some symbol to say 'we really don't care, but this is really really big'.
On last thing, infinity is closely related to zero. Which we will leave to another maths lesson, at which time I will be able to think of some witty and wonderful maths about zero

Infinity is not intrinsically silly. It only seems silly is because we humans have gotten used to using numbers and counting as the basic concepts of our mathematics.

If a person starts with finite quantities (1,2,3) and tries to count up to infinity, he will get frustrated because it will take forever. Then he might conclude that there is something mysterious and not quite right about the concept of infinity. The same thing happens if he tries to apply the mathematics of finite quantities to things that are NOT finite -- it just doesn't work right.

Looking up into the night sky, there is every reason to believe that infinity is a basic, intuitive concept that deserves as much reverence as 1,2,3. Had things gone differently, our mathematics might have been based on infinity and cross-sections of infinity instead of numbers. Then, if a person starts with infinity and tries to cross-section his way down to get to finite quantities like 1,2,3, he will get frustrated because it will take forever. Then he might conclude that finite quantities like 1,2,3 are silly.

Infinity is a concept that's quite difficult to grasp when it comes to mathematics. Most people don't realize that there are entire fields built up around the concept of infinity. Things like 2*infinity have some difficulty making sense unless you're a little clearer about what you mean by that ambiguous statement.

If you're discussing cardinality, multiplying two by any infinite quantity is just going to get you back that number again. The same thing happens if you use the infinity of the extended real numbers.

If you're discussing limits, which is where most students first encounter infinity and choke on it, it all depends on how you're getting to infinity. Are we talking about the limit as n goes to infinity of n or of n squared?

When you get into surreal numbers, things get even more odd.

I'm thinking about writing an entire entry on the mathematics of infinity. There's a lot of good stuff to be discussed there (Hilbert's
Hotel, aleph-nought, calculus, etc.)

Greetings and salutations!
Infinity is just the mathematical version of "lots", only bigger.

And rounder.

Tee-hee

For more on infinty, you could look at the introduction to the Guide (the real one). All that stuff about it being a long way down the street to the chemist etc.

I always like problems like if any number to power zero equals one, and any number multiplied by zero is zero, what is zero to power zero? Answer - a pretty graph, or 'undefined' (cop-out) or a headache. Zero divided by zero faces the same problems. Also, is there a good reason why zero factorial should equal one? I've heard some pretty lame reasons in my time, and I would like to be convinced.

[email protected]

The reason why people call zero divided by zero undefined actually does exist. It's not just a cop out. The problems arise when you start dealing with limits. If you take the limit as x goes to infinity of x/x, you obviously get 1. But if you take the limit as x goes to infinity of (x^2)/x, you get zero. The limit as x goes to infinity of x/(x^2) is infinity. So if you were to give any definition for zero divided by zero, you should be allowed to plug those numbers into each of those limits and get the same answer, which is clearly absurd.

I believe the same problems arise with zero to the zeroth power, but I haven't actually come across any nice examples like the ones I gave for 0/0.

As for 0!, there are many reasons for assigning it a value of 1. One reason is that there exists a function, known as the gamma function, that behaves almost identically like the factorial except that it's continuous (meaning that you can find the gamma function of 3.7, whereas 3.7! is meaningless). It's also off by one, but if you add one to the number before evaluating using the gamma function, you always get the factorial. The actual formula for the gamma function is a really ugly integral taken over an infinite domain. Not happy stuff.

There are other reasons for setting 0!=1, but the one that convinced me best of all was something I came up with myself after learning a lot of more advanced mathematics. It's mostly an asthetic argument, but I find it highly convincing.

First ask yourself, what is nothing as far as numbers are concerned? Your instinctive answer is zero, which is why you would guess that when you take zero factorial, you are multiplying no numbers together and you should get zero. But when you're dealing with multiplication, you have a different sort of nothing. As far as multiplication is concerned, the number that does nothing is one. Zero does an awful lot when you multiply by it. One, however, might as well not be there in a multiplication problem. So when you look at the pattern of factorials (4*3*2*1, 3*2*1, 2*1, 1,...), you expect the next number to be no numbers multiplied together, which is one. It's the same reason that 4^0 is one. You're not multiplying any numbers together, and as far as multiplication is concerned, no numbers is one!

Hope I've been able to answer your questions.

Thankyou for your help. When I said 'undefined' was a cop-out, I didn't mean the concept, merely the word. This was all that was given to me when I first asked (as well as a pretty graph) and I was not filled with the thrill of discovery.
The demonstrations you gave are similar to ones I found after that depressing incident. Thanks anyway!
As for 0!, I remain unconvinced. I understand, of course, that 1 is the identity for multiplication, but I am not convinced that this is basis enough for the claim. I have no trouble believing that 0!=1 - the number patterns alone are good enough to show it should be - but I would like to see a good proof. I have been told (I think) that there is one somewhere with nasty calculus, but my teacher refused to elaborate. He was probably referring to this gamma thing. Intriguing!

Do you think there are a lot of Mathematicians on h2g2? Does this mean that mathematicians enjoy the style of hhgttg? Or could it be that reading the guide when young turns one into a mathematician? There are quite a few mathsy refs in it... I wonder where DA stands on this whole maths thing? Would he be able to help me with 0! ?

I've long held the beleif that 1/0=infinity.

To find 12/2:
Division (i.e a/b) is finding out how what be must be multiplied by to get a so you can work it out like this:
1*2=2 - 2 is less than 12 so the answer is more than 1
2*2=4 - higher
4*2=8 - higher
6*2=12 - Answer is 6
now try it with 1/0:
1*0=0 - Higher
2*0=0 - Higher
100000*0=0 - Higher
Etc...
So no matter how high you go to try to solve this problem, the answer will always be higher hence it is infinite.

What you've been doing with your step-by-step proccess is quite similar (although not as rigid) as the method of limits. The problem with saying that 1/0 = infinity is not that the answer doesn't make sense, but simply that infinity as you've mentioned it is NOT a number. This is a bit of a problem.

You can't treat infinity as if it were a number. If that's a little hard to understand, then think about it in this method. Suppose 1/0 = inf (I'm using "inf" as short for "infinity"). Now that means that 0 * inf = 1, correct? Now using your same argument, you can deduce that 2/0 = inf as well (try it). This, however, means that 0 * inf =2. Of course that lands you with the result 1=2.

Oops.

The problem with your demonstration is that you don't seem to be getting any closer to your target number as you go along. What you've proven is not that 1/0 = inf, but that as the number in your denominator gets closer to zero, the fraction gets bigger and bigger. This is the same as saying that the limit as x goes to inf of 1/x diverges to infinity.

There really isn't any argument to be made about whether 0! = 1 or not. It's simply a matter of definition, made for purely practical purposes. Mathematicians like the fact that 2! = 2 * 1!, and 3! = 3 * 2!, so they decide that they want 1! = 1 * 0!. Or in other words, 0! = 1! = 1. Note that this doesn't solve the problem of (-1)! I don't think there's any standard definition of negative factorials.

But this is only a convenient definition. n! wasn't handed down by God or anyone. It doesn't even occur naturally in nature. Factorials, like just about every mathematical concept, are inventions of human beings, and so can be defined however we want. A famous physicist (whom I forget) once said that 0,1,e, and pi are the only God-given mathematical objects.

Secondly, if x = 1/0, then 1 = 0*x, but 0*x = 0, right?? So therefore 1 = 0, which is absurd. Therefore, 1/0 doesn't make any sense. You can't divide by zero, because if you did, the whole universe would fall apart.

And I hope HippieChick does get around to an article on infinity. As a grad student in math, I'm really happy to see some math articles get written here!

Well, I'd been waiting to write another article until I got my latest one declined/approved. It's been in limbo for 3 months. I'll write one soon. Promise. Just as soon as I get back home.

Actually, I can think of a good reason why 0! is 1. When you're thinking about addition, nothing is zero because 0 + x = x. However, when you're thinking about multiplication, nothing is 1 since x * 1 = x. So if you're building up chains of addition, when you've got nothing in your chain, you're left with 0, as in 0*5=0, a chain of no fives. However, if you're building up a chain of multiplication, an empty chain means one, as in 5^0 = 1 or 0!=1.

Hey Hippie Chick! How's it going? I'm putting a link to this Entry in the Scientific History section of the h2g2 Historical Society, which can be found here:

http://www.h2g2.com/A240058

Apart from an Entry on Zero, the only other numbers are dates.


Cheers,

Mustapha

Thanks. I appreciate the link. I'm curious. Is there much of an interest in other mathematical articles? If so, I'd be glad to start work on an "arithmetic" article. And then maybe an "algebra" or a "set theory" article.

Sure, if they have a historical angle. It doesn't have to be a huge angle, say, Who first came up the idea of algebra (eg it's an Arabic word, al-jabr = the bone-setting, mathematical reduction) or Did any past cultures use set theories?

The most any Entry needs to receive a link is "invented by Joe Sixpack in 1675". Apart from that, you can mathematically go to town!

Take time=distance/speed. If you go at infinite speed, you will take no time in getting to your destination. Therefore x/infinity=0 for all values of x.
If you have an infinite distance to travel, you will take forever to get there. Therefore infinity/x=infinity for all values of x.
However, if you go an infinite distance at infinite speed, does it take you no time, an infinite time, or a time of 1?
In other words, does infinity/infinity=infinity, or does infinity/infinity=1 like x/x=1 for any other value of x? Therefore, is infinity a real number?*

Similarly, if you have no distance to go, you will take no time to get there, even if your speed is zero. Therefore 0/x=0 for all values of x, and also 0/0=0. Somebody tell me this doesn't make sense, and explain it to me, or I don't believe you.

However (except for this example) I agree with HippieChick. Zero is the unique number for which x/x=1 is false, because otherwise 0=1, because the above statement must be true.

*My personal belief is that portrayed by Hilbert's Hotel, ie. that infinity is unique in being the only value of x for which x/x=1 is false. Note I didn't say 'the only number...'

PS. You could substitute, for example, Boyle's Law and get the same results.

Certainly infinity is not a real number. It isn't even really a number in any sense that anyone other than a mathematician dealing with transfinites could make sense of.

The equation you should be dealing with is dist=speed*time. You can solve for time by dividing both sides by speed only if the speed is not zero. So the equation you were starting with: time=dist/speed is flawed if you want to consider a speed of zero.

The problem is that saying things like infinity*0 is that multiplication isn't defined for the concept infinity. It's like saying banana*0. Somewhere in the back of your head, something wants to say that banana*0=0, but only because you've been remembering multiplication tables by thinking 0*anything=0, when in fact it's only 0*(any complex number)=0.

What you _can_ talk about is limits. You can talk about what the trend of things is when numbers get really big. Now there are technical definitions of limits that are all nice and rigid and well-defined, but that's a discussion in and of itself.

For instance, let's take your time=dist/speed equation. First, let's make sure we say that speed is not 0 or the equation is invalid. When the distance gets bigger and you don't change the speed, the time gets bigger proportionally. No matter how big you want the time to be, I can find you a distance long enough so that it takes at least that long to cover. In _that_ sense, you can say that it takes an infinite amount of time to cover an infinite distance at a constant, _finite_ speed.

Similarly, let's let the speed get really big (really big is not zero, so we can use the equation)and keep the distance constant. If the distance is a positive number, then as the speed gets bigger and bigger, dist/speed gets smaller and smaller. No matter how short of time you want to make the trip in (except 0 seconds), I can find a speed at which you can make it in that time (relativity ignored). In that sense, one can say that travelling at an infinite speed over a constant, finite distance takes no time. HOWEVER, if the distance is zero, no matter how big you make the speed, dist/speed is STILL zero...and since that never changes, one can say that the trend is for the time to be always zero. In that case, the limit is still 0, but for a different reason.

So now here comes the important question. What happens if you let both the speed _and_ the distance get big? Well, that all depends on how fast they get big in relationship to each other. The limit as x gets big of x/x is 1, but the limit as x gets big of x^2/x is infinity, and the limit as x gets big of x/x^2 is 0. That's why things get confusing. You can't just toss around phrases like infinity/infinity because division doesn't really make sense for a concept like infinity.

You've mentioned Hilbert's Hotel, so I'll use some of those same metaphors. Keep in mind that the main purpose of Hilbert's Hotel is to illustrate what I keep saying; that infinity isn't a number and doesn't behave like one. You can't just do arithmetic with it and expect it to make sense. So what is division? x/y just means that we take x things and divide them up into y pieces and count how many there are in each piece, right? So lets do that with the infinite number of rooms in Hilbert's Hotel. I'm going to divide the hotel up into pieces that contain one room each. So there, an infinite number of rooms divided into an infinite number of groups, each containing 1 room. inf/inf=1, right? Wrong. I could also divide it up so that there are 2 rooms in each group. I'll take 1,2 together, 3,4 togeher, and so on forever. I can do this forever, so there are still an infinite number of groups, right? But now there are _2_ rooms in each group! So inf/inf=2, of course!

But I can get even sneakier! Let's take the infinite number of rooms we have and I'm going to take every other room and put it in group 1. How many rooms are in group 1? Infinitely many. Ok, how many are left? Infinitely many. So take what's left and only take every other room and put it in group 2. How many rooms in group 2? Infinitely many! How many left? Infinitely many!! Repeat the process. How many groups can I make like this? Infinitely many!!! So I have an infinite number of groups, each with an infinite number of rooms in each group! So inf/inf=inf, right?! How many exclamation points were in that sentence? Well, not infinitely many, but a lot!

Do you begin to see why infinity is not a number? It's not just because the mathematicians didn't want it to be one. It's because it doesn't behave like a number. Arithmetic just doesn't make sense.

I hope this helps.

Thinking about zero and infinity yesterday (before I read your post) I realised that, whatever you multiply zero by, you can't get it to equal anything, because there isn't anything to multiply. This surely applies even for infinity. Of course, infinity doesn't obey the other laws of numbers so it isn't a number (real or imaginary - I forgot about i when I posted) in the conventional sense, or in fact possibly any sense.

Wizard said that to find the answer to x/y you find what y has to be multiplied by to get x, and (quote) 'no matter how high you go to try to solve this problem, the answer will always be higher hence it is infinite.' But all of these equasions have the same solution: zero. In other words, those attempted values of x converge to infinity, but the solution is constant. There is no solution to x/0=1. It simply can't be done.

Query: what does the term 'aleph' mean? What are aleph-zero, aleph-one, aleph-two, aleph-infinity? Does x/0=aleph-infinity=nonsense?

I think you're getting the hang of it. When you say "whatever you multiply zero by, you can't get it to equal anything", you were on the right track. You can get it to equal something, but you can only get it to equal zero. And no, it surely does not apply even for infity, no more than it applies to "banana". Zero times banana doesn't make any sense, just like zero times infinity doesn't make any sense.

One thing that some people confuse is the difference between not getting an answer and getting an answer of zero. When I ask how many guys does it take to have a child, there is no reasonable answer because you need a woman to have a child (barring some bizarre future biological advancement). The answer is not zero, because that would imply that if you have no men, you can have a baby. When I ask how many noses my computer has, the answer is zero. It's a sensible, if silly, question, the answer just happens to be that there aren't any noses on my computer.

Wizard (whoever that is) is correct in saying "it is infinite". This does NOT mean that the answer is infinity. Saying that x/y tends to the infinite as y tends toward zero does not mean that x/y=infinity is the answer when y=0.

True. There is no solution to x/0=1. There is also no solution to x/0=0, although this is much harder to see, and I won't go into it at the moment.

WARNING: THIS IS ABOUT TO GET REALLY CONFUSING

"aleph" is the first letter in the Hebrew alphabet. It is used by mathematicians to talk about sets that are of infinite size. Talking about the "size" of a set can be confusing, so we make some definitions to aid us in talking about sets.

The first thing to define is what we call the "cardinality" of a set. Now if the set is of finite size, this is easy. The cardinality of a finite set is defined to be the number of elements in the set. For instance, the set {apple, 78, ***} has cardinality 3. The set of all even prime numbers (i.e. {2} ) has cardinality 1. The set of all hats that are the size of planets (i.e. the empty set, {} ) has cardinality 0.

But we can talk about sets that have infinite size. The first set we like to talk about is the natural numbers, which consists of all integers from 0 on up towards infinity (notice I said "towards infinity" not "to infinity"; infinity is not something included in this set). Anyway, the set of natural numbers (I'll call it N={0,1,2,3,...} from now on) clearly is not finite in size. If you doubt this, think about it thus: if it's finite, then there is some positive number that is the size of the set, call it n. Well our set N certainly contains all the non-negative integers smaller than n, and there are n of them, plus some more! {0,1,2,...n-1,n,n+1,n+2...}
So that would imply that the size of N is bigger than n, which is a contradiction. Anyway, most people believe me when I say that there are not a finite number of natural numbers.

So we need a way to define the cardinality of our set N. We straight up define the cardinality of the natural numbers to be aleph-nought (sometimes pronounced aleph-zero if you happen to be taught by an American; it's written as the letter aleph with the subscript zero). That's a strict definition.

Now if you can find a one-to-one correspondence between all the elements of your happy little set (whatever it is) and the elements of the natural numbers, then your happy little set also has cardinality aleph-nought. For example, take the set of all even non-negative numbers. For every even number, I can assign a natural number, and vice versa. I can do so like this: assign 0 to 0, 1 to 2, 2 to 4, 3 to 6, ... and for every natural number x, I assign the even number 2x. And for every even number y, I assign the natural number y/2. This way every even number has a natural number assigned to it and every natural number has an even number assigned to it. In this way, the two sets have the same cardinality, namely aleph-nought.

It might seem that one could make such a 1-1 correlation between any two infinite sets, but that's not true. For example, it can be proven that the set of all real numbers has no such correlation with the natural numbers. I know the proof, and it's not too complicated, but my message is now long enough as it is, so if you want it, ask.

Also, the set of all sets of natural numbers has no one-to-one correlation with N. This set, called the Power Set of N ( which we writte P(N) ) is a special set. It can be proven that there are no sets that are have "too few" elements in a mathematical sense to match up with P(N) and that also have "too many" to match up with N. In other words, there are no sets of a cardinality "between" aleph-nought and aleph-one, which is the name we give to the cardinality of P(N). This is hard to prove, but it is true.

Aleph-two is the cardinality of P(P(N)), and so on and so forth.

Here's where it gets REALLY weird. It's easy to show that the set of real numbers ( R ) can't have cardinality aleph-nought. However, it is IMPOSSIBLE to prove that R has cardinality aleph-one. It's also impossible to prove that R has cardinality BIGGER than aleph-one. R isn't too big or too small or anything, it's just too weird. It can be metamathematically proven that such a proof cannot exist. If such a proof existed, things wouldn't be consistent in mathematics, which is annoying, but a necessary problem due to the fact that mathematics is as complicated as it is. According to Godel's Theorem, (in a rough sense) either your math system is incomplete (for instance, no proof about the cardinality of R) or there's an inconsistency. We'd rather have an incomplete system than an inconsistency, which would allow the proof of absolutely anything.

Infinity is not an easy thing to get your brain around.

HC

R not having no correlation with N sounds reasonable, but R not having any defined cardinality? That's hard to swallow.

What's the cardinality of the set of the square roots of natural numbers? There is not an exact correlation since each natural number (except zero) has two square roots, one positive and one negative. Surely 2*aleph-nought makes no sense, despite being the obvious value?