Infinity

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jestingrabbit
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Postby jestingrabbit » Thu Apr 05, 2007 10:17 am UTC

sin2beta wrote:Nice links and a very nice proof in the edit. Seems your links did answer the question regarding inf*0 = indeterminate


They answered it after a little bit of work. I was afraid that the OP would just look at them and see dogmatic assertions.

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Postby Drostie » Thu Apr 05, 2007 10:37 am UTC

You guys talked much more about the issue than I would have...

My explanation: what is meant by 0 * ∞ is not the same as what is meant by 3*4, because ∞ is not a number. Once we start talking about ∞, we must start talking about limits, and once you're talking about limits, you're not saying 0*∞, you're saying (tiny number)*(huge number) -- which, for different tiny and huge numbers, could limit to any number you'd like.

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Postby SpitValve » Thu Apr 05, 2007 11:16 am UTC

warriorness wrote:
SpitValve wrote:
warriorness wrote:I know, and the reason I'm mentioning it is because using limits is the only way to represent the value of a function at a point where it's indeterminate. You can't evaluate it.


You can't replace 1/infinity with "limit of 1/x as x approaches infinity". They're different concepts.


But it's the only way to express it. You can't compute 1/infinity. It's like saying "what's the square root of banana?" The function is simply not applicable to the argument.

That's why we have limits. They make infinity make more sense; they let us express stuff that we previously wouldn't be able to express, such as the 1/x limit you just said.


I think I might just be arguing over semantics, but hey :)

lim 1/x as x->infinity is the closest we can get to a concept of 1/infinity, without breaking maths. However, I see them as distinct because 1/inifinity=0 doesn't make sense, while (lim 1/x as x-> infinity)=0 does make sense.

So I wouldn't say you can represent 1/infinity with limits, because 1/infinity doesn't make sense. But what you can say is "when you say 1/infinity I think you're really trying to say limit of 1/x as x goes to infinity". It's not that the limit is equivalent to 1/infinity, but that the limit is the real concept you're trying to express by saying 1/infinity.

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Re: Infinity

Postby yy2bggggs » Thu Apr 05, 2007 11:51 am UTC

mwace wrote:But the god of this world is a cruel one, and nearly every math professor in my institution tells me this value is undefined, and I'm pretty confident the rest of modern mathmatics is backing them up on it. If I'm wrong (I don't want to say I am), its probably because my definition of multiplication is in error. If you believe this is true, then please supply the actual definition of multiplication before you correct me. Aside from that, who can help me?

The problem here is an incorrect intuition. Imagine a practical example. Draw a line segment--a unit section of the number line. Consider:
  • The line consists of a set of points
  • There are infinitely many points
  • The width of each point is 0
  • The width of the line, therefore, should be "infinity*0"=0

But the last point is clearly incorrect. The width of the line is 1.
Last edited by yy2bggggs on Sat Apr 07, 2007 2:16 am UTC, edited 1 time in total.

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Postby Solt » Thu Apr 05, 2007 11:52 am UTC

You people are insane. Unless you're planning to be a pure mathematician or a theoretical physicist, it doesn't matter.

Just treat infinite in a way that makes sense- as a really really big number. Think of it as determinate, it doesn't make a difference in the calculations, as long as you keep in mind that it is much bigger than any other number you could possibly write down.

oo * 0 = 0 because anything times 0 equals 0.
Any (x|x != 0) * oo equals oo (+ or -) because anything times a really huge number will eventually get huge itself.

The reason we use infinite is to understand behavior as the variables become very large. Just treating infinite as any arbitrary large number that is simply too large to write down will always suffice. Any number that gets closer to this large number than it does to 0 also might as well be written down as oo as well, because the difference will inevitably be negligible.

sin2beta wrote:If x/infinity = 0 for all x then infinity * 0 = x for all x.


The only thing untrue about this statement is "for all x."

If x/oo = 0, then x = 0.

If x != 0, then x/oo != 0. Rather, it equals a very very (very) small number. So small that you might as well call it 0, but it's technically not.

oo/2 is still pretty damn large. Because oo was so large in the first place, you still cannot write down oo/2, so you might as well call it oo. As a matter of fact, the only division that can make oo small enough to allow you to write it down is oo. oo/oo = 1 because any (determinate) number divided by itself is 1, duh. Mathematicians would tell you that oo/oo is indeterminate, but they would also tell you that it is 1. They say it has something to do with a hospital. Yea, a mental hospital. Mathematicians are pathologically insane, don't listen to them.
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Postby Torn Apart By Dingos » Thu Apr 05, 2007 12:25 pm UTC

There are different kinds of infinity (the "concept" in analysis which isn't a number, infinity in the extended reals or complexes, cardinals, ordinals, hyperreals, etc), and there are different kinds of multiplication (especially noncommutative, like matrix multiplication).

If your definition "A*B=A sets of B" means A*B={(a,b)|a in A, b in B}, then 0*X=X*0=0 for all, even infinite, sets X, where 0={}. This definition of multiplication doesn't work for rational or real numbers though, and there's now infinitely many infinities of infinitely many different cardinalities.

You can't make multiplication work for 0*banana. Since all of mathematics is built upon set theory, you'd like a set-theoretic definition, but there is no set of all sets (it can impossibly exist) in standard set theory, so you can't define multiplication for arbitrary objects. You could define it for particular objects, eg banana, but then it would just be a definition, and you can't multiply banana with anything else, so it wouldn't mean anything.

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Postby SpitValve » Thu Apr 05, 2007 9:43 pm UTC

Solt wrote:You people are insane. Unless you're planning to be a pure mathematician or a theoretical physicist, it doesn't matter.


'course, in this forum we have plenty of both :)

Solt wrote:Just treat infinite in a way that makes sense- as a really really big number. Think of it as determinate, it doesn't make a difference in the calculations, as long as you keep in mind that it is much bigger than any other number you could possibly write down.

oo * 0 = 0 because anything times 0 equals 0.
Any (x|x != 0) * oo equals oo (+ or -) because anything times a really huge number will eventually get huge itself.

The reason we use infinite is to understand behavior as the variables become very large. Just treating infinite as any arbitrary large number that is simply too large to write down will always suffice. Any number that gets closer to this large number than it does to 0 also might as well be written down as oo as well, because the difference will inevitably be negligible.

sin2beta wrote:If x/infinity = 0 for all x then infinity * 0 = x for all x.


The only thing untrue about this statement is "for all x."

If x/oo = 0, then x = 0.

If x != 0, then x/oo != 0. Rather, it equals a very very (very) small number. So small that you might as well call it 0, but it's technically not.

oo/2 is still pretty damn large. Because oo was so large in the first place, you still cannot write down oo/2, so you might as well call it oo. As a matter of fact, the only division that can make oo small enough to allow you to write it down is oo. oo/oo = 1 because any (determinate) number divided by itself is 1, duh. Mathematicians would tell you that oo/oo is indeterminate, but they would also tell you that it is 1. They say it has something to do with a hospital. Yea, a mental hospital. Mathematicians are pathologically insane, don't listen to them.


I don't like your arguments... they sound too much like the people who argue "0.9999... is not really equal to 1, but it's so close that it doesn't matter". Which is a loose and untrue statement...

Infinity is not just a really big number. It's a concept beyond that...

Also, infinity/infinty is undefined, but the limit a(x)/b(x) where a->infinity as x->x0 and b->infinity as x->x0 may be definied, depending on the nature of a and b. These are distinct concepts. It's important to keep such things in mind even when doing applied maths - because you can't just rely on intuition in maths, it's a bad way to go.

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Re: Infinity

Postby gmalivuk » Thu Apr 05, 2007 10:33 pm UTC

mwace wrote:Thus, it would be impossibe for the value of ∞*0 to be anything except zero.


Why couldn't it be infinity?

After all, 3*∞ = ∞, 5.3*∞ = ∞, etc. So why not 0*∞ = ∞, as well?

You've got two trends going here. Zero times most things is zero, and infinity times most things is (+/-) infinity. So infinity times zero is either zero or infinity (or perhaps something in between). This is (a simplified explanation of) why it's indeterminate.

Here's a rundown of the definition of multiplication (which can from here be extended to other sets of numbers, of course)
For n and m integers:
n * 0 = 0 * n = 0
n * (m + 1) = n * m + n
n * -m = -(n * m)
1/m = that fraction which, when multiplied by m, gives 1 (I'm not going to go all the way into the construction of the rationals from the integers here.)
m * 1/n = m/n
for q rational
q * (m/n) = (q * m) / n (From the first couple properties above, you can prove associativity, so I'm getting perhaps a bit lazy here)
Finally, for r and s real
r * s = Lim(q in Q -> r) q * s

The problem is that nowhere in any of these properties does it say anything about infinity. Infinity is not, as generally used, a specific number. As such, even while we usually treat infinity * n as infinity, even this isn't exactly determinite.

Now, you can define multiplication on sets that include infinity in various ways, some of which are described in links other people have posted. Thing is, you have to define multiplication by infinity before you can use it. It isn't something that pops out of the more basic definitions and properties of multiplication.

So sure, you could just say that infinity * nonzero = infinity and infinity * 0 = 0, but then you've made infinity a specific number and would have to start using another symbol for the usual limit-based concept that most mathematicians are talking about when they say infinity.
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Postby fjafjan » Fri Apr 06, 2007 12:11 am UTC

I just wanted to point out why "5 * banana" doesn't make sence is because you can't multiply 'things'. you can multiply numbers, you can't take "banana * apple*, you can take "number of bananas * number of apples* or a variety of such things, but the reason "zero sets of banana" doesn't make any sence is because "5 devided by banana" doesn't make any sence, banana is not a number. (even if you could probably describe a banana numerically a variety of ways)
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Postby warriorness » Fri Apr 06, 2007 12:22 am UTC

fjafjan wrote:I just wanted to point out why "5 * banana" doesn't make sence is because you can't multiply 'things'. you can multiply numbers, you can't take "banana * apple*, you can take "number of bananas * number of apples* or a variety of such things, but the reason "zero sets of banana" doesn't make any sence is because "5 devided by banana" doesn't make any sence, banana is not a number. (even if you could probably describe a banana numerically a variety of ways)


Right, that was my whole point about arithmetic only being defined for certain things (everything for all numbers, except the second argument of division can't be 0). Without using limits, trying to use infinity in the elementary mathematical functions is just as silly as saying "5 divided by banana" or "what's the logarithm of apple".

Some people seem to be confusing this with units, which is incorrect. "apple" can be a perfectly legitimate unit, just like "kilometer". You could say "Jane drives 300 kilometers per hour" and "Billy-Bob can eat 5 apples per minute" - that's totally different than actually trying to use non-numbers in arithmetic operations.

You can't use infinity in arithmetic (and can't divide by zero) without incorporating limits.
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Postby Strilanc » Fri Apr 06, 2007 12:29 am UTC

The answer here is your definition of infinity is wrong. Infinity is not just "really, really, really big", infinity is just a mathematical concept stated like so:

(x = infinity) if and only if (for every number c, x is greater than c)

For example:
Consider the limit of 1/x as x approaches 0 from the positive direction
We say this equals infinity because for any c you can give me, I can make x close enough to 0 such that 1/x > c.

How does this proper definition help us with our problem? It makes it clear what we mean by infinity*0. You have to make it explicit what you mean by infinity (for example, we can use our 1/x limit).

lim[x->0+] (1/x*0) = lim[x->0] (0) = 0

However, what if we mix things up a bit, and make that 0 a function of x as well?
consider: lim[x->0+](1/x*x)
notice that, in the limit, 1/x is infinity and x is 0. Therefore we are multiplying infinity*0. But, this time the answer is not 0, the answer is 1 because 1/x*x = 1.

And that's why you can't assign a value to infinity times 0, because not all limits of the form 0*infinity are equal (however, it is true, for example, that 1/infinity = 0 because all limits of the form 1/infinity equal 0).
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Postby tallest » Sat Apr 07, 2007 9:29 am UTC

I think this has been beaten to death by now, but I thought I would throw my two cents in.

Infinity is not a number, only a concept used in limits. If you want to start plugging "infinities" and "infinitesimals" into equations and expressions you will probably want to use something like non-standard analysis. The hyperreal number system introduces classes of numbers called i-large and i-small that, while still real numbers, can be used as infinities and infinitesimals.

The interior construction of the hyperreals pretty much boils down to a formal way of defining numbers that are "large enough" (i-large) and "small enough" (i-small).

If I can recall this correctly from the seminar I saw last semester, it goes something like this. Because the human race will exist for a finite time, only a finite number of numbers (I know, bad word choice) will be considered by the human race. Thus there are numbers that are larger than any number we will ever use (i-large) and numbers that are smaller than we will ever use (i-small). Clearly there is no way to actually give an i-small or i-large number. Even if you could find the maximum of all numbers considered in human history, adding 1 would simply increase that maximum, not give an i-large number. So how is this useful?

It can be proven that, if a is an i-large number then 1/a is an i-small number; similarly, if b is an i-small number then 1/b is an i-large number; and so on and so forth. Thus we can use infinitesimals and infinities in mathematical expressions without the need for a limit.

Using this formulation, it is easy to see that if a is an i-large number then 0*a=0 because a is a real number. Little can be said about what a*b equals if a is i-large and b is i-small, though.

Interestingly, it can be proven that nothing can be proven with i-large or i-small numbers (or any use of the hyperreals) that couldn't be proven with limits. Thus the hyperreals are simply a different way of approaching limits, one which many people find more intuitive.

NOTE: clearly I have not given any formality to the definition of hyperreals, this is just what I took away from a math seminar last semester. I apologize for my rambling.
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Postby SpitValve » Sat Apr 07, 2007 11:40 am UTC

He appears to be from the 19th century. He must be right!

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Postby the tree » Sat Apr 07, 2007 11:44 am UTC

tallest wrote:Infinity is not a number, only a concept used in limits.
This is the nail's head, Tallest has hit it good.

You can't apply arithemeticesque operations to things that aren't numbers, that's all you need to know.

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Postby Solt » Sat Apr 07, 2007 11:47 am UTC

the tree wrote:
tallest wrote:Infinity is not a number, only a concept used in limits.
This is the nail's head, Tallest has hit it good.

You can't apply arithemeticesque operations to things that aren't numbers, that's all you need to know.



x + oo = oo
x - oo = -oo
x * oo = oo
x / oo = 0
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Postby SpitValve » Sat Apr 07, 2007 11:58 am UTC

What if x is infinite or zero?

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Postby LE4dGOLEM » Sat Apr 07, 2007 12:46 pm UTC

SpitValve wrote:What if x is infinite or zero?

You can not divide by zero. Infinity is infinity, it doesn't matter.
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Postby jestingrabbit » Sat Apr 07, 2007 2:36 pm UTC

the tree wrote:
tallest wrote:Infinity is not a number, only a concept used in limits.
This is the nail's head, Tallest has hit it good.


I don't think tallest's statement has any meaningful content. What is a number? Seriously, do you have a definition? At one time or another people have said that 0, negatives, irrationals, the imaginary number i and whole slew of other constructs aren't numbers, and they have provided a whole slew of arguments for why that is, but most of us here acknowledge these things are numbers every other day.

There is no set "the set of all numbers". There is nothing like it. If I told you I was working with octonions, would you say that I was working with numbers?

The affinely extended reals, the projectively extended reals, the hyperreals, the surreal numbers, all have well defined arithmetics, all are considered numbers by somebody and all have numbers which are bigger than any number in the reals. Infinity works fine as a number for a great many people. In the hyperreals 0*I (where I is bigger than any of the images of real numbers, and there are many such numbers in the hyperreals) is 0, because the hyperreals are a field. 0*infinity in the affinely extended reals is undefined because they are an attmept to formalise certain facts about limits.

How you deal with infinity is a question of what you are trying to do. If you just want to do real analysis and have infinity hanging around like a bogey man, go for it. I won't stop you. But please don't say something of the form "blah is not a number". Its not a statement with content.

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Postby tallest » Sat Apr 07, 2007 4:44 pm UTC

@ jestingrabbit: What is a number????

The real number line is very well defined. Two distinct sets, the rational numbers and the irrational numbers. Banana is not a real number, ocotnions are not real numbers, pi is a real number, 2.3456723 is a real number, and sqrt(2) is a real number.

There are, of course, other number systems. These number systems have different algebras (although some times they are very similar). I am restricting my discussion to real numbers. I don't see the point of discussing what 7*banana is unless you define an algebra for the set of all objects. This is why you cannot evaluate 0*infinity using the algebra of the real number line. Infinity is not a real number. Of course, there are other number systems that include infinity as a number, but these number systems have rules of algebra and the answer to all your questions lies in those rules.

Hence, if you want to ask what 0*infinity is, then you have to say what number system you are using. I just used the internally defined hyperreals as an example of one way to think about infinity being a number, there are obviously other ways.

The reason I chose the internally defined hyperreals as an example is because you don't actually add any numbers to the real number line. i-large and i-small numbers are real numbers, just like any other. Mathematicians have just defined some numbers to be big enough or small enough. It's a little complicated because these numbers are defined axiomatically, but they're still real numbers.

I don't actually like the hyperreals, as I'm fairly used to just using limits. But when you just use limits and the real numbers, infinity is not a number and we may only say that as x->infinity, x*0->0.
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Postby SpitValve » Sat Apr 07, 2007 9:28 pm UTC

Solt wrote:x + oo = oo
x - oo = -oo
x * oo = oo
x / oo = 0


LE4dGOLEM wrote:
SpitValve wrote:What if x is infinite or zero?

You can not divide by zero. Infinity is infinity, it doesn't matter.


I wasn't dividing by zero anywhere... I was pointing out the flaw in Solt's argument that ? can be used that simply.

(p.s. I love Macs... option-5 is ?... what's the simplest way to make that symbol on Windows or Linuxy things?)

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Postby tallest » Sat Apr 07, 2007 9:52 pm UTC

My last post was made without quite enough sleep, and I think I can express myself in a better fashion. I think jestingrabbit and I are making similar points, but I don't think I've been specific enough in my posts for a discussion about mathematics.

Math is very well defined. If we wan to evaluate infinity*0 then we have to work in a number system in which this makes sense. The real numbers do not include infinity and hence infinity*0 has no meaning in the real number system, instead we have to use lim_{x->infinity}(x*0)=0.

There are number systems that do include infinity. These numbers systems will have rules for dealing with infinity and the expression 0*infinity will be well defined. The list of equations involving infinity that Solt posted is actually a subset of the additional rules of arithmetic that the affinely extended real numbers use to evaluate expressions with infinity.

As SpitValve has mentioned, if we want to talk about mathematics, an intuitive understanding is not good enough.
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Postby EvanED » Sun Apr 08, 2007 11:29 pm UTC

SpitValve wrote:(p.s. I love Macs... option-5 is ?... what's the simplest way to make that symbol on Windows or Linuxy things?)


Um, shift+'/'?

Is that symbol NOT supposed to look like a question mark to me?

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Postby SpitValve » Mon Apr 09, 2007 1:23 am UTC

bugger :)

? is infinity for me.. I blame phpBB, it doesn't like non-English characters all that much either

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Postby warriorness » Mon Apr 09, 2007 1:30 am UTC

The infinity symbol in the original post ("∞") works fine for me, but not whatever the Mac guy posted. His must be some crazy unicode deal.
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Re: Infinity

Postby Pathway » Tue Apr 10, 2007 3:19 am UTC

0*∞ = ∞*0 is necessary (multiplication in the real numbers is commutative) so you can't use an order-dependent definition.

Even if you could, infinity doesn't have the properties of numbers so you can't treat it like one. Normal numbers are elements of an ordered set, meaning there are numbers greater than and less than any given number. This would imply that for any a in the real numbers, a+1 > a > a-1.

But does that work with infinity? Nope. Infinity plus one is... what? If you say infinity it's not a number. If you say it's just infinity plus one, you run into a problem as well.

Now, multiplication and addition and all that stuff are defined to be operations on numbers. If you want to multiply with anything else you'd better define what multiplication means for that thing.

Solt wrote:The only thing untrue about this statement is "for all x."



That means it's all untrue.

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Postby Aoeniac » Tue Apr 10, 2007 4:11 am UTC

Infinity sucks.

They should just take Infinity out of math.

Give it its own department so people who don't want to deal with it don't have to.
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Postby AntonT » Tue Apr 10, 2007 7:33 pm UTC

Hi guys, first post on these forums. :) I got introduced to xkcd by a friend a few weeks ago and I just love the comic. Anyway, I was browsing through the forum and picked up on this interesting discussion. Hats off to Alky for what's probably the most solid post in here so far. I think you really hit the nail on the head there. Anyway, to the reason I registered.

tallest wrote:It can be proven that, if a is an i-large number then 1/a is an i-small number; similarly, if b is an i-small number then 1/b is an i-large number; and so on and so forth. Thus we can use infinitesimals and infinities in mathematical expressions without the need for a limit.


I'd like to see this proof because I think I can see a bit of a hole in it:

Firstly, by your definition of i-small and i-large, I have to assume that you're only talking about positive, real numbers, (excluding zero). Seeing as i-large and i-small are by definition, real numbers, real number arithmetic must apply to them.

Hence, any x that is less than a (some really small value approaching 0) is i-small or in mathematical notation:
for x < a, x is i-small
and
any x that is greater than b (some really large value approaching infinity) is i-large or in mathematical notation
for x > b, x is i-large

Now let some variable, y, approach a from 0. Then 1/y will be i-large according to what you've posted.

Let another variable, z, approach b from infinity. Then 1/b will be i-small according to what you've posted.

These, in turn must mean that a = 1/b and b = 1/a, from what my logic is telling me, and this means that the smallest value that will ever be used by people will be the reciprocal of the largest value that will ever be used by people (which isn't necessarily true).... or am I looking at this all wrong? :roll:

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tallest
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Postby tallest » Tue Apr 10, 2007 10:58 pm UTC

Yes, I know. I realized this shortly after I posted it and just hoped no one would notice. :)

This is my failing though, and not the internally defined hyperreal's. Like I said, I was only recalling this information from a seminar I saw last semester. I can't remember the correct formulation, but the theory is sound and is used in practice by crazy number theorists.
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