Why is infinite technically not a number.

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Why is ?(infinite) not considered a number?

I mean, if zero is considered a number, it sounds reasonable to consider infinite a number too.

Edited by gasto

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It depends on what field of mathematics you're talking about. There is a field that works with infinity as a "number." Considering the Arabic basis of numbers, a system of powers of ten placeholders, zero indicates that the number represented has no component of the power of ten at the position the zero occupies. I.e., 1023 = 1*1000 + (nothing)*100 + 2*10 + 3*1. In that context, a sequence of multiples of powers of ten, infinity is of no use. It would not have meaning as a power of ten in a specific position in an Arabic base number.

In Roman numerals, there is no concept of zero as it is not a placeholder-type system.

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For example, in set theory,

M??=?
Set M union universal discourse equals universal discourse.

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Infinity is a concept and doesn't work like a number. If you add 1 to infinity you still have infinity. On the other hand if you add 1 to 0 you get 1.

Look at it a different way - zero has a specific place on a number line. Where is infinity? You would say the number of points between zero and one is infinite and not a specific point.

Actually, if you add one to infinity you get ?+1=?

Edited by gasto

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In a certain sense, Infinity can't be *a* number, because it is all numbers at once. There is no single value for infinity, There are also multiple infinities, and some infinities are demonstrably larger than other infinities. Eg, take the set of all odd integers and the set of all even integers. Clearly they are equal in size, but if you take the set of all integers, it is the sum of the odd integers set and even integers set, and is thus larger than either of them.

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"Infinity" means several different things, some of which allows you to think of it as a number:
* If you are measuring how big sets are, you define two sets to be as big as each other if there is a bijection between them, and you say a set is infinite if it is as big as one of its proper subsets (in other words, if you add one element that wasn't in the set, you get a set exactly as large as the original). But thinking of the cardinality of an infinite set as being simply "infinity" doesn't really cut it, because there is a large collection of different sizes of infinite sets.
* If you are thinking of the behavior of sequences of real numbers in the limit, it is convenient to extend the real line to include +infinity and -infinity, so you can say that a sequence has limit +infinity or -infinity, meaning for any bound you propose, there is a point from which all the elements of the sequence are beyond your proposed bound.
* If you are dealing with the slope of a straight line, it makes sense to think of it as a real number or infinity (a single infinity, in contrast with the previous bullet point). In this case we are dealing with the projective real line. There is also an analogous "projective complex line", which consists of the complex numbers plus a number called infinity (this is also known as the Riemann sphere).
* Any topological space can be extended into a compact topological space by adding a single point called "infinity" (Alexandroff one-point compactification).
* The surreal numbers contain a number "omega" that you could call "infinity" instead (they call it "omega" because it is one very specific type of infinity: the first infinite ordinal). However, these surreal numbers are kind of weird, and they contain notions like "infinity minus 5", "infinity times 3", "infinity squared", and even "square root of infinity".

I would say the notions of infinity from the second, third and fifth bullet points fit well into considering it as a number. Notice how the floating-point representations of numbers on a computer typically include +infinity and -infinity, which leads me to believe they are based around the notion of infinity from the second bullet point.

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If you think like a mathematician, you can make your own definitions (as long as you take the consequences). So you can define infinity as a number if you like (e.g. as the sum of a sufficient number of ones so that it becomes larger than any finite number). And you'll end up with the surreal or hyperreal numbers. You may ask, are those numbers real? Well, are the real numbers real? And are the imaginary numbers just imaginary? Hmmm...

As for the original question:

Assume x is a positive integer that is greater than any other integer. Let y = x + 1. Then y is greater than x (by rules of addition). But x was supposed to be greater than y. Contradiction. Assumption was false. So there can't be such an integer x.

Edited by Felix Ungman

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My favorite is what happens when you try to add all the integers -- 0 + 1 + 2 + 3... all the way to infinity.

The intuitive answer is infinity, but it's also possible to get the answer of -1/12th

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There are ideas how to extend the field of real numbers to include infinity:

http://en.wikipedia.org/wiki/Extended_real_number_line

(The German article is more elaborate: http://de.wikipedia.org/wiki/Erweiterte_reelle_Zahl)

However, the usual arithmetic rules would no longer hold in such a field, because this set is no longer an ordered field:

http://en.wikipedia.org/wiki/Ordered_field

So in practice it seem to be not very useful to work with such a definition.

Related question:

From a mathematical point of view: If you die young, are you longer dead?

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You can easily place "0" on a ruler, right?  So it is easy to see that "0" is a number.

If infinity was a "number", where would you put that on the ruler?

For infinity to be a "number", it had to have a fixed position on this ruler, right?  To have a fixed position, Infinity would need to have an upper limit -- but no matter how big of a number you can think of, there is always a number that is bigger.  In fact, there is an infinity of numbers bigger than any numbers you can think of.  Infinity just means that something goes on and on forever, never ending.  Infinity is not a number, but just something to describe that something tends to grow or shrink without bounds, when used in mathematics.  It is an entirely different concept altogether.

Edited by aregee

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Assume x is a positive integer that is greater than any other integer. Let y = x + 1. Then y is greater than x (by rules of addition). But x was supposed to be greater than y. Contradiction. Assumption was false. So there can't be such an integer x.

Or the conclusion could be that this particular definition of infinity is not very good. :)

If there is such a thing as "infinity" in a set of numbers, it most definitely won't be a positive integer. The smallest infinite ordinal "omega" is greater than any integer. But of course "omega" itself is not an integer: http://en.wikipedia.org/wiki/Ordinal_number

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One thing that hasn't been mentioned much (besides Álvaro) is which infinity you're talking about. Not all infinities are the same. For example, there's Aleph-0* and Aleph-1**.

So when you say "is infinity a number" one valid response is "which infinity are you talking about?"

It's also important to note that in spite of us sometimes treating infinity like a number, it doesn't match our definition of number: "an arithmetical value, expressed by a word, symbol, or figure, representing a particular quantity and used in counting and making calculations and for showing order in a series or for identification." There isn't a specific, particular quantity represented by infinity.

If you just go back and look at the core definitions of number and infinity, you'll see that infinity doesn't quite match the requirements to be a number.

*Aleph-0 is the number of integers (and, interestingly, there are the same number of positive integers as there are negative and positive integers).
**Aleph-1 may or may not be the number of real numbers; we cannot prove nor disprove this, but it's important to note that Aleph-1 is greater than Aleph-0.

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My favorite is what happens when you try to add all the integers -- 0 + 1 + 2 + 3... all the way to infinity.
The intuitive answer is infinity, but it's also possible to get the answer of -1/12th

Perhaps indicating some kind of overflow in the substrate of the universe, or that something akin to floating point error exists even for the humble integer when the values are extreme?

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My favorite is what happens when you try to add all the integers -- 0 + 1 + 2 + 3... all the way to infinity.
The intuitive answer is infinity, but it's also possible to get the answer of -1/12th

Perhaps indicating some kind of overflow in the substrate of the universe, or that something akin to floating point error exists even for the humble integer when the values are extreme?

Nah, it's a bug in the universe's FPU, similar to the Pentium FDIV bug. Send bug reports to your nearest church/chapel/synagogue/mosque/temple/etc.

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Infinitely smaller if we divide by infinity.

Infinitely larger if we multiply by infinity.

Both results can be compared if one is larger than the other, so both must be numbers if we are able compare if infinitely small is less than infinitely large...right?

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My favorite is what happens when you try to add all the integers -- 0 + 1 + 2 + 3... all the way to infinity.

The intuitive answer is infinity, but it's also possible to get the answer of -1/12th

This is in fact, not true. If you evaluate the sum of all the natural numbers (1 + 2 + 3 + 4...) it is infinitely large. To evaluate this sum to have a value of -1/12th is not really correct. =/

If you want to learn the mathematical reasoning behind why the answer is infinity, but why you could evaluate a similar looking sum to have a value of -1/12th, I suggest looking up the zeta function.If you have an interest in maths, I really recommend it, there are some surprising results and really beautiful mathematics to be found there. If you don't want to learn the maths, just take it as 1 + 2 + 3 + 4... = infinity

Both results can be compared if one is larger than the other, so both must be numbers if we are able compare if infinitely small is less than infinitely large...right?

No, not right. Infinities can be compared to each other, but this does not mean they must be a number. Think of it like this - I can compare two shirts and select my favourite, but that does not mean I am evaluating each to a numerical value. Not everything that can be compared must be a number.

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This is in fact, not true. If you evaluate the sum of all the natural numbers (1 + 2 + 3 + 4...) it is infinitely large. To evaluate this sum to have a value of -1/12th is not really correct. =/
If you want to learn the mathematical reasoning behind why the answer is infinity, but why you could evaluate a similar looking sum to have a value of -1/12th, I suggest looking up the zeta function.If you have an interest in maths, I really recommend it, there are some surprising results and really beautiful mathematics to be found there. If you don't want to learn the maths, just take it as 1 + 2 + 3 + 4... = infinity

I'm not a mathematician, but according to wikipedia and wolframalpha, ?(?1) = -1/12

It might not be true under every system of mathematics, but it's certainly a correct answer under some of them. It's even used in physical calculations where the mathematical prediction matches up correctly with observations!

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This is in fact, not true. If you evaluate the sum of all the natural numbers (1 + 2 + 3 + 4...) it is infinitely large. To evaluate this sum to have a value of -1/12th is not really correct. =/
If you want to learn the mathematical reasoning behind why the answer is infinity, but why you could evaluate a similar looking sum to have a value of -1/12th, I suggest looking up the zeta function.If you have an interest in maths, I really recommend it, there are some surprising results and really beautiful mathematics to be found there. If you don't want to learn the maths, just take it as 1 + 2 + 3 + 4... = infinity

I'm not a mathematician, but according to wikipedia and wolframalpha, ?(?1) = -1/12

Mats1 is actually kind of correct. The thing is that the sum of the natural numbers is indeed infinite. In order to get -1/12 you have to use a different concept of numbers, called p-adic numbers.

For the curious, this question and answer give good a good introduction to the subject.

Anyway, this is further complicated by the fact that we aren't actually talking about 1 + 2 + 3 + ... = -1/12. What we're really talking about are limits and convergence, which isn't necessarily the same (or as strict) as equality. Because it's a limit we're computing, there are more ways to show 1 + 2 + 3 + ... = -1/12 than just the zeta function. So you might say 1 + 2 + 3 + ... is infinity just as much as you might say it's -1/12. Edited by Cornstalks

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I don't know what 1+2+3+4+...=-1/12 has to do with p-adic numbers. Care to explain?

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I'm not a mathematician, but according to wikipedia and wolframalpha, ?(?1) = -1/12

It might not be true under every system of mathematics, but it's certainly a correct answer under some of them. It's even used in physical calculations where the mathematical prediction matches up correctly with observations!

?(?1) = -1/12

This is the zeta function. Zeta of -1 DOES evaluate to - 1/12. The simple sum of writing 1 + 2 + 3 + 4 etc does NOT evaluate to - 1/12.

The zeta of -1 is the sum:

1/1^-1 + 1/2^-1 + 1/3^-1 + 1/4^-1

Edited by Mats1

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Isn't 2-1 equivalent to 1/2, so 1/2-1 would be 1/(1/2), which would be equivalent to 2? So 1/1-1 + 1/2-1 + 1/3-1 + 1/4-1 ... = 1+2+3+4...?

Like I said, I'm not good at the math, so it's interesting why two seemingly equivalent statements aren't equivalent.

i.e. sum n where n=1..? != sum 1/(1/n) where n=1..?

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Isn't it simply because they use different summation methods? I similar problem occurs when measuring an area with riemann integrals vs lebesgue integrals, some areas can be measured with one but not the other.