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 
Why is infinite technically not a number.
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?
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.
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
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.
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.
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?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
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.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?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
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?
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.
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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