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sheld
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Is infinity a imaginary number?
sheld said:Is infinity a imaginary number?
1,2,3,4,5,6,7,8,9...Pretty real to me. But I've had many an argument over infinity.sheld said:Is infinity a imaginary number?
Instead of being presented with a list of all the natural numbers, we see here a list of numerals and an ellipsis. That's a couple of levels of abstraction away from being "real".BL4CKB0X97 said:1,2,3,4,5,6,7,8,9...
You know what the ellipsis means. I do not have an infinite amount of time spare to write all of the sequence,I'm afraid.jbriggs444 said:Instead of being presented with a list of all the natural numbers, we see here a list of numerals and an ellipsis. That's a couple of levels of abstraction away from being "real".
Math_QED said:No. Infinity is not a number. It's a concept.
BL4CKB0X97 said:1,2,3,4,5,6,7,8,9... Pretty real to me.
UsableThought said:Agree with @Math_QED and don't understand what @BL4CKB0X97 is trying to say. What constitutes "pretty real" when talking about an abstraction?
The question has some interest for me for this reason: I'm finishing up a course on Coursera titled "Introduction to Mathematical Thinking" (basically, predicate logic for doing proofs); we have been looking at examples from the naturals, integers, and reals. In the last couple of weeks we learned about intervals (those of us who didn't already know about these), and the question of "what's infinity" came up; and more specifically, someone asked on the course forum, "Is infinity a number?"
I believe what prompted the question was that we had just learned a notation for intervals where the right side of the interval can point to ##\infty## or the left side can point to ## - \infty##.
The answer I gave was that generally, the definition of a number presupposes that if it is not represented by an unknown, then it can be described as a point on a number line. By contrast, infinity cannot be represented as a point on a number line. If you read more math, I guess you'll find out the infinities can be discussed as sets, along the lines proposed by Cantor; and so far as I know, sets aren't numbers either. See http://mathworld.wolfram.com/Infinity.html and http://mathworld.wolfram.com/InfiniteSet.html
P.S. I also got a bit smart-alecky in my answer on that other forum and said that "If the number line were a train line, infinity would be the last stop . . . which would never be reached." But that's just being cute.
BL4CKB0X97 said:What I was trying to say is that you can just keep counting and never stop. As per your train line analogy, you can get to the last stop, but then build another. It's a real number.
It is not. And your counting or the train will never reach it.BL4CKB0X97 said:It's a real number.
The "real numbers" have an agreed-upon definition within mathematics. They are the members of a complete, ordered, archimedean field. "Infinity" does not qualify.BL4CKB0X97 said:What I was trying to say is that you can just keep counting and never stop. As per your train line analogy, you can get to the last stop, but then build another.
It's a real number.
alan2 said:It all depends on what you're talking about and defining a mathematical object and checking the logical self consistency of the system in which you are working is very different from the normal casual concept of real.
UsableThought said:I guess this is why, in a funny way, the question appeals to me. As someone who has returned to learning some math after many decades of total absence, the notion of abstraction is one of the more difficult yet intriguing concepts I've started to pick up. E.g. it gets mentioned a lot early on in Tim Gowers's little book Mathematics: A Very Short Introduction. I think it makes math more appealing.
PeroK said:Mathematics is an abstraction. That's a fundamental point about it.
UsableThought said:Referring back to the Gowers book ...
PeroK said:I wouldn't get too hung up on this stuff.
That what bugs me. Just because we can't reach it, doesn't mean it's not there.jbriggs444 said:The "real numbers" have an agreed-upon definition within mathematics. They are the members of a complete, ordered, archimedean field. "Infinity" does not qualify.
If by "real" you have in mind something more physical, then the evidence to date indicates that you cannot just keep counting and never stop. You die first.
BL4CKB0X97 said:That what bugs me. Just because we can't reach it, doesn't mean it's not there.
BL4CKB0X97 said:That what bugs me. Just because we can't reach it, doesn't mean it's not there.
Just a question to build off of this:PeroK said:No one is saying that infinity does not exist, as a mathematical abstraction. But, it is not a number - by definition. If you try to make it a number, then one problem is that you can no longer do algebra with all numbers (as infinity would not obey the normal rules of addition and multiplication).
Comeback City said:Just a question to build off of this:
Infinity + 1 = ?
I have seen a few answers to this, including "infinity" and "undefined"
Can anyone clarify this?
Ah, I always loved a good mind trick!PeroK said:It's not a number, so you cannot use it in arithmetic operations. The sum is s undefined, therefore.
In fact, "infinity -1" is more interesting, since that must be the whole number you got "just before" you finally got to infinity! What number might "infinity -1" be? If it's a "normal" whole number ##n##, then so is ##n+1## and hence ##n+1## can't be infinity. And, if it's an another infinite number, then how did you reach that?
PeroK said:No one is saying that infinity does not exist, as a mathematical abstraction. But, it is not a number - by definition. If you try to make it a number, then one problem is that you can no longer do algebra with all numbers (as infinity would not obey the normal rules of addition and multiplication).
Comeback City said:Just a question to build off of this:
Infinity + 1 = ?
I have seen a few answers to this, including "infinity" and "undefined"
Can anyone clarify this?
Does that not contradict @mfb and his notion of infinity + 1 in the hyperreals? Or is it the same?alan2 said:As I mentioned above, existence depends on what you're talking about. In the real number system infinite numbers may not exist but they certainly do exist in a hyperreal number system and do obey the normal rules of addition and multiplication. In that case, an infinite plus one is infinite.
mfb said:It is defined in the hyperreal numbers, for example, where it is simply infinity + 1.
Comeback City said:Does that not contradict @mfb and his notion of infinity + 1 in the hyperreals? Or is it the same?
alan2 said:As I mentioned above, existence depends on what you're talking about. In the real number system infinite numbers may not exist but they certainly do exist in a hyperreal number system and do obey the normal rules of addition and multiplication. In that case, an infinite plus one is infinite.
It is "not finite", but ##w+1 \neq \omega## - they are different hyperreal numbers.alan2 said:As I mentioned above, existence depends on what you're talking about. In the real number system infinite numbers may not exist but they certainly do exist in a hyperreal number system and do obey the normal rules of addition and multiplication. In that case, an infinite plus one is infinite.
That is not a sequence of real numbers, and if the question is not about real numbers, it is unclear what "unbound" means. In the hyperreal numbers, and assuming ##\infty## means ##\omega##, the sequence is bound (e.g. by ##\omega##).PeroK said:When I was a graduate student I had to mark undergraduate homework. One question asked for an example of an unbounded sequence. The answer given was:
##1, 2, \infty, 4, 5 \dots##
How would you assess that?
Comeback City said:Does that not contradict @mfb and his notion of infinity + 1 in the hyperreals? Or is it the same?
PeroK said:Invoking the hyperreals in a "B" level thread is the maths equivalent of invoking the stress-energy tensor to explain the SHM of a pendulum!
The hyperreals are at an advanced undergraduate level and depend on a solid grasp of real analysis. They are not suitable for a "B" level thread, IMHO.
mfb said:It is "not finite", but w+1≠ωw+1 \neq \omega - they are different hyperreal numbers.
alan2 said:I couldn't disagree more.
Is it safe for me to assume that you were simply speaking from the "real number" point of view in this answer?PeroK said:Infinity + 1 is undefined
He actually said the opposite...alan2 said:He did say infinity plus one equals infinity
mfb said:It is "not finite", but w+1≠ωw+1≠ωw+1 \neq \omega - they are different hyperreal numbers.