No. Infinity is not a number. It's a concept.Is infinity a imaginary number?
1,2,3,4,5,6,7,8,9.....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".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.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".
No. Infinity is not a number. It's a concept.
Agree with @Math_QED and don't understand what @BL4CKB0X97 is trying to say. What constitutes "pretty real" when talking about an abstraction?1,2,3,4,5,6,7,8,9..... Pretty real to me.
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.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.
So if I can paraphrase, you say that since we can count toward infinity (even if we can never get there), that makes it a number, yes?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.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.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.
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.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.
Mathematics is an abstraction. That's a fundamental point about it.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.
Yes, I know. What I'm saying is (a) I like this, and (b) nobody ever bothered to tell me about it when I was a kid! Maybe that's why high school algebra was so dull back then.Mathematics is an abstraction. That's a fundamental point about it.
I wouldn't get too hung up on this stuff. To me the abstraction of numbers comes from 1) recognising the abstract thing that links 12 lions with 12 coins (both of these are very real) but the notion of 12 as a thing in itself is abstract; and 2) manipulating numbers so that 12 + 7 = 19, hence (in the real world) 12 of anything plus 7 of anything is 19 of anything.Referring back to the Gowers book ...
There is no "hung up" here; I fear you have either completely misunderstood me, or aren't interested in what I have to say, or for some reason want to lecture me against some peril that exists in your mind but not in mine. Whatever the reason, I appreciate your efforts to be helpful, but would rather that you not try; we don't seem simpatico.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.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.
Maybe we can make an analogy. Say you're in the U.K. and you want to get to the a point in the Himalayas. You'd head roughly east, right? And eventually, if everything went well, you'd get to your destination.That what bugs me. Just because we can't reach it, doesn't mean it's not there.
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).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: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).
It's not a number, so you cannot use it in arithmetic operations. The sum is s undefined, therefore.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!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?
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).
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.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?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 defined in the hyperreal numbers, for example, where it is simply infinity + 1.