Is infinity a imaginary number?
No. Infinity is not a number. It's a concept.
Pretty real to me. But I've had many an argument over infinity.
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".
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.
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.
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.
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?
I was curious enough to look up the topic on Wikipedia (which isn't a terrible source for math-related topics). Here is that link - https://en.wikipedia.org/wiki/Infinity - and here are the lead two paragraphs.
Infinity (symbol: ∞) is an abstract concept describing something without any bound or larger than any number. Philosophers have speculated about the nature of the infinite, such as Zeno of Elea, who proposed many paradoxes involving infinity, and Eudoxus of Cnidus, who used the idea of infinitely small quantities in his method of exhaustion. Modern mathematics uses the concept of infinity in the solution of many practical and theoretical problems, such as in calculus and set theory, and the idea also is used in physics and the other sciences.
In mathematics, "infinity" is often treated as a number (i.e., it counts or measures things: "an infinite number of terms") but it is not the same sort of number as natural or real numbers.
The second paragraph would seem to capture your argument, but nonetheless insists there is a distinction still to be observed. Of course the only useful purpose for defining infinity as one thing or another is for doing math. And I'm not at the point where I'm doing math that involves infinities.
It is not. And your counting or the train will never reach it.
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.
You really have to be clear about what you're asking. Is it imaginary? No. Imaginary numbers are well defined and do not include a number called infinity. Is it real? No, the real numbers are also well defined and do not include infinity. But no number is "real" in a philosophical sense, they are concepts. In the natural number system rationals do not exist. In the system of rationals, irrationals do not exist. However, in the hyperreal number system infinite and infinitesimal numbers do exist. 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. That's a question for philosophers.
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.
Mathematics is an abstraction. That's a fundamental point about it.
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.
Referring back to the Gowers book I just mentioned, here is an early spot in the book where Gowers starts to explain what he means by "abstraction"; the bold is mine:
The abstract method in mathematics, as it is sometimes called, is what results when one takes a similar attitude to mathematical objects. This attitude can be encapsulated in the following slogan: a mathematical object is what it does.
He then points out that once we get past the few very small natural numbers that we can subitize (understand at a glance), we are into a realm that most of us aren't even conscious of any more: we don't understand numbers as pure objects, e.g. as we do "3", but as the result of operations; even though we do these operations very quickly:
However, when we consider larger numbers [than 5], there is rather less of this purity. Figure 8 gives us representations of the numbers 7, 12, and 47. Perhaps some people instantly grasp the sevenness of the first picture, but in most people’s minds there will be a fleeting thought such as, ‘The outer dots form a hexagon, so together with the central one we get 6 + 1 = 7.’ Likewise, 12 will probably be thought of as 3 × 4, or 2 × 6. As for 47, there is nothing particularly distinctive about a group of that number of objects, as opposed to, say, 46. If they are arranged in a pattern, such as a 7 × 7 grid with two points missing, then we can use our knowledge that 7 × 7 − 2 = 49 − 2 = 47 to tell quickly how many there are. If not, then we have little choice but to count them, this time thinking of 47 as the number that comes after 46, which itself is the number that comes after 45, and so on.
In other words, numbers do not have to be very large before we stop thinking of them as isolated objects and start to understand them through their properties, through how they relate to other numbers, through their role in a number system. This is what I mean by what a number ‘does’.
It's also interesting to me to compare abstraction in mathematics with abstraction in other fields - for a laundry list see https://en.wikipedia.org/wiki/Abstraction#As_used_in_different_disciplines.
And then again it's interesting to realize that mathematical abstraction does indeed arise out of the intuitive physical basis of having a body and a mind shaped by evolution; e.g. we understand things like "collections", "inside/outside," and "above/below" at this level. That's a whole other philosophical debate, whereas playing with abstraction in math seems like just that, play.
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.
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'm glad that authors such as Tim Gowers exist who do enjoy talking & sharing ideas about such concepts as abstraction. He is one of several authors who make math enjoyable for me, along with people like Gelfand, Kevin Devlin, etc.
That what bugs me. Just because we can't reach it, doesn't mean it's not there.
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.
But say you just decide to head east . . . and keep going east, as you would keep going toward infinity. You could travel east, more east, and more east. You could keep going around the globe, ever eastward, until you died; but there would be no point where you could stop; no "destination" for you. Only more traveling east.
So just as traveling forever east means there is no destination at the end, traveling forever toward infinity means there is no number at the end. To take a Gertrude Stein quote out of context, "There is no 'there' 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).
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?
It is not defined in the real numbers as infinity is not a real number.
It is defined in the hyperreal numbers, for example, where it is simply infinity + 1.
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?
Ah, I always loved a good mind trick!
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.
Does that not contradict @mfb and his notion of infinity + 1 in the hyperreals? Or is it the same?
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