B Is infinity an imaginary number?

[sheld]

Is infinity a imaginary number?

 

[member 587159]

Is infinity a imaginary number?

No. Infinity is not a number. It's a concept.

 

[BL4CKB0X97]

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.

 

[jbriggs444]

1,2,3,4,5,6,7,8,9...

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]

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.

 

[UsableThought]

No. Infinity is not a number. It's a concept.

1,2,3,4,5,6,7,8,9... Pretty real to me.

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]

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.

 

[UsableThought]

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.

 

[mfb]

It's a real number.

It is not. And your counting or the train will never reach it.

 

[jbriggs444]

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.

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.

 

[alan2]

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.

 

[UsableThought]

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.

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]

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.

 

[UsableThought]

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.

 

[PeroK]

Referring back to the Gowers book ...

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.

 

[UsableThought]

I wouldn't get too hung up on this stuff.

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.

 

[BL4CKB0X97]

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.

That what bugs me. Just because we can't reach it, doesn't mean it's not there.

 

[UsableThought]

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."

 

[PeroK]

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).

 

[Comeback City]

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?

 

[mfb]

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.

 

[PeroK]

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'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?

 

[Comeback City]

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!

 

[alan2]

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?

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.

 

[Comeback City]

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?

It is defined in the hyperreal numbers, for example, where it is simply infinity + 1.

 

[UsableThought]

Does that not contradict @mfb and his notion of infinity + 1 in the hyperreals? Or is it the same?

See https://en.m.wikipedia.org/wiki/Hyperreal_number

I understand about every third word, but it does look interesting.

 

[PeroK]

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.

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?

 

[mfb]

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.

Here is an illustration.

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?

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]

@mfb
@alan2

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.

 

[Comeback City]

Then someone better switch that tag up to an A+... I'm ready to learn me some more hyperreals!

 

[alan2]

Does that not contradict @mfb and his notion of infinity + 1 in the hyperreals? Or is it the same?

No, I think we said essentially the same thing. He did say infinity plus one equals infinity but there is no number called infinity, there are infinite numbers. So, technically, an infinite number plus one is another infinite number. If you're interested, Jerome Keisler has made his intro calculus text available for free. Sections 1.5 and 1.6 contain a discussion of the hyperreals.

https://www.math.wisc.edu/~keisler/calc.html

 

[alan2]

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.

I couldn't disagree more.

 

[alan2]

It is "not finite", but w+1≠ωw+1 \neq \omega - they are different hyperreal numbers.

I never said they were the same, I said they were both infinite.

 

[PeroK]

I couldn't disagree more.

A Basic level thread is one that is suitable for High School students and assumes the appropriate knowledge. Hyperreals do not come into that category.

We have rules on PF because it's a serious science and maths forum. You need to adhere to these rules.

 

[Comeback City]

@PeroK ...
In the thread "How can the universe grow if it is infinite", you noted...

Infinity + 1 is undefined

Is it safe for me to assume that you were simply speaking from the "real number" point of view in this answer?

He did say infinity plus one equals infinity

He actually said the opposite...

It is "not finite", but w+1≠ωw+1≠ωw+1 \neq \omega - they are different hyperreal numbers.

 

[alan2]

A Basic level thread is one that is suitable for High School students and assumes the appropriate knowledge. Hyperreals do not come into that category.

We have rules on PF because it's a serious science and maths forum. You need to adhere to these rules.

Again, I disagree. Hyperreals have been used consistently and successfully in teaching calculus to high school students for decades. I'm not sure what your objection is. They are also taught irrationals and use and understand them successfully without any thought to the construction of the reals which most of them will never see.

 

[PeroK]

@PeroK ...
In the thread "How can the universe grow if it is infinite", you noted...

Is it safe for me to assume that you were simply speaking from the "real number" point of view in this answer?

Yes..

 

[FactChecker]

There are different levels of "infinity" which are represented by cardinal numbers, $\aleph$₀, $\aleph$₁, $\aleph$₂, ...
They can be used to measure the size of objects that we would consider "real" (the size of the set of natural numbers is $\aleph$₀ and the size of the set of real numbers is $\aleph$₁). They have mathematical properties and can be used in mathematical proofs like transfinite induction.

With all of that, I would say that they are not imaginary. They measure things that really exist, just like natural numbers do.

 

[UsableThought]

There are different levels of "infinity" which are represented by cardinal numbers, $\aleph_0, \aleph_1, \aleph_2, ...$ . . . I would say that they are not imaginary. They measure things that really exist, just like natural numbers do.

I'm confused here & may be misunderstanding; but I would have assumed that we wouldn't want to conflate the human act of counting (i.e. the natural numbers, or any other number system we invent) with things that exist in nature - if that's what you mean by "really exist"? (Though maybe you mean something else?)

I.e., nature produces phenomena that we can count; that doesn't mean nature counts. Nature does what it does but there is no God somewhere keeping track by counting. And both counting and numbers I would say are imaginary - that is, they are acts of the imagination.

As to the question of whether infinities exist in nature, this article gives the differing perspectives of several cosmologists; the general thrust seems to be that although we can speculate or even assume that some aspects of nature are infinite, as yet we have no actual way of knowing: https://plus.maths.org/content/do-infinities-exist-nature-0

 

[FactChecker]

I am not so concerned about what mere humans can verify. Do we have to physically count to a number in order to agree on its existence? I accept the number 10^{1,000,000,000} as a valid number even though no one and nothing has never counted that high, one-by-one. Nature and reality are not limited by capabilities of humans and their machines. I accept the set of all natural numbers as a real thing even though there are infinitely many of them. I designate its size as $\aleph$₀. Others don't have to, but I think they are being needlessly stubborn. I guess I am just not as philosophical about this as others are.

 

[Randy Beikmann]

We know that pi exists as a real number, and yet no one has ever written it out completely, and never will. Therefore "not ever reaching the end of a series" is not a valid argument against something existing. (I realize that the series that produce pi do converge though.)

And while infinity + 1 = infinity indicates it is not truly a number, we say that 1/infinity = 0, so there we do use it as a number.

 

[jbriggs444]

And while infinity + 1 = infinity indicates it is not truly a number, we say that 1/infinity = 0, so there we do use it as a number.

We do not say that 1/infinity = 0. At least not in the field of real numbers (or hyper-reals for that matter).

 

[Randy Beikmann]

We do not say that 1/infinity = 0.

OK, I'll bite. What do we call it? I believe the limit of 1/x as x -> infinity is zero. Are you making that distinction?

 

[Comeback City]

OK, I'll bite. What do we call it? I believe the limit of 1/x as x -> infinity is zero. Are you making that distinction?

Are you thinking of infinity in a sense of (n/0) where n is any real number, and thus...
1/∞ = 1/(n/0) = 1 (0/n) = 0/n = n
?

 

[jbriggs444]

OK, I'll bite. What do we call it? I believe the limit of 1/x as x -> infinity is zero. Are you making that distinction?

There is a distinction between a limit and a quotient, yes. There is no rule that says that the limit of the quotient must be equal to the quotient of the limits when one of the two fails to exist.

Edit: Note that when we write "as x approaches infinity", this is an abuse of notation that can be better understood as "as x increases without bound". The infinity that is being approached in this case is not a number. It is a notional limit point. While one can take these notional limit points and add them to the set of real numbers (see Compactification), the resulting set of objects is no longer a field. It is missing some useful properties.

 

[Randy Beikmann]

There is a distinction between a limit and a quotient, yes. There is no rule that says that the limit of the quotient must be equal to the quotient of the limits when one of the two fails to exist.

So my question is, still, how much is 1/infinity?

 

[jbriggs444]

So my question is, still, how much is 1/infinity?

Since "infinity" is not a number, 1/infinity is not defined.

 

[Randy Beikmann]

Since "infinity" is not a number, 1/infinity is not defined.

In that case I, and many other people, have done many problems incorrectly.

 

[jbriggs444]

In that case I, and many other people, have done many problems incorrectly.

You need to understand something basic about mathematics. Mathematics is (by and large) about careful definitions, precise axioms and their logical consequences. There are lots of possible sets of definitions and no one true "right" set.

If you write down "1/infinity", that is simply a meaningless sequence of ASCII characters until you identify a context where we can find definitions for the relevant concepts. One commonly used context is arithmetic on the so-called real numbers. "infinity" does not denote any real number.

Another possible context is the extended real line -- the reals augmented by +oo and -oo. One can define an arithmetic on these extended reals. And in this arithmetic, 1/oo is defined to be 0.

When you ask "what is 1/infinity" without identifying a context, I can correctly object that it is undefined. You have not clarified which "infinity" you mean (or which arithmetic). That does not mean that I am calling you an incorrect problem solver.

 

[Randy Beikmann]

I am an engineer, and not a mathematician, so I understand I do not speak in the same precise terms as one. But I struggle to think of an infinity that can't be inverted, and, when it is, would not equal zero. Is there one?