# B Is infinity an imaginary number?

#### PeroK

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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.

$1, 2, \infty, 4, 5 \dots$

How would you assess that?

#### mfb

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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.

$1, 2, \infty, 4, 5 \dots$

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

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@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

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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

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@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

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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

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#### FactChecker

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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 101,000,000,000 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.

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#### Randy Beikmann

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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

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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

Gold Member
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

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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.

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#### Randy Beikmann

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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

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So my question is, still, how much is 1/infinity?
Since "infinity" is not a number, 1/infinity is not defined.

#### Randy Beikmann

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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

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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

Gold Member
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?

#### UsableThought

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.
I like the above post, including the qualification I have quoted here. This agrees entirely with what I have been reading lately in a few different books, including the Tim Gowers book I mentioned, https://www.amazon.com/dp/0192853619/?tag=pfamazon01-20. Gowers, by the way, is a prof at U. of Cambridge in the U.K.. who seems pretty highly credentialed; all I know about him is that I like his writing.

Anyway if folks can stand another long quote, here is another point Gowers makes, again still early in the above book, about the supposed "reality" of numbers (which is a question this thread seems to have swerved into temporarily). I have bolded a couple of relevant sentences:

A further difficulty with explaining subtraction and division to children is that they
are not always possible. For example, you cannot take ten oranges away from a bowl
of seven, and three children cannot share eleven marbles equally. However, that does
not stop adults subtracting 10 from 7 or dividing 11 by 3, obtaining the answers −3
and 11/3 respectively. The question then arises: do the numbers −3 and 11/3 actually
exist, and if so what are they?

From the abstract point of view, we can deal with these questions as we dealt with
All we need to know about −3 is
that when you add 3 to it you get 0, and all we need to know about 11/3 is that when
you multiply it by 3 you get 11. Those are the rules, and, in conjunction with earlier rules,
they allow us to do arithmetic in a larger number system. Why should we wish to extend
our number system in this way? Because it gives us a model in which equations like
$x + a = b$ and $ax = b$ can be solved, whatever the values of $a$ and $b$, except that $a$ should
not be 0 in the second equation. To put this another way, it gives us a model where subtraction
and division are always possible, as long as one does not try to divide by 0.​

The last paragraph especially appeals to me. I have read the same sort of thing over & over in books that examine the history of mathematics & its development and various extensions over time. Deemed irrelevant here is the question of whether numbers are somehow "real" in the sense of having representations out in the universe. That question - "does the universe use math"? - is thus a separate one from "what is mathematics?" It's much easier to see how math has evolved & what the game is & how it is played - which is what @jbriggs444 is pointing to. The philosophical debates that arise in math now & then have all seemed to relate to its utility and logical coherence, that is, making sure that further extensions are consistent & usable & don't create backwards contradictions or difficulties. That is very, very different from saying that "numbers are real" in some sense. That's a much trickier question, from what I have read, and there isn't widespread agreement on whether it can be answered, let alone how.

A really short way to put this: The fact we can do math says nothing about whether its objects (e.g. numbers) "actually exist" outside our imagination.

I should add that many folks don't seem interested in ideas such as abstraction, cognition, language, etc. I find them very interesting. So do many, many mathematician/authors. There is Gowers, but also Keith Devlin; Devlin wrote an entire book, The Math Gene, in which he speculated that the human ability to do math may be derived in part from our ability to use ordinary language and to understand relations within large social networks.

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