Is infinity a number?

[NotDarkYet]

A minor thought.

 

I say infinity isn't a number.  It's a direction.  Wherever you're at, keep going.

 

If infinity were a number, how could some infinities be bigger than others?

 

X can never not equal X.

[ParaSait]

Yeah, it's not a number, it's just a concept to denote that whatever X may be, Y is always bigger (or smaller)

[Existing Alternatives]

It's not. Here is what our friends at Wikipedia have to say...

Infinity (symbol: ∞) is an abstract concept describing something without any limit and is relevant in a number of fields, predominantly mathematics and physics. In mathematics, "infinity" is often treated as if it were a number (i.e., it counts or measures things: "an infinite number of terms") but it is not the same sort of number as the real numbers.

[WasatchMan]

"Number" is an abstract concept.

 

Is infinity a number? Is the fraction 1/0 a number? Is the square root of -4 a number? I don't know. It would seem to be how you define "number" and how you define "infinity". 

 

The one thing that definitely can be said is that imaginary numbers, complex numbers, matrices, infinity, etc are all useful mathematical concepts for dealing with numerical relationships, and therefore the concept "infinity" would be meaningless without the concept "number", and "numbers" do go on forever and therefore you cannot have the concept "number" without the concept "infinity".  In other words, you can't have one without the other, so what comes first? The chicken or the egg?

[Kevin Beal]

Javascript says that it is, in fact, a number:

// 'NaN' stands for "Not a Number",
// which means it's a double negative below
isNaN(Infinity)
=> false

PHP says that it's not a number:

// var_dump outputs something
// is_nan() is the same idea as above
// +INF is a representation of positive infinity
var_dump(is_nan(+INF));
=> bool(false)

But I think if it can be bigger or smaller than a number, it is itself a number.

[J. D. Stembal]

For some reason, I thought of this:

 

 

“If the doors of perception were cleansed every thing would appear to man as it is, Infinite. For man has closed himself up, till he sees all things thro' narrow chinks of his cavern.”
― William Blake, The Marriage of Heaven and Hell

[Rainbow Dash]

Although infinity is not a real number, it is a hyperreal number, which is still a number. http://en.wikipedia.org/wiki/Hyperreal_number

[Sal9000]

If infinity were a number, how could some infinities be bigger than others?

 

 

Because numbers and sets are constructed using sets. If a set has the same number of elements like another one, it has the same cardinality. Lets say you want to find out if there are more natural numbers than uneven numbers. How do you go about that? You look at the sets and compare the sets. Then you will see that for every natural numbers there is an uneven number:

 

1 --> 1

2 --> 3

3 --> 5

4 --> 7

...

So there are as much uneven numbers as natural numbers. You can show just as easily that there are just as many rational numbers like there are natural numbers:

 

1 --> 1/1

2 --> 1/2

3 --> 2/1

4 --> 1/3

5 --> 3/1

...

 

If you can show that there is a set that is bigger than all natural numbers, this set is bigger. The set of real numbers is one of those, thus, there are more real numbers than natural numbers.

 

For more see http://en.wikipedia.org/wiki/Cantor%27s_diagonal_argument

 

 

[Jack]

As a mathematical concept, infinity seems to have the same validity as any other "number." Where infinity is different is that it is not observable. I can see that there are no eggs in my basket, I can see that there are a billion eggs in my basket, but I cannot see if there are infinite eggs. 

 

As for some infinities being larger than others, I remember from limits that one part of an equation would dominate another because it would approach infinity faster than the other. It wasn't that one infinity was bigger than the other, it's that one of the variables or ends of the equation would always be greater than the other, and you could cancel out an infinity that way. 

 

The definition of a number according to google is, "an arithmetical value, expressed by a word, symbol, or figure, representing a particular quantity and used in counting and making calculations and for showing order in a series or for identification." Since the definition of a number doesn't include that it needs to be observed, and since infinity doesn't contradict any part of that definition, I can conclude that yes, infinity is a number. It is different from every other number, but it is still a number. 

[Pepin]

Honestly, this subject is a bit beyond the scope of FDR as it takes a huge expertise in many different mathematical theories to answer.

[FriendlyHacker]

There are different types of infinities, and yes some infinities are larger than others.

 

There are infinity of natural numbers:

 

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11..... to infinity

 

There is the integer numbers infinity:

 
to infinity...-11, -10, .9, -8, -7, -6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11..... to infinity

There is the infinity that includes fractions and negative fractions:

to infinity... -4/3, -3/4, -3/2,-2/3, -2/1............1/2, 2/1, 2/3, 3/2, 3/4, 4/3…to infinity

The interesting thing is that infinite fractions will never actually reach a single whole number, and you have infinite fractions for infinite whole numbers, and there are probably more infinities than these.

Another thing about infinity, is that real objects can have infinite numbers. EX: Pi is an infinite number that can be calculated from real objects.

[Sal9000]

The infinity of rational numbers and of natural numbers is the same. See my post above

 

Pi is an infinite number that can be calculated from real objects.

 

 

The distinction here is irrational and transcendental irrational. Any irrational number cannot be written as a fraction, like the squareroot of 2. Transcendental irrational numbers cannot be written in a finite polynomial form, like pi.

Squareroot of Pi is irrational, because it has a finite polynomial form: x^2 = 2. You can't do that for Pi, e, or many other important numbers, hence they are transcendental irrational. I hope that helps

[JanneW]

May I ask y'all ... why do you care? What would change in your life, if you found out that infinity is a number, or that it is not? I ask not to shut down the discussion, which could spark an interest in certain areas of mathematics and of thinking about the infinite, which could certainly be very valuable for some of you.

 

I ask because your answer to the original question could depend on your answer to this question. For some intents and purposes, it can be useful to include certain kinds of infinities in your stock of numbers, for a lot of other purposes you'll only need natural numbers or real numbers or irrational or complex or what-have-you.

 

I don't think this to be one of the questions with a universally true answer, because it does not deal with the circumstances of human action, but rather hinges on definitions in the made-up (but important) world of mathematics. If you're interested in the latter, I recommend you look into aleph numbers or transfinites. If you can grok diagonal arguments like that given by Sal9000, you should be able to push through. I'd recommend you look for good lectures on this by experts in the field, because I fear that the discussion here might lead to confusion, not clarity because the personal backgrounds of the participants are so diverse.

 

Maybe anyone can recommend a good online resource to learn about these topics? I attended a superb course on the introduction to mathematical philosopy on coursera, which I found through this board, but sadly, the course's contents are not publicly available and there's no future date given for the course.

[Sal9000]

This (https://archive.org/details/SurrealNumbers) is a good introduction for laypersons on how numbers are constructed these days. A good read for non-mathematicians. If you are interested, any introduction to number theory will cover the issue, but at a price. Unlike humanities, you have to work hard to understand the issues covered.

[FriendlyHacker]

May I ask y'all ... why do you care? What would change in your life, if you found out that infinity is a number, or that it is not? I ask not to shut down the discussion, which could spark an interest in certain areas of mathematics and of thinking about the infinite, which could certainly be very valuable for some of you.

 

Math Nerd

A person who loves math, and enjoys doing hard math problems for fun or for challenging him or herself.

 

Example:

James: "Math Rocks"

Student 1: "Nerd"!

Student 2: "Bum!"

Student 3: "Math sucks nerd!"

Stduent 4: "You are retarded!"

Student 5: "Math sucks!"

James: "You guys only think math sucks cause you can't do a simple math problem!"

Student 3: "No, we just don't wanna do math problems and do other crappy stuff! We have lifes you know!"

Student 3: "Your a Math Nerd!"

[shirgall]

Great Courses made a chapter on Mathematical Riddles on this (see Chapters 6 & 7):

 

http://www.thegreatcourses.com/courses/mind-bending-math-riddles-and-paradoxes.html

[Frosty]

Depends how you define number, but the basic answer is yes some infinities can be bigger than other infinities, the infinity of all integers is a larger set than the infinity of the set of all the decimals.

 

if you're interested in this kind of then I'd recommend subscribing to numberphile on youtube here - https://www.youtube.com/user/numberphile

 

This video in particular will answer your question in detail - 

[Donnadogsoth]

A minor thought.

 

I say infinity isn't a number.  It's a direction.  Wherever you're at, keep going.

 

If infinity were a number, how could some infinities be bigger than others?

 

X can never not equal X.

 

You're describing what's known as bad infinity, an infinity of endless counting.

 

The alternative is transfinite, something Cantor talked about, related to what some others here have talked about with taking the set of rational numbers between 0 and 1, etc., but applied deeper.  I don't understand it much, but I acknowledge that it's present in the mathematical world.

[Magnetic Synthesizer]

It's not. Here is what our friends at Wikipedia have to say...

 Can there be infinity in a system with limitations?

 

Infinity (symbol: ∞) is an abstract concept describing something without any limit and is relevant in a number of fields, predominantly mathematics and physics. In mathematics, "infinity" is often treated as if it were a number (i.e., it counts or measures things: "an infinite number of terms") but it is not the same sort of number as the real numbers.[

 

In other words, can you have an abscence of limits within limits? Example: The abscence of limits to the amount of 3s in answer to the operation of 10/3 within the limits of mathematical logic, meaning nothing besides a 3 can follow a 3 and perhaps a 3 MUST follow a 3. (must being a limitations)

''MUST'' is an absolute and therefore a limitation?

Is infinity a paradox?

is _ having to not be limited_ a limitation?

 

Wait, what if _ having to not be limited _ =/= without any limit.

 

scenario:

the object is without any limit, it is infinite.

'To be considered infinitem it must be without limits' is not a limitation it's like saying:

It must be infinite to be infinite

 

It is only a limitaiton within the language used to talk about  the logic.

 

I just disentigrated to the bottom hole of a circular reasoning, the kind that determines what is reality and logic starting with nothing but senses and cognition (or just coginition)

[bugzysegal]

Javascript says that it is, in fact, a number:

// 'NaN' stands for "Not a Number",
// which means it's a double negative below
isNaN(Infinity)
=> false

PHP says that it's not a number:

// var_dump outputs something
// is_nan() is the same idea as above
// +INF is a representation of positive infinity
var_dump(is_nan(+INF));
=> bool(false)

But I think if it can be bigger or smaller than a number, it is itself a number.

What's really confusing is that some infinities can be bigger than others!