• ### Announcements

GameDev.net and CRC Press have teamed up to bring a free ebook of content curated from top titles published by CRC Press. The freebook, Practices of Game Design & Indie Game Marketing, includes chapters from The Art of Game Design: A Book of Lenses, A Practical Guide to Indie Game Marketing, and An Architectural Approach to Level Design. The GameDev.net FreeBook is relevant to game designers, developers, and those interested in learning more about the challenges in game development. We know game development can be a tough discipline and business, so we picked several chapters from CRC Press titles that we thought would be of interest to you, the GameDev.net audience, in your journey to design, develop, and market your next game. The free ebook is available through CRC Press by clicking here. The Curated Books The Art of Game Design: A Book of Lenses, Second Edition, by Jesse Schell Presents 100+ sets of questions, or different lenses, for viewing a game’s design, encompassing diverse fields such as psychology, architecture, music, film, software engineering, theme park design, mathematics, anthropology, and more. Written by one of the world's top game designers, this book describes the deepest and most fundamental principles of game design, demonstrating how tactics used in board, card, and athletic games also work in video games. It provides practical instruction on creating world-class games that will be played again and again. View it here. A Practical Guide to Indie Game Marketing, by Joel Dreskin Marketing is an essential but too frequently overlooked or minimized component of the release plan for indie games. A Practical Guide to Indie Game Marketing provides you with the tools needed to build visibility and sell your indie games. With special focus on those developers with small budgets and limited staff and resources, this book is packed with tangible recommendations and techniques that you can put to use immediately. As a seasoned professional of the indie game arena, author Joel Dreskin gives you insight into practical, real-world experiences of marketing numerous successful games and also provides stories of the failures. View it here. An Architectural Approach to Level Design This is one of the first books to integrate architectural and spatial design theory with the field of level design. The book presents architectural techniques and theories for level designers to use in their own work. It connects architecture and level design in different ways that address the practical elements of how designers construct space and the experiential elements of how and why humans interact with this space. Throughout the text, readers learn skills for spatial layout, evoking emotion through gamespaces, and creating better levels through architectural theory. View it here. Learn more and download the ebook by clicking here. Did you know? GameDev.net and CRC Press also recently teamed up to bring GDNet+ Members up to a 20% discount on all CRC Press books. Learn more about this and other benefits here.
Followers 0

# Why is infinite technically not a number.

## 39 posts in this topic

Isn't 2-1 equivalent to 1/2, so 1/2-1 would be 1/(1/2), which would be equivalent to 2? So 1/1-1 + 1/2-1 + 1/3-1 + 1/4-1 ... = 1+2+3+4...?

Like I said, I'm not good at the math, so it's interesting why two seemingly equivalent statements aren't equivalent.

i.e. sum n where n=1..? != sum 1/(1/n) where n=1..?

It's because you can only do arithmetic on limits of series if they are absolutely convergent (i.e. the absolute value of each term, summed up, converges to a finite limit). Otherwise you can expect nonsense in general.

0

##### Share on other sites

Isn't 2-1 equivalent to 1/2, so 1/2-1 would be 1/(1/2), which would be equivalent to 2? So 1/1-1 + 1/2-1 + 1/3-1 + 1/4-1 ... = 1+2+3+4...?

Like I said, I'm not good at the math, so it's interesting why two seemingly equivalent statements aren't equivalent.
i.e. sum n where n=1..? != sum 1/(1/n) where n=1..?

The problem is that you can't sum infinitely many things. So when you write something like 1 + 1/2 + 1/4 + 1/8 + ... = 2, you are actually saying something like "the limit of the sequence (1, 1 + 1/2, 1 + 1/2 + 1/4, 1 + 1/2 + 1/4 + 1/8, ...) is 2". That is by far the most common way to interpret an "infinite sum", and according to that one 1 + 2 + 3 + 4 + ... = infinity.

There are other interpretations of "infinite sums". See the Wikipedia page on divergent series for details.

2

##### Share on other sites

But writing the terms as k or 1/k^-1 shouldn't matter, should it? Same values in the same order, same series, right?

0

##### Share on other sites

The zeta function is only equal to the summation if the argument (corresponding to the power in the sum) is greater than 1. (Well, it is also true if the argument is complex and the real part is > 1).

0

##### Share on other sites

The series in the definition of the zeta function is convergent (in the usual sense) for complex numbers with real parts greater than 1. We may however give different definitions of this function with a greater domain and this analytic continuation is what we use to compute the zeta function of -1.

0

##### Share on other sites

I don't know what 1+2+3+4+...=-1/12 has to do with p-adic numbers. Care to explain?

Because using the "normal"/simple definitions of summation and integers, that summation does diverge. The natural numbers are closed over addition, and yet -1/12 is not a natural number, which breaks that closure of natural numbers. So in order to make sense of this contradiction, alternative/fancier definitions of summations and numbers must be used. Specifically, p-adic numbers, which converge for large values rather than diverge. Once you're using p-adic numbers, you're not using the natural numbers and aren't restricted to the closure of natural numbers, and so can achieve 1 + 2 + 3 + ... = -1/12.

Isn't 2-1 equivalent to 1/2, so 1/2-1 would be 1/(1/2), which would be equivalent to 2? So 1/1-1 + 1/2-1 + 1/3-1 + 1/4-1 ... = 1+2+3+4...?

Like I said, I'm not good at the math, so it's interesting why two seemingly equivalent statements aren't equivalent.
i.e. sum n where n=1..? != sum 1/(1/n) where n=1..?

As others have said, you can't treat infinity like a variable and do algebra with it. You can do some things (which actually involve evaluating a limit), but there are several things one might be tempted to do with infinity that would seem valid, but in reality aren't.

But maybe I'm not understanding, as I can't see the contradiction/inequality in sum n where n=1..? != sum 1/(1/n) where n=1..?. Edited by Cornstalks
1

##### Share on other sites
I am going to do a point-by-point refutation of Conrstalks's post, and I know that can come through as rude. I just want to say I don't mean to be rude or sound upset in any way.

I don't know what 1+2+3+4+...=-1/12 has to do with p-adic numbers. Care to explain?

Because using the "normal"/simple definitions of summation and integers, that summation does diverge. The natural numbers are closed over addition, and yet -1/12 is not a natural number, which breaks that closure of natural numbers. So in order to make sense of this contradiction, alternative/fancier definitions of summations and numbers must be used. Specifically, p-adic numbers, which converge for large values rather than diverge. Once you're using p-adic numbers, you're not using the natural numbers and aren't restricted to the closure of natural numbers, and so can achieve 1 + 2 + 3 + ... = -1/12.

I understand what you are saying, but it doesn't apply in this case. Even when working with p-adic numbers, an infinite series can only converge if the terms have limit 0 (it turns out in p-adic numbers this condition is also sufficient). That means that you can compute things like 1*1 + 2*2 + 3*4 + 4*8 + 5*16 + 6*32 + ... in the 2-adic numbers (I think the sum is 1). But the general term in 1 + 2 + 3 + 4 + ... does not converge to zero even using the p-adic norm.

Isn't 2-1 equivalent to 1/2, so 1/2-1 would be 1/(1/2), which would be equivalent to 2? So 1/1-1 + 1/2-1 + 1/3-1 + 1/4-1 ... = 1+2+3+4...?

Like I said, I'm not good at the math, so it's interesting why two seemingly equivalent statements aren't equivalent.
i.e. sum n where n=1..? != sum 1/(1/n) where n=1..?

As others have said, you can't treat infinity like a variable and do algebra with it. You can do some things (which actually involve evaluating a limit), but there are several things one might be tempted to do with infinity that would seem valid, but in reality aren't.

But maybe I'm not understanding, as I can't see the contradiction/inequality in sum n where n=1..? != sum 1/(1/n) where n=1..?.

1 + 2 + 3 + 4 + ... and 1/(1/1) + 1/(1/2) + 1/(1/3) + 1/(1/4) + ... are the exact same series, because their terms are the same. Of course their sums can't be different, no matter how you define them.
0

##### Share on other sites

Isn't 2-1 equivalent to 1/2, so 1/2-1 would be 1/(1/2), which would be equivalent to 2? So 1/1-1 + 1/2-1 + 1/3-1 + 1/4-1 ... = 1+2+3+4...? Like I said, I'm not good at the math, so it's interesting why two seemingly equivalent statements aren't equivalent.i.e. sum [font=courier new']n[/font] where [font=courier new']n=1..?[/font] != sum [font=courier new']1/(1/n)[/font] where [font=courier new']n=1..?[/font]

As others have said, you can't treat infinity like a variable and do algebra with it. You can do some things (which actually involve evaluating a limit), but there are several things one might be tempted to do with infinity that would seem valid, but in reality aren't.But maybe I'm not understanding, as I can't see the contradiction/inequality in sum [font=courier new']n[/font] where [font=courier new']n=1..?[/font] != sum [font=courier new']1/(1/n)[/font] where [font=courier new']n=1..?[/font].
1 + 2 + 3 + 4 + ... and 1/(1/1) + 1/(1/2) + 1/(1/3) + 1/(1/4) + ... are the exact same series, because their terms are the same. Of course their sums can't be different, no matter how you define them.
That question of mine was in response to Mats1's post above it, where he told me that one of those sums did equal -1/12, but the other did not. Seeing they're equivalent, I was asking why this is so.

As for treating infinity as a regular algebraic variable - yeah I've seen that used to "prove" that 0=1, etc...
However, wasn't the calulus-type method before calculus to introduce, say x/? for a large part of your working (allowing you to work with infinitely small 'differentials'), but then at the end make the assumption that these terms are 0 and remove it. IIRC, this is no where near as formally robust as modern calculus, but in many situations can give you the same results. I heard there were two camps at the time, those who rejected calculus out of hand as a stupid trick that doesn't make sense, and those who were willing to treat infinity as a workable value...

Don't physicists also end up with a lot of ?'a in their algebra, and have to 'normalize' them away using different methods?

• n.b. I really don't know what I'm on about here, just a curious layman!
0

##### Share on other sites

That question of mine was in response to Mats1's post above it, where he told me that one of those sums did equal -1/12, but the other did not. Seeing they're equivalent, I was asking why this is so.

The answer is that you are correct: Those two things are the same, and however you define one of them, you should get the same answer as for the other.

As for treating infinity as a regular algebraic variable - yeah I've seen that used to "prove" that 0=1, etc...

Those "proofs" usually include invalid manipulation. Notions of number that include infinity often will break some common features that you might expect. For instance, in ordinal numbers, 1 + omega = omega, but omega + 1 is greater than omega. In cardinal numbers, 1 + aleph_0 = aleph_0 = aleph_0 + 1, and even aleph_0 * 2 = aleph_0. However, 2^aleph_0 > aleph_0.

The use of those funky names for infinities is not capricious: One has to be very precise when dealing with infinities, or you'll end up with paradoxes all over the place.
2

##### Share on other sites

The answer is that you are correct: Those two things are the same, and however you define one of them, you should get the same answer as for the other.

I thought however, that when presented with the naive sum 1 + 2 + 3 + 4... This will diverge. However, a value can be assigned to this series, after some manipulation, that is - 1/12, but it does not reflect the simple summation of the terms of the sequence.  I'm no expert of the zeta function, but I was pretty sure this is how it goes down.

0

##### Share on other sites

@Mats1: As I said ealier, the result has been obtained using more complicated methods based on complex analysis and not using some kind of series manipulation. When you sum an infinite number of terms you have to be very careful to what you do. Most manipulations gives in fact wrong results.

0

##### Share on other sites

A number must fullfill some rules to be considered a number. Eg zero is the unit value of the additive operation, like 1 is the unit value of multiplication operation. Therefor

Addition:
(1) x + 0 = x = 0 +x
and
(2) x + (-x) = 0 = (-x) + x



Now lets check infinit

Addition:
(1) inf + 0 = inf = 0 + inf

(2) inf + (-inf) = 0

If for a infinit number inf = inf + X (eg inf = 1+inf) would hold true and inf would be a valid number, then following should work
inf + (-inf)
= 1 + inf + (-inf)
= 1 +     0
= 1
!= 0
= inf + (-inf)



Even simpe add operations would not work in a mathematically way with such a definition of infinit.

Edited by Ashaman73
0

##### Share on other sites

A number must fullfill some rules to be considered a number. Eg zero is the unit value of the additive operation, like 1 is the unit value of multiplication operation. Therefor

Addition:
(1) x + 0 = x = 0 +x
and
(2) x + (-x) = 0 = (-x) + x



Now lets check infinit

Addition:
(1) inf + 0 = inf = 0 + inf

(2) inf + (-inf) = 0

If for a infinit number inf = inf + X (eg inf = 1+inf) would hold true and inf would be a valid number, then following should work
inf + (-inf)
= 1 + inf + (-inf)
= 1 +     0
= 1
!= 0
= inf + (-inf)



Even simpe add operations would not work in a mathematically way with such a definition of infinit.

Check out surreal numbers, which have been mentioned in this thread before. They contain many infinite numbers, but they satisfy all the properties you mention.

0

##### Share on other sites

It is weird no one has linked to this interesting video yet:

0

##### Share on other sites
Vi Hart just released a video about different types of infinity. If you were interested in this thread, you'll probably enjoy it. I'm sure you'll enjoy some of her other videos too.
2

## Create an account

Register a new account