The Consequences of Making Infinity a Number in Mathematics

[med17k]

let \infty = 1/0
then 1 =0 * \infty
0 = 0 * 1
then 0= 0 * (0*\infty)
then 0 = (0* 0 )* \infty
= 0* \infty
=1
so 0 = 1
There is nothing called infinity

 

[tushmath]

Hey first of all u are making too many conceptual mistakes in ur calculations...watch it out... We may call infinity a quantity that has no bound or we can't limit its magnitude...

 

[Landau]

You have proven that 1/0 does not obey the usual rules of computing with fractions. This should not be too surprising, and doesn't have much to do with 'infinity'.

 

[I like Serena]

IEEE has created a standard which most computer-chip implementations follow.
In particular:

1 / 0 = infinity,

but:

0 x infinity = NaN

where NaN represents "not a number".

 

[lavinia]

let \infty = 1/0
then 1 =0 * \infty
0 = 0 * 1
then 0= 0 * (0*\infty)
then 0 = (0* 0 )* \infty
= 0* \infty
=1
so 0 = 1
There is nothing called infinity

This just means that infinity can not be included in arithmetic. But in non-standard analysis it may be possible to have infinite numbers - not sure.

 

[micromass]

let \infty = 1/0
then 1 =0 * \infty
0 = 0 * 1
then 0= 0 * (0*\infty)
then 0 = (0* 0 )* \infty
= 0* \infty
=1
so 0 = 1
There is nothing called infinity

No, you have shown that 1/0 doesn't make any sense in our system. It doesn't mean that infinity doesn't make sense. You need to be very, very careful about infinity!

 

[Nebuchadnezza]

Infinity is not a number but a limit... Thats why it makes no sense to treat infinity as a number in your calculations...

 

[Hurkyl]

Please read

Projective Real Numbers​

and

http://en.wikipedia.org/wiki/http://en.wikipedia.org/wiki/Extended_real_numbers​

before making comments on arithmetic systems containing numbers called "infinity".

(aside: these uses of "infinity" have nothing to do with the idea of cardinality -- e.g. they have nothing to do with the idea of the "size of an infinite set")

 

[DaveC426913]

Yep. As if it need to be said again:
1] 1/0 does not equal infinity. 1/0 is undefined. 1/0 does not result in a unique number.
2] Who told you you could perform arithmetical calculations using a non-number?

 

[Mark44]

Interesting quote!

...a bit of lemon, just a pinch of lemon pepper seasoning, and some scrapings from the arm of some sort of, I'm guessing, mammal that was apparently electrocuted on the power lines outside the house.

 

[Char. Limit]

let \infty = 1/0
then 1 =0 * \infty
0 = 0 * 1
then 0= 0 * (0*\infty)
then 0 = (0* 0 )* \infty
= 0* \infty
=1
so 0 = 1
There is nothing called infinity

Rather, there is no real number called infinity. The arithmetic you're using, you've defined for real numbers.

 

[HallsofIvy]

As several people have said, there is no "real number" called "infinity" and the operations your were trying to do are real number operations and so invalid when you include "infinity". In particular, you cannot "let \infty= 1/0".

There are in fact, many different ways of defining "infinity" depending upon what kind of system you want or what kind of problem you are doing.

 

[Antiphon]

Also search for "Chuck Norris can divide by zero".

 

[Deveno]

i see things like this all the time.

it's not so illogical...we learn that negative numbers are an enlargement of natural numbers that allow for us to evaluate things like the formerly "forbidden" 3 - 5, and then we learn that rational numbers are an enlargement of the integers that allow us to say things like 1 is divisible by 4, so it seems like a natural progression to wonder:

"why can't we enlarge the real numbers, so as to allow for expressions like 1/0 having meaning?"

and, in point of fact, we can do this, in several ways. the trouble is, doing so changes the rules of the game, unlike the other enlargements. and you can't just apply the "old rules" to the "new system".

this doesn't mean we have to throw all of the rules "out the window". from a spatial point of view, for example, there is no problem with regarding ∞ as just another point.

but there are different ways of regarding ∞ as the extension of a real number, since real numbers have so many interesting properties. and keeping some properties comes at the expense of others.

for example: naively, it seems that 2*∞ ought to equal 1*∞ (or else we need a 2nd ∞).

but if that's true, then we can no longer say that ax = bx implies a = b, our number system loses its cancellative property. which in turn, is going to make equations MUCH harder to solve. of course, ax = bx doesn't imply a = b if x is 0, so it's not surprising that we feel intuitively as if ∞ should in some sense be the "opposite" of 0, and share some exceptional qualities. it turns out it is easier to just leave ∞ out of the number system entirely, than to create a lengthy list of except when... (whatever involving ∞)'s and ... (something here)'s if x ≠ ∞. making exceptions for 0 turns out to be complicated enough, without adding another "weird-acting number" to the mix.

but that hasn't stopped mathematicians from trying: cantor wrote a book on how to extend induction to certain infinite sets (and here, again, we lose a rule, for example, in his system ∞+1 ≠ 1+∞ (he did not actually use "∞", but rather an ordinal number ω), so addition isn't commutative, anymore, kind of disturbing), and perfectly well-defined extensions of real numbers exist with "transfinite" elements, which ARE reciprocals of "infinitesimals" (which have a "standard part" of 0, but are not themselves 0). in one of those systems, the equations the original poster made would be:

1/ε = ∞
1 = ε*∞
0 = 0*1 = 0*(ε*∞)
0 = (0*ε)*∞
0 = 0*∞
0 = 0 see the difference? the infinitesimal difference between ε and 0, keeps 1/ε from "cancelling out the zeros".

 

[I like Serena]

Well, IEEE standardised the concepts of -0, +0, ±∞, and ±NaN.

The modern floating point processors all support these concepts.
The nice thing is that you can calculate mathematical algorithms without worrying over much whether your calculations are allowed. If it doesn't work out, it will simply yield ±∞ or ±NaN, whatever is most appropriate.

I'd say that incorporating ±∞ and ±NaN in the number system and in all mathematical operations definitely has interesting applications, although you have to be aware of course that strange and unexpected things may happen.

 

[Deveno]

well, sure, one can say (for example) for any finitary sytem, that ∞ is a number which exists, but is "out of range", and NaN covers the algorithms where you get inconsistent results.

the mathematics you can express in such a system, is not ALL of real analysis. with a non-zero real number, you can always divide, and this proves to be useful. computers, even the best ones, can only carry calculations so far...outside of their capabilities (such as with VERY large numbers) they lose their correlation with the real number system. as a practical matter, this doesn't hurt very much, as a theoretical one, it is problematic.

there are some clever ways around this: there are transformations that can be expressed with terminating algorithms that "scale down" infinite ranges to finite ones, such as:

v(+)w = (v+w)/(1 + (vw/c^2))

the system of hyperreal numbers (or perhaps i should say "a" system) does a little better, because everything which is a true statement (with certain caveats, due to logical constraints) about real numbers becomes a true statement about hyperreal numbers (but not, unfortunately, vice versa), one can logically deal with "several infinities at once", which is more flexible that just designating ONE ∞.

but the point is, if it's arithmetic/algebra/analysis we want to do, having a complete ordered field is a nice environment to do it in. and if you let ∞ be a real number, you break that system, which makes arithmetic/algebra/analysis no longer very "nice". there is a certain sense in which the real number system has "maximal niceness", as many properties that we desire, and very few of the ones we don't.

of course, if you're trying to solve a polynomial, real numbers AREN'T very nice. and then you have to trade the "orderedness" off to gain "algebraic completeness". we lose some "linearity" in the topology, and gain a different kind of completeness. and which "kinds" of infinities then make sense?

the nice thing about mathematics, is that there is a wealth of available structures to analyze behaviors with. but the inner consistency of each structure has ramifications, we can't just "append" any old rule we like. if you want to make "infinity" a number, the impact upon the structure HAS to be taken into account, you cannot just use the old rules without consequence.