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**sqrt squared****Member**- Registered: 2014-05-15
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I have a question. Is this possible:

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
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Hi;

Yes, a or b or both are infinity.

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not for real numbers!

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**sqrt squared****Member**- Registered: 2014-05-15
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many times it leads to an error if infinity is directly inserted into equations, for example, consider:

if you insert

you will get

so it is somehow "illegal" to treat infinity as a number, and write that

so perhaps also it is "illegal" to think that

or

in my equation

So perhaps there are other solutions instead of a or b or both being infinity.

*Last edited by sqrt squared (2014-05-15 22:14:30)*

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
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Infinity is not a number.

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**sqrt squared****Member**- Registered: 2014-05-15
- Posts: 6

bobbym wrote:

Infinity is not a number.

True.

So infinity can't be directly inserted into equations without making an error.

Wouldn't also

make infinity a number?

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
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Hi;

Nothing makes infinity a number.

Wikipedia wrote:

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.

Please read this.

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**sqrt squared****Member**- Registered: 2014-05-15
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but also in the above limit we cannot insert

otherwise we would get

and division by zero is not allowed.

Is there a "legal" way to "arrive at" infinity, if we cannot divide by zero?

I am proposing the equation

*Last edited by sqrt squared (2014-05-16 00:14:56)*

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**bobbym****Administrator**- From: Bumpkinland
- Registered: 2009-04-12
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Hi;

Did you read the first link?

We do not need to derive infinity from arithmetic operations. Arithmetic is defined for numbers both real and complex. Division by 0 is undefined.

The limit of 1 / n as n approaches infinity can be calculated.

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**ShivamS****Member**- Registered: 2011-02-07
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]Infinity is not a member of the set of real numbers. We can often adjoin infinite quantities to R, but these infinite things do not behave like ordinary real numbers and not every calculation with infinite things is allowed. For example, 2∞ = ∞, but we can't divide both sides by 2 and get 2=1. When asked to solve a problem in the set of real numbers, then working with infinity is very often not allowed (because again: infinity is not a real number). If you do want to work with it, then you need to mention this explicitly and you need to be very careful about the rules you use!

sqrt squared wrote:

Is there a "legal" way to "arrive at" infinity, if we cannot divide by zero?

I am proposing the equation

You need to understand that there is no unique definition of "infinity" in mathematics. Sometimes, infinity is just a symbol, like in limits or the order of an element in a group. However, while the above are simply symbols, it is often very useful to give them some kind of meaning anyway. We do this by adjoining some infinite quantities to our original set. This way, the ∞-notation in limits becomes an actual limit. Furthermore, we are often allowed to do all kinds of arithmetical operations on the infinite quantities.

The extended real line is R∪{+∞,−∞}.

The projective real line is R∪{∞}.

The Riemann sphere is C∪{∞}.

In nonstandard analysis, there are infinite numbers and infinitesimal numbers.

bobbym wrote:

Division by 0 is undefined.

Apart from the projective line, where a/0 = infinity

But, in any case, these systems weren't built for doing arithmetic, they were built for doing calculus and other geometric things. They are not the relevant systems when speaking about arithmetic questions.

*Last edited by ShivamS (2014-05-16 02:59:51)*

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**sqrt squared****Member**- Registered: 2014-05-15
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bobbym wrote:

Hi;

Did you read the first link?

Yes I did. On your link appears the formula

this can be written also as

Now, lets assume for a while that infinity is a number, lets say it is

so that

Let's insert this into the above equation and get

which is the same as

Now we have two "values" for a:

this should not surprise us, since they are nothing but what we started with, also

What is interesting is that the equation

is the same as my equation

if b=1

The question remains the same: does this equation mean that the variable a must be equal to infinity?

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**bobbym****Administrator**- From: Bumpkinland
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Hi;

let's stop right at the beginning. If you start using infinity as number and then doing algebra on it you will run into all kinds of contradictions.

this can be written also as

But why not this

Subtract infinity from both sides.

this is of course incorrect.

You can never reason about or treat infinity as number, something you can do algebra on. It will not work.

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**ShivamS****Member**- Registered: 2011-02-07
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sqrt squared wrote:

bobbym wrote:Hi;

Did you read the first link?Yes I did. On your link appears the formula

this can be written also asNow, lets assume for a while that infinity is a number, lets say it is

so that

Let's insert this into the above equation and get

which is the same asNow we have two "values" for a:

this should not surprise us, since they are nothing but what we started with, alsoWhat is interesting is that the equation

is the same as my equation

if b=1

The question remains the same: does this equation mean that the variable a must be equal to infinity?

This is a common mistake and happened to me too before I switched to better books.

Please read my previous post. If you are talking about the "usual" real number system, the are NO "operations with infinity" because "infinity" is not a real number. And there are several different ways to create number systems which include "infinity" as a number. Which are you talking about? They make sense in some number systems equipped with infinity, but not in others. For example, on the affine real line, the above operations are true, see http://en.wikipedia.org/wiki/Extended_real_number_line

But on the projective real line, they are false see http://en.wikipedia.org/wiki/Real_projective_line

There are many other systems which allow an infinity and where the above might make sense or not, so you need to specify.

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**sqrt squared****Member**- Registered: 2014-05-15
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ShivamS wrote:

Please read my previous post. If you are talking about the "usual" real number system, the are NO "operations with infinity" because "infinity" is not a real number. And there are several different ways to create number systems which include "infinity" as a number. Which are you talking about? They make sense in some number systems equipped with infinity, but not in others.

It is confusing that sometimes there is a need to treat infinity as a number.

Why is this accepted, is infinity here a number or is it not:

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**bobbym****Administrator**- From: Bumpkinland
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Infinity there is not a number like a 1034 or .2

I still think of it as a concept. You can not do algebra on it like the Reals.

MIF wrote:

Just think "endless", or "boundless".

Any real number is not endless or boundless, so if you add 1 to it is is larger than before. Since infinity is endless, ∞+1 = ∞, ∞ + ∞ = ∞ etc.

There are systems as Shivam is pointing out where infinity may be defined differently.

**In mathematics, you don't understand things. You just get used to them.**

**If it ain't broke, fix it until it is.**

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