what is infinity?

Question

It is states that ,Infinity is the notation used to denote greatest number.

And $\infty + \infty = \ infinity$ When my brother and i has discussed about it we have the following argument.

1. I say "infinity is not a real number".but my brother arguree with me And my proof is like the one which follows

$$\infty + \infty = \ infinity$$ And $$\infty + 1= \ infinity$$ Since infinity not equal to zero,if infinity is a real number,then by 'cancellation law' $$\infty + \infty = \infty +1$$.

Therefore $$\infty=1$$ but it is not true .hence it is not a real number. Is it right proof?

2.As the discussion goes on my brother ask "why we say $\infty + \infty = \ infinity$" He give a proof like this

If infinity is a greatest number then $\infty + \infty $ is again a greatest number so we called it as infinity"

But my stand is "if p is a greatest number then p+p = 2p .therefore ,2p is the greatest number.then how you call p as a greatest number.

Again he say to consider a statement " if $\infty$ is a greatest number then $\infty + \infty = \ infinity$"

Since infinity is undefined the statement is true.I accept it but I won't know whether it is exactly true. I want to know 3. What is infinity? At last I was very confused about infinity .please someone explain the three question that I ask.Very thanks in Advance

Answer 1

Infinity is not a real number. The real numbers form a field $\Bbb R$ under the well-known addition and multiplication, and in such a field $x+x=x$ implies $x=0$, so there cannot be another real number $\infty$ with the same property.

If you want to enlarge $\Bbb R$, you will definitely loose some of the nice properties it has. For example, if you enlarge it to the field $\Bbb C$ of complex numbers, you loose the linear order. This doesn't mean that $\Bbb C$ is useless, of course. On the other hand if you enlarge $\Bbb R$ by adding a symbol $\infty$ (or two symbols $+\infty$ and $-\infty$) you get some nice properties (e.g., you can handle some classes of otherwise divergent sequences consistently), but you loose the field properties. Most notably, it is hard to come up with a consistent definition of $0\cdot \infty$ or $\infty-\infty$.

Answer 2

Infinity is not a real number. (For your own reference, it is an extended real number, not the same as a real number.)

This "proof," i.e.,

$$\infty + \infty = \infty$$ And $$\infty + 1= \infty$$ Since infinity not equal to zero,if infinity is a real number,then by 'cancellation law' $$\infty + \infty = \infty +1$$.

Therefore $$\infty=1$$

operates under the assumption that $\infty$ is a real number (cancellation works with real numbers), which it's not. So it's flawed.

Also,

As the discussion goes on my brother ask "why we say $\infty + \infty = \infty$" He give a proof like this

If infinity is a greatest number then $\infty + \infty $ is again a greatest number so we called it as infinity"

Again, this operates under the assumption that $\infty$ is a real number, which it's not.

Answer 3

If you add $\pm\infty$ to the real numbers it is no longer what we call a "field" and therefore the cancellation law no longer necessarily holds. You can extend the real numers to what we call the extended real numbers but you cannot extend all algebraic operations as you might like. That's the root of all such contradictions - which are common we see such arguments on a daily basis.

Answer 4

In the ordered set $\overline {\Bbb R}=\Bbb R\cup\{-\infty,\infty\}$ the symbols $\pm\infty$ are considered as "numbers" which are respectively the greatest element and the smallest element. This is a compactification of $\Bbb R$ for its usual topology.

On the other hand, there are infinitely many objects which are infinite but with different kinds of infinitude (the transfinites deduced in set theory and based in the fact that a set has strictely less cardinality that its set of subsets).

Answer 5

Good that you raised this question: What is infinity? The fact that the symbol $\infty$ appears so frequently in calculus textbooks in the notations like $x \to \infty$ and $n \to \infty$ seems to suggests that it is to be treated on the same footing as $1,2, 3, \pi$ etc (i.e. treated as a real number like we use the notation $x \to 1$ or $x \to a$ for a real number $a$).

First and foremost, we need to get rid of the myth that the notation $x \to a$ or $x \to \infty$ has a meaning in isolation. Sorry! this notation has no meaning in isolation. A notation $x \to a$ always comes as a part of a bigger notation like $$\lim_{x \to a}f(x) = L$$ or as part of the phrase $$f(x) \to L\text{ as }x \to a$$ and note that in the above notations both $L, a$ can be replaced by symbols $\infty$ or $-\infty$. Same remarks apply to the notation $n \to \infty$.

The symbol $\infty$ has a meaning in a specific context and the meaning of $\infty$ in that context is given by a specific definition for that context. There is no meaning of the symbol $\infty$ by default in absence of a context and the related definition applicable to that context.

Adding symbols $\infty, -\infty$ to the set of real numbers to form extended real number system is a device used for technical convenience (mainly to reduce typing effort and writing concise books thereby reducing their understandability). This approach does not serve any purpose for a beginner in calculus who is trying sincerely to develop concepts of calculus. It is however suitable for those experienced in the art of calculus because they can do away with some extra effort of typing.

As a beginner of calculus one should first try to learn about all the contexts where the symbol $\infty$ is used and then study very deeply the definition of that context. Unless you do this $\infty$ will always remain a confusing concept. Unfortunately most textbooks don't try to handle $\infty$ in that manner and rather start giving rules like $\infty + \infty = \infty$.

I will provide a context here for use of $\infty$ and give its definition:

Let $f$ be a real valued function defined for all real values of $x > a$ where $a$ is some specific real number. The notation $\lim_{x \to \infty}f(x) = L$ where $L$ is a real number means the following:

For every given real number $\epsilon > 0$ there is a real number $N > 0$ such that $|f(x) - L| < \epsilon$ for all $x > N$.

The same meaning is conveyed by the phrase $f(x) \to L$ as $x \to \infty$. Another context for infinity is the phrase $f(x) \to \infty$ as $x \to a$ whose meaning I will provide next.

Let $f$ be a real valued function defined in a certain neighborhood of $a$ except possibly at $a$. The phrase "$f(x) \to \infty$ as $x \to a$" means the following:

For every real number $N > 0$ there exists a real number $\delta > 0$ such that $f(x) > N$ for all $x$ with $0 < |x - a| < \delta$.

The same meaning is conveyed by the notation $\lim_{x \to a}f(x) = \infty$ but in this case I prefer to use the phrase equivalent as I hate to see the operations of $+,-,\times, /, =$ applied to $\infty$.

You will notice that understanding these definitions is a challenge. And it requires reasonable amount of effort to really understand them. Having a copy of Hardy's A Course of Pure Mathematics would be a great help here because it explains these things in very great detail in a manner suitable for students of age 15-16 years.

Now here is an exercise. Using both the contexts try to give the definition for the phrase $f(x) \to \infty$ as $x \to \infty$. And if you can do this then the next step would be to provide similar definitions for the contexts in which $-\infty$ occurs.

The treatment of $n \to \infty$ happens slightly differently because by convention $n$ is assumed to be a positive integer unless otherwise stated. If you are able to supply the definitions required in last paragraph then you will also be able to supply the definition for the context $\lim_{n \to \infty}s_{n} = L$ where $s_{n}$ is a sequence (i.e a real valued function whose domain is $\mathbb{N}$).