What is Infinity?

Infinity has no end

Infinity is the idea of something that has no end.

In our world we don't have anything like it. So we imagine traveling on and on, trying hard to get there, but that is not actually infinity.

So don't think like that (it just hurts your brain!). Just think "endless", or "boundless".

If there is no reason something should stop, then it is infinite.

Infinity does not grow

Infinity is not "getting larger", it is already fully formed.

Sometimes people (including me) say it "goes on and on" which sounds like it is growing somehow. But infinity does not do anything, it just is.

Infinity is not a real number

Infinity is not a real number, it is an idea. An idea of something without an end.

Infinity cannot be measured.

Even these faraway galaxies can't compete with infinity.

Infinity is Simple

Yes! It is actually simpler than things which do have an end. Because when something has an end, we have to define where that end is.

More Examples:
{1, 2, 3, ...}	The sequence of natural numbers never ends, and is infinite.
	OK, 1/3 is a finite number (it is not infinite). But written as a decimal number the digit 3 repeats forever (we say "0.3 repeating"): 0.3333333... (etc) There's no reason why the 3s should ever stop: they repeat infinitely.
0.999...	So, when we see a number like "0.999..." (i.e. a decimal number with an infinite series of 9s), there is no end to the number of 9s. You cannot say "but what happens if it ends in an 8?", because it simply does not end. (This is why 0.999... equals 1).
AAAA...	An infinite series of "A"s followed by a "B" will NEVER have a "B".
	There are infinite points in a line. Even a short line segment has infinite points.

Big Numbers

There are some really impressively big numbers.

A Googol is 1 followed by one hundred zeros (10¹⁰⁰) :

10,000,000,000,000,000,000,000,000,000,000,000,
000,000,000,000,000,000,000,000,000,000,000,
000,000,000,000,000,000,000,000,000,000,000

A Googol is already bigger than the number of elementary particles in the known Universe, but then there is the Googolplex. It is 1 followed by Googol zeros. I can't even write down the number, because there is not enough matter in the known universe to form all the zeros:

10,000,000,000,000,000,000,000,000,000,000,000,000, ... (Googol number of Zeros)

And there are even larger numbers that need to use "Power Towers" to write them down.

For example, a Googolplex can be written as this power tower: 
That is ten to the power of (10 to the power of 100),

But imagine an even bigger number like  (which is a Googolplexian).

And we can easily create much larger numbers than those!

Using Infinity

We can sometimes use infinity like it is a number, but infinity does not behave like a real number.

To help you understand, think "endless" whenever you see the infinity symbol "∞":

The most important thing about infinity is that:

Which is mathematical shorthand for 
"negative infinity is less than any real number,
and infinity is greater than any real number"

Here are some more properties:

Example: Isn't ∞ / ∞ equal to 1?

No, because we really don't know how big infinity is, so we can't say that two infinities are the same. For example ∞ + ∞ = ∞, so

∞∞ = ∞ + ∞∞

which  means that:

11 = 21

And that doesn't make sense!

I could have also made 1=3 and so on ... so we say that ∞ / ∞ is undefined.

Infinite Sets

If you continue to study this subject you will find discussions about infinite sets, and the idea of different sizes of infinity.

That subject has special names like Aleph-null (how many Natural Numbers), Aleph-one and so on, which are used to measure the sizes of sets.

But that is an advanced topic, and goes beyond the simple concept of infinity we discuss here.

Conclusion

Infinity is a simple idea: "endless". Most things we know have an end, but infinity does not.

The original article from mathisfun.com can be found HERE