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.
Example: in Geometry a "Line" has infinite length ... it goes in both directions without an
end.
If it has one end it is called a Ray, and if it has two ends it is called a Line Segment, but
that needs extra information to define where the ends are.
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.
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.
0.999...
AAAA...
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).
An infinite series of "A"s followed by a "B" would 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 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 "∞":
For example: ∞ + 1 = ∞
Which says that infinity plus one is still equal to infinity.
When something is already endless, we can add 1 and it is still endless.
Here are some more properties:
Special Properties of Infinity
∞+∞=∞
-∞ + -∞ = -∞
∞×∞=∞
-∞ × -∞ = ∞
-∞ × ∞ = -∞
x+∞=∞
x + (-∞) = -∞
x - ∞ = -∞
x - (-∞) = ∞
For x>0 :
x×∞=∞
x × (-∞) = -∞
For x<0 :
x × ∞ = -∞
x × (-∞) = ∞
Conclusion
Infinity is a simple idea: "endless". Most things we know have an end, but infinity does not.