Infinity is the idea of something that has no end.
In our world we don't have anything like it ... so we imagine travelling on and on, trying hard to get there, but that is not actually infinity, just an attempt to reach it.
So don't think like that ... it just hurts your brain. Just think "endless". You will never reach it, so don't try.
Examples:
The distance to the "end" of a circle (there is no end!)
1111...
An infinite series of "1"s followed by a "2" would NEVER have a "2".
So, when you see a number like "0.999..." (ie 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.
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 if something has an end, you have to define where that end is.
Example: a "Line" has infinite length- it goes in both directions without an end. If it does have an end it is either a Ray (one end) or a Line Segment (two ends).
Big Numbers
There are some really impressively big numbers.
A Googol is 1 followed by one hundred zeros (10100) :
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, becuase 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,... etc (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
And you can easily create much larger numbers than those!
Finite
All of these numbers are "finite". Which means they have an end ... there is a limit to how big they are.
But none of these numbers are even close to infinity. Because they are finite, and infinity is ... notfinite!
Using Infinity
We can sometimes use infinity like it is a number, but infinity does not behave like a real number.
For example: ∞ + 1 = ∞
Which says that infinity plus one is still equal to infinity.
The most important thing about infinity is that:
-∞ < x < ∞
Where x is a real number. So minus infinity is less than any real number and infinity is greater than any real number.
Here are some more properties:
Special Properties of Infinity
∞ + ∞ = ∞
-∞ + -∞ = -∞
∞ × ∞ = ∞
-∞ × -∞ = ∞
-∞ × ∞ = -∞
x + ∞ = ∞
x + (-∞) = -∞
x - ∞ = -∞
x - (-∞) = ∞
For x>0 :
x × ∞ = ∞
x × (-∞) = -∞
For x<0 :
x × ∞ = -∞
x × (-∞) = ∞
Undefined Operations
All of these are "undefined":
"Undefined" Operations
0 × ∞
0 × -∞
∞ + -∞
∞ - ∞
∞ / ∞
∞0
1∞
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 would
mean that:
1
=
2
∞
∞
1
1
And that doesn't make sense! I could have also made 1=3 and so on ... so we say that ∞ / ∞ is undefined.
Different Sizes of Infinity
If you continue to study this subject you will find that there are different sizes of infinity.
For example, there are infinitely many whole numbers {0,1,2,3,4,...}, but there are morereal numbers (such as 12.308 or 1.111111) because any real number could have an infinite number of decimal places.
Conclusion
Infinity is a simple idea: "endless". Most things we know have an end, but infinity does not.
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