(no subject)
This is long, and about numbers. If that doesn't interest then you stop here.
Infinity is an awesome concept. It is so absolutely mind bendingly big that it is completely impossible to describe. In fact, to paraphrase Douglas Adams, it is so mind bendingly huge that something finite but really really big gives a much better idea of what infinity actually is than the idea of infinity itself. Sort of an understanding by proxy.
Here today, we will attempt to gleen some sort of idea of how big infinity really is. We are going to use numbers, because nothing else in the universe exists in suffucent quantity to satisfy this illustration. In particular, we're going to look at Graham's Number. Graham's Number is the largest number that's even been used in a mathematical proof. It's the lowest known upper bound on a really, really weird mathematical problem that I won't repeat here because it's confusing, I don't understand it, and it's not really relivent to what I'm talking about anyway.
Since Graham's Number is really really big, we're going to need some (probably) new notation to express it, since trying to use scientific notation is just not possible for numbers this big. (There would be more digits in the exponant of 10 than there are atoms in the universe). Plus it helps illustrate just how bloody big this number really is if we work from the ground up.
The notation that we will be using is called Kunth's up arrow notation, because (you've probably figured this out already) it uses up arrows. It's sort of like an extension of exponants, for when exponants just aren't big enough for the number you want to express. In fact, in their simplest form, Kunth's up arrows are exponants; so in actuality up arrows aren't an extension of exponants but rather exponants are a simplification of up arrows, which brings us to a perfect place to start the discussion. (I'm going to use ! in place of an up arrow, becasue my laptop lacks an up arrow key).
To begin:
a!b = a^b = (a * a * a ... [b copies of a])
As I said, in their simplest form up arrows are simply exponants. a!b simply means the product of a copies of b.
But you can quickly get more complicated:
a!!b = a!a!...[b copies of a]...!a = a^a^a^a^[b copies of a]
And even more complicated:
a!!!b = a!!a!!a!!...[b copies of a]...!!a, where of course a!!a means a^a^[a copies of a]
Some examples:
3!!3 = 3!3!3 = 3!(3!3) = 3^(3^3) = 3^27 = 7625597484987
3!!!3 = 3!!(3!!3) = 3!!7625597484987 = 3^3^[7625597484987 copies of 3]
3!!!!3 = 3!!!(3!!!3) = 3!!!(3^3^[7625597484987 copies of 3]) = 3!!3!!...(3^3^[7625597484987 copies of 3] copies of 3)..!!3 = I don't even want to try and write this out.
If that still doesn't make any sense it's not really that important. The important thing is that you have some idea of how big up arrows make numbers.
Now, remember earlier when I mentioned Graham's number and said that we couldn't express it using scientific notation? Well we can't express it using this notation either. Still too big, you'd be scrolling for days if I tried. But we can express it as a specific term in a recursive series which can be expressed using up arrows.
Consider the following:
g(1) = 3!!!!3 (See above for some idea of how big this is).
g(n) = 3!...!3 where there are g(n-1) !. Let that sink in. That means in the second number in the sequence there are 3!!!!3 up arrows.
Now, finally, Graham's Number is the value of this sequence when n=64.
If you read that, I'm impressed.
Infinity is an awesome concept. It is so absolutely mind bendingly big that it is completely impossible to describe. In fact, to paraphrase Douglas Adams, it is so mind bendingly huge that something finite but really really big gives a much better idea of what infinity actually is than the idea of infinity itself. Sort of an understanding by proxy.
Here today, we will attempt to gleen some sort of idea of how big infinity really is. We are going to use numbers, because nothing else in the universe exists in suffucent quantity to satisfy this illustration. In particular, we're going to look at Graham's Number. Graham's Number is the largest number that's even been used in a mathematical proof. It's the lowest known upper bound on a really, really weird mathematical problem that I won't repeat here because it's confusing, I don't understand it, and it's not really relivent to what I'm talking about anyway.
Since Graham's Number is really really big, we're going to need some (probably) new notation to express it, since trying to use scientific notation is just not possible for numbers this big. (There would be more digits in the exponant of 10 than there are atoms in the universe). Plus it helps illustrate just how bloody big this number really is if we work from the ground up.
The notation that we will be using is called Kunth's up arrow notation, because (you've probably figured this out already) it uses up arrows. It's sort of like an extension of exponants, for when exponants just aren't big enough for the number you want to express. In fact, in their simplest form, Kunth's up arrows are exponants; so in actuality up arrows aren't an extension of exponants but rather exponants are a simplification of up arrows, which brings us to a perfect place to start the discussion. (I'm going to use ! in place of an up arrow, becasue my laptop lacks an up arrow key).
To begin:
a!b = a^b = (a * a * a ... [b copies of a])
As I said, in their simplest form up arrows are simply exponants. a!b simply means the product of a copies of b.
But you can quickly get more complicated:
a!!b = a!a!...[b copies of a]...!a = a^a^a^a^[b copies of a]
And even more complicated:
a!!!b = a!!a!!a!!...[b copies of a]...!!a, where of course a!!a means a^a^[a copies of a]
Some examples:
3!!3 = 3!3!3 = 3!(3!3) = 3^(3^3) = 3^27 = 7625597484987
3!!!3 = 3!!(3!!3) = 3!!7625597484987 = 3^3^[7625597484987 copies of 3]
3!!!!3 = 3!!!(3!!!3) = 3!!!(3^3^[7625597484987 copies of 3]) = 3!!3!!...(3^3^[7625597484987 copies of 3] copies of 3)..!!3 = I don't even want to try and write this out.
If that still doesn't make any sense it's not really that important. The important thing is that you have some idea of how big up arrows make numbers.
Now, remember earlier when I mentioned Graham's number and said that we couldn't express it using scientific notation? Well we can't express it using this notation either. Still too big, you'd be scrolling for days if I tried. But we can express it as a specific term in a recursive series which can be expressed using up arrows.
Consider the following:
g(1) = 3!!!!3 (See above for some idea of how big this is).
g(n) = 3!...!3 where there are g(n-1) !. Let that sink in. That means in the second number in the sequence there are 3!!!!3 up arrows.
Now, finally, Graham's Number is the value of this sequence when n=64.
If you read that, I'm impressed.