In February (on the 26th, to be precise, since it's a good day to celebrate, num' sein?) of 1998, a chemistry teacher gave a set of lines to one of his students, Adam Clarkson. The lines read, "I must always tuck my shirt in whilst participating in a Chemistry Lesson". He had to do a hundred of them. However, the clever part was that if he didn't do them by the next day, they would double, and if they weren't done by the day after that, they would double again. We pointed out that the lines would become too great to do pretty soon, but this didn't stop the teacher giving them. In September 1998, 7 months late, due to a confused and old man apparently mishearing us (we tried to inform him of the vastness of the Lynz, but he didn't want to hear, and so he said "If they're not on my desk by tomorrow, they'll square!"), the lines were squared every day they weren't done. To this day, they still haven't been done.
On September 17th when they started to square, they were 10^63. We chose an arbitrary time in the morning so that the lynz would be exactly 10^63 (to make calculations easier), although this may be a few digits out because of the inaccuracy of Excel. The next day, they were (10^63)^2, and so on. At the time of writing, the lynz (as they are commonly known) are 10^298084483796761000000000000000000. A second set of lynz were given on July 13th (I believe they said "I must do my lines"), but they never started squaring, so they remain considerably smaller than the 1st set of lynz. The 2nd set of lynz was, at the time of writing, 1.41 x 10^53. Whenever we refer to The Lynz, we refer to the 1st set plus the 2nd set.
It has been demonstrated that by August 9th, 1999 (at 6:34:15am), the 1st set of lynz will be bigger than a googolplex, which was thought to be a pretty hefty number in the first place.
He will never be able to do the lynz, as it would require more time than the universe is long. This number is extremely unweildy, and because of it's dynamic nature, is expressed as K, or Kappa.
As you can imagine, K is big, and by the time you've read the w in the next word, it will already by unimaginably bigger. As a matter of fact, by tomorrow, it will have squared, and squaring is a very powerful function. I'm going to carry on writing right now just to give you an idea of how huge and vast it really is, but I don't think I can describe it. Try looking at it for a few minutes and understanding what all those 0's mean. They don't just mean 'nothing' - they make the number 10 times bigger with each one you add. Am I getting this across? This number is big, Big, BIG. It's bigger than the number of elementary particles in the universe. It's bigger than a googol (and will soon be bigger than a googolplex). It's bigger than the odds of rolling a bucket of dice and them all landing on 6, then picking a million aces from packs of cards, and then flipping every single coin in the world, and them all landing on heads.
It's big, OK? But it's nothing compared to...
To explain how to get to the Clarkkkkson, we need to define the Hyperfactorial. It is based on a normal factorial, for example:
5! = 5 * 4 * 3 * 2 = 120
4! = 4 * 3 * 2 = 24
3! = 3 * 2 = 6
2! = 2 = 2
The hyperfactorial has three inputs, and is written like this: hypf(class, operator, number)
Class 1 factorial of 5: 5! = 5 * 4 * 3 * 2 =
120
Class 2 factorial of 5: 5!! = 5! * 4! * 3! * 2!
= 120 * 24 * 6 * 2 = 34,560
Class 3 factorial of 5: 5!!! = 5!! * 4!! * 3!! *
2!! = 34560 * 288 * 12 * 2 = 238,878,720
Class 4 factorial of 5: 5!!!! = 5!!! * 4!!! *
3!!! * 2!!! = 238878720 * 288 * 24 * 2 = 3,302,259,425,280
Class 5 factorial of 5: 5!!!!! = 5!!!! * 4!!!! *
3!!!!* 2!!!! = 3302259425280 * 331776 * 48 * 2 =
105,178,600,615,842,938,880
As you can see, higher class factorials generate vast numbers, even with a low input number like 5.
(somebody else called it 'hyper', by the way)
Addition = 1st operator
Multiplication = 2nd operator
Exponentiation = 3rd operator
Hyper4 = 4th operator
The addition of 4 and 5 is 4+5=9
The multiplication of 4 and 5 is 4*5=20
The exponentiation of 4 and 5 is 4^5=1024
The hyper4 of 4 and 5 is 4^^5=(((4^4)^4)^4)^4 or
in other words, 4 to the power itself, with five 4's. It works out
as 1.34 x 10^154. Hyper4 is often refered to as tetration.
Higher hyper operators include hyper5, hyper6, hyper7 and so on, e.g.:
Hyper1(addition) of 4 and 3 = 4+3 = 7
Hyper2(multiplication) of 4 and 3 = 4*3 = 12
Hyper3(exponentiation) of 4 and 3 = 4^3 = 64
Hyper4 of 4 and 3 = 4^^3 = 4^4^4 = 4294967296
Hyper5 of 4 and 3 = 4^^^3 = (4^^4)^^4 = 4^4^4^4^4^4^4^4 = 1.42 x
10^9864
Hyper6 of 4 and 3 = 4^^^3 = 4^^^4^^^4 = 4^^4^^4^^4^^4^^4^^4^^4 =
4^4^4^4^4^4^4^4^4^4^4^4^4^4^4^4^4^4^4^4^4^4^4^4^4^4^4^4^4^4^4^4
= ????
Large hyper operators make almost stupidly large numbers. This is all the more apparant when you consider that 6 isn't a particularly large number, and yet hyper6 and stumped my best calculator. Can you imagine how big hyper100 would be?
Another way of thinking of them is like this:
4 * 5 is the same as 4 plus itself, four times, 4+4+4+4+4
4 ^ 5 is the same as 4 times itself, four times, 4*4*4*4*4
4 ^^ 5 is the same as 4 to the power itself, four times,
4^4^4^4^4
4 ^^^ 5 is the same as 4 hyper4 itself, four times,
4^^4^^4^^4^^4
Note that you do the operation 4 times, you are in fact doing it on 5 numbers (because the one you started with counts, doesn't it?), which is where the 5 fits in.
The only thing about hyper operators is that they are not applicable to non-integers. For example, 2^^^2.5 has no meaning. High hyper operators generate huge numbers like high class factorials. The hyperfactorial combines them with
hypf(class, operator, number)
For example:
hypf(1,2,4) = 4! * 3! * 2! = 288
hypf(2,2,4) = 4!! * 3!! * 2!! = 288 * 12 * 2 = 6912
hypf(3,2,4) = 4!!! * 3!!! * 2!!! = 6912 * 24 * 2 = 331776
hypf(3,3,4) = 4!!! ^ 3!!! ^ 2!!! = 6912 ^ 24 ^ 2 = 1.99 x
10^184
As you can see, the real power of the hyperfactorial comes when we use higher operators. Observe:
hypf(3,4,4) = 4!!! ^^ 3!!! ^^ 2!!! = 6912^^24^^2 = ???????
even this number is unimaginable as it would require(6912^6912^6912^6912^6912^6912^6912^6912^6912^6912^6912^6912^6912^6912^6912^6912^6912^6912^6912^6912^6912^6912^6912^6912)^ (6912^6912^6912^6912^6912^6912^6912^6912^6912^6912^6912^6912^6912^6912^6912^6912^6912^6912^6912^6912^6912^6912^6912^6912)
which is vast.
A number such as hypf(5,5,5) would on first impressions seem small, but because it uses hyper5, it is even more insanely large than hypf(3,4,4).
Now the second part...
Remember the lynz? Remember how we simply called them K? The first part of The Clarkkkkson is defined as... wait for it...
hypf(K,K,K)
... but that isn't all. Another function is used. The Clarkkkkson function ck() is an extension of the hyperfactorial function. It goes like this:
ck(class, operator, number, repeats)
The repeats part means how many times the answer is fed back into the hypf() function, for example:
ck(1,2,4,2) =
hypf(1,2,4) = 4! * 3! * 2! = 288
hypf(1,2,288) = 288! * 287! * 286! ......... 3! * 2! = ??????
The 288 from the first hypf() is fed into the second hypf(). Only two hypf()'s are used, because we are only repeating twice. If we went even to three repeats, the answer would be uncomputable (probably). So get ready for the big one. The Clarkkkkson is worked out like this:
ck(K,K,K,K) = A1
<--- the class K factorial, with hyperK operator, of K, repeated
K times
ck(A1,A1,A1,A1) =
A2
<--- the previous answer, fed into the hypf() function (i.e.
class A1 factorial, hyper A1 operator
etc.)
ck(A2,A2,A2,A2) =
A3
<--- and so on... and so on....
... and so on, until ck(AK,AK,AK,AK), which is the actual number. You may wish to pause and try to imagine how large that is. 'illions don't even come close to being able to describe the size of it.
Don't try to calculate an answer to it. It is larger than anybody could ever possibly ever have use for ever. All other numbers pale into insignificance when compared to it. However, a final, unbelievable fact is that when compared to infinity, The Clarkkkkson is tiny.
Finally, if you should ever want to refer to it in an equation (I cannot for one second imagine which equation requires a number so huge or so pointless), we use the symbol ¥, because it looks good.
4! = 24
4!! = 4!*3!*2! = 24*6*2 = 288
4!!! = 4!!*3!!*2!! = 288*12*2 = 6912
4!!!! = 4!!!*4!!!*4!!! = 31776
3!! = 3!*2! = 12
3!!! = 3!!*2!! = 12*2 = 24
3!!!! = 3!!!*2!!! = 24*2 = 48
Class n factorial of 2 will always be 2
Class 5 factorial of 5: 5!!!!!
Class 4 factorial of 5: 5!!!!
Class 3 factorial of 5: 5!!!
Class 2 factorial of 5: 5!!
Class 1 factorial of 5: 5!
Class 0 factorial of 5: 5
Furthermore, negative (Class -n) factorials can be defined as the reciprocal of the positive (class n) factorial, for example:
Class -6 factorial of 5: 1/(5!!!!!!)
Class -7 factorial of 5: 1/(5!!!!!!!)
Hyper0 of any numbers can be defined as 0, because there is no
operator.
Also, negative hyper operators such as hyper-n are just the
reciprocal of hypern, for example:
Hyper-3 of 4 and 5: 1/4^5
Hyper-4 of 4 and 5: 1/4^^5
So basically, hypf(0,0,0) is 0. Class 0 factorial of anything is itself, Hyper0 of anything is 0, so longhand, it would be 0 hyper0 0 = 0
Yes. Like so:
6!! = 6*(5^2)*(4^3)*(3^4)*(2^5) = 6 * 25 * 64 ^ 81 ^ 32 = 24,883,200
This is becuase 6!! is actually:
6!! = 6! * 5! * 4! * 3! * 2!
= (6*5*4*3*2) * (5*4*3*2) *
(4*3*2) * (3*2) * 2
Who knows? But in a similar fashion to the mind blowingly vast Clarkkkkson, here's a good analogy for infinity:
"Imagine a ball of steel the size of the sun. Imagine that every trillion years, a fly lands on it. When the ball of steel has eroded away from the friction of all the flies landing, infinity will not even have begun."
As far as I know, you can't. Well, there's always the simple Clarkkkkson+1, which is 1 bigger, but that doesn't count, as you're not allowed to use a previously defined number, unless you defined it. And you didn't. So you can't. But there's no reason why you shouldn't get a bigger number. Just don't even think about using a calculator to display it. The trick is not to start with a big number, but to find a way of making a number VERY BIG, very quickly, i.e. in only a few iterations. The rhypf() function is the best way of doing this at the moment.
Have a look at this. It canexxors:
1x1=1
11x11=121
111x111=12321
1111x1111=1234321
11111x11111=123454321
111111x111111=12345654321
1111111x1111111=1234567654321
11111111x11111111=123456787654321
111111111x111111111=12345678987654321