The UpArrow Notation
The Problem With Potentiation
Potentiation in Math is usually raising something to a power, for instance if X is potentiated by Y then you have X^Y
For instance: X² = X times X.
So if we have 2 in the power of 2 we have:
2 in which it’s 2 times 2, then 2² = 4.
The general formula is: a^n
Potentiation is enough to non-large numbers such as in scientific notation. We have for instance avogrado’s number: 6 x 10^-23.
In my last Article I wrote about the number of stars in observable universe and I also wrote something about 10¹⁶, which is ten Quadrilions.
Even Googol or a Googolplex can be demonstrated by potentiation:
Googol = 10¹⁰⁰
Googolplex = 10¹⁰^¹⁰⁰
Size of a Proton = 10^-15 of a meter
Plank Lenght: 6 x 10^-35 meters
Weight of Earth in Kilograms = 5.9722 x 10²⁴ Kg
We also have Millillion: 10³⁰⁰³ (thats huge but not bigger than a googolplex)
Or even Milli-Millillion: 10³⁰⁰⁰⁰⁰³.
So, as we can see for Scientific notation, Potentiation is enough to demonstrate how big numbers can get, yes? Well, NO.
When we talk about really huge numbers such as skewes’s number we realize that potentiation begins to hamper our calculations!
Skewes’s Number was know as the biggest useful number know until 80s.
Skewes’s Number is: 10¹⁰^¹⁰^³⁴.
Ok, thats a large number.
Its bigger than a Googolplex.
But where did this number come from?
Well, the Skewes’s number is the number Sk1 above which π(n) < li(n) must fail assuming that the Riemann hypotesis is true where π(n) is the prime counting function and li(n) is the logarithmic integral.
Isaac Asimov featured the Skewes’s number in his science fact article Skewered! (1974)
In 1912, Littlewood proved that Sk1 exists(Hardy 1999, p17) and the upper bound was subsequently found by skewes(1933).
Looking for a number bigger than Skewes? Well we have Skewes two(Sk2): 10¹⁰^¹⁰^⁹⁶⁴.
In Physics the larger applicable number is know as “Longest Time”
(poincare recurrence) which is the time it will take to the Universe reset it self!
PR = 10¹⁰^¹⁰^¹⁰^¹⁰^¹.¹
As you can see we are getting more and more potentiations as we go through large numbers.
The Point is that Potentiation is useless when we are introduced to EXTREME-LARGE-NUMBERS so we need a new system of notation, right?
So now lets talk about UpArrow Notation. Get ready because I’m going to a totally new level.
Knuth Arrow Notation
Let me introduce you a new kind of notation: The UpArrow Notation, also know as Knuth Arrow Notation, was introduced by Donald Knuth(1976)
Donald Knuth is a computer scientist, he invented the UpArrow Notation.
As we saw previosly we are having some issues to work with potentiation notation for really large numbers.
So Knuth invented a system of notation not-complex and easy to understand that grow fast to measure really large numbers.
So for instance we have A and B
If we want to represent A in the power of B:
A^B
Using UpArrow Notation:
A↑B
Then, A^B = A↑B
2² = 2↑2
General Formula is:
A↑(n) = A↑(n-1) times A↑(n-1) times A↑(n-1)… B times
For instance:
3↑2 = 9 ’cause 3² = 9
3↑3 = 27 ’cause 3³ = 27
But now comes why Knuth Notation is better than Potentiation Notation:
We, for instance, can do this:
3↑↑3 = ?
Okay, what’s happening here?
Basically this: 3↑↑3 = 3↑(3↑3)
Yeah, things start to change at this point.
Let’s see what that means:
(3 to power of 3) to the power of 3.
So: 3↑↑3 = 3^(3³) = 3²⁷ = 7.6 X 10¹² (7.6 Trillions)
If we for instance: 3↑↑4 = 3↑3↑(3↑3)
so we have a stack of 4 threes: 3³^³^³
Now, what if we add one more arrow?
3↑↑↑3 = 3↑↑(3↑↑3)
Which (3↑↑3) is: 3↑(3↑3), right?
So: 3↑↑↑3 is equal to 3↑↑(7.6 trillions)
Humm, things are getting interesting here.
So now we have a tower of 7.6 trillions of Three.
Something like: 3³^³^³^³^³^³^³^³^…(7.6 X 10¹²)
The size of this stack of threes is 7.6 X 10¹².
Can you imagine something like this?
The fun fact about 3↑↑↑3 is that if each three has a meter we will have a tower of three that would reach the surface of the SUN!
The distance between the earth and our star is like 149.600.000 Km
in meters = 149.600.000.000 (Billions).
Yeah that’s how big this number is!
But what’s the value of 3↑↑↑3???
Well, I don’t know.
But it’s already bigger than a googol, googolplex, googolduplex…
You may be thinking that this number is large right?
Well we can still add the last arrow:
3↑↑↑↑3 = 3↑↑↑(3↑↑↑3)
We know that (3↑↑↑3) is equal to that crazy tower of threes, but what 3↑↑↑↑3 is equals to ????
3↑↑↑(3³^³^³^³^³^³^³^³^³^… (7.6 X 10¹² times).
It’s like you try to perform (3↑↑3 … 3↑↑3) and this space between is that crazy tower. So you would have to perform 3↑↑3 for 7.6 X 10¹² times.
So basically now we have 3 up to a insane power of threes…now we are starting to measure really large numbers…
We usually don’t use 5 UpArrows(something like: 3↑↑↑↑↑3)
Cause it would be insane and would lose the sense of having a simple notation. After 4 UpArrows the Knuth Notation would result in the same problem we had with potentiation, right?
So to go further we will need a new notation…
Graham’s Number
Graham’s Number is an enormous number that arises as an upper bound on the answer of a problem in the mathematical field of Ramsey Theory. It was discovered by Ronald Graham(1977)
The Ramsey Theory is: “How many elements of some structure must there be to guarantee that particular property will hold?”.
Ron Graham describes Ramsey Theory as a “Branch of combinatorics”.
The Graham’s Extreme-Large Applicable Number.
It’s such a large number that if you could imagine how big that number would be, your brain would be such a massive cause of synaptic connections and would become a Black Hole.
We would need a Googolplex of Universes to write each digit of this number in a single atom and maybe it wont be enough…
But can we measure this number using potentiation? Well, not if you wanna see how big this number is this century… however using the Knuth Notation we might have an interesting measurement(and it wont take a few centuries at all, just a few minutes)
We already had a tiny-tiny conception of how big this number actually is.
I wrote about 3↑↑↑↑3.
This number is a “crazy tower of threes” 3³^³^³^…(7.6 X 10¹² size) plug into 3↑↑↑3^(This “crazy power of threes”)
That’s huge, that’s insane!
Even for a normal-computer it would take a few hours to process 3↑↑↑↑3. Maybe Super-Computers would take a few minutes or not even it.
But this value is astronomical, right?
Well however 3↑↑↑↑3 is not Graham’s Number at all.
Let’s call 3↑↑↑↑3 = g1.
Well if this insane number is g1 what is g2?
g2 = 3↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑…(g1 Times)…↑↑↑↑3
No words can express how big this number is.
The Number of arrows is g1.
So is g2 equals to Graham’s Number? No! NOT EVEN CLOSE!
What about g3 = 3↑↑↑↑…(g2 Times)…↑↑↑↑3
Still not even close to Graham’s Number.
And not even g4, g5, g6, g7…
at g12 we have “Little Graham’s Number”.
But not even g20 is good enough.
Not even g32 or g48 but then we reach g64.
g64 = 3↑↑↑↑…(g63 times)…↑↑↑↑3
and g64 is equal G, and G is the Graham’s Number.
Conclusion
As we can see here to reach really-large numbers we needed a new kind of notation called Knuth Notation. At this point Potentiation is obsolete.
In the future we will see that even Knuth Notation will become obsolete and I’m gonna show you extremely-larger-applicable-numbers.
Graham’s Number has turned into pop culture and it’s mainstream now.
However, there are more underground numbers that would make Graham’s Number(g64) look small.
References
Skewered Number: https://goo.gl/cN9tqd
Knuth Notation: https://goo.gl/21W19f
Graham’s Number: https://goo.gl/23M464
