Fundamentals of Googology I
Nathan Richardson
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SOURCE
Sbiis Saibian
CONTRIBUTORS
Nathan Ho
Aarex
EDITOR
Rohan Ridenour
AUTHOR
Nathan Richardson
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Printed: May 19, 2019
iii
Contents
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Contents
1
Exponentiation
1.1
Concept 1.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2
Concept 1.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.3
Concept 1.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2
Functions and Notations
2.1
Concept 2.1 . . .
2.2
Concept 2.2 . . .
2.3
Concept 2.3 . . .
2.4
References . . . .
3
iv
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2
7
9
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13
16
26
Numbers
3.1
Concept 3.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2
Concept 3.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3
Concept 3.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Chapter 1. Exponentiation
C HAPTER
1
Exponentiation
Chapter Outline
1.1
C ONCEPT 1.1
1.2
C ONCEPT 1.2
1.3
C ONCEPT 1.3
Welcome to Googology! This textbook is purposed to educated beginners on the fundamentals of Googology 1. In
Googology 1, there are 3 concepts (or units), each having 3 objectives. So, in Googology 1, there are 9 objectives.
Being familiar with all of these better prepares you for the test at the end of the year.
At the end of every section, there are 5 review questions. These review questions are sometimes placed randomly to
give examples of questions that you may encounter in regards to the topics just mentioned. If you can solve these
accurately, you can expect to do well throughout the course; the questions placed in are sometimes hard, sometimes
easy.
All of the textbook pages are on the right side of the page. The left side is for you to take notes from additional
information taught in class, or even for showing your work from the review problems. You teacher can only
encourage you to fill this up, but it is extremely beneficial to write things down.
Here is the course for googology; from all of the fellow googologists, we hope you have fun learning about large
numbers!
Concept 1 Objectives
1. Be familiar with the history and etymology of googology
2. Know the first 10 powers of 2, 3, 4, 5, and properties of these values
3. Identify the hyperoperators, know the rules for exponentiation and be able to exponentiate with large bases.
1
1.1. Concept 1.1
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1.1 Concept 1.1
Objective
Be familiar with the history and etymology of Googology
Googology is the study and nomenclature of large numbers.
One who studies and invents large numbers and large number names is known as a googologist. A mathematical
object relevant to googology is known as a googologism; the term googolism is similar, but only applies to numbers.
Googology is known for the rather comic names given to the googolisms, such as "meameamealokkapoowa ommpa",
"a-ooga", and "wompogulus".
Googology is not to be confused with googlology, the study of the Google search engine and its various other
services.
The antithesis to googology is ultrafinitism, which states that large numbers simply do not exist.
Etymology
The term was coined by Andre Joyce, formed by combining googol (the classic large number) + -logos (Greek
suffix meaning "study"). Joyce’s googology involved devising a system of names for numbers based on wordplay
and whimsical extrapolation. Ironically, the term does not appear to be well-known even among its own practitioners,
and few "googologists" use the term to describe themselves.
History
Although the term "googology" is modern, the subject has existed for as long as humans have been fascinated by
large numbers.
The earliest known work by a "googologist" is probably the ”Sand Reckoner” written by Archimedes, a Greek
polymath, sometime in the 3rd century B.C. In it, he develops a system of numbers extending to 108 x 10^16. There
are other examples in ancient history that illustrate mankind’s fascination, and even adeptness, with large numbers.
Some religious texts contain some very large numbers. Although the Bible contains no definite numbers greater than
108 , it uses figurative language in many places to describe very large numbers such as "the stars in the sky" or "the
sands of the sea."
With the advent of modern mathematics and the impending invention of the computer, mathematicians of the 19th
and 20th centuries had access to numbers larger than ever before. This fascination was relayed to the laymen
through popular books on mathematics. "Googol", "googolplex", and "mega" were all introduced in books of popular
mathematics, written by mathematicians who wanted to explain to the laymen what mathematicians meant when they
invoked ”infinity.”
Eventually, the fascination of large numbers spread to a class of amateurs who took it upon themselves to extend
the ideas hinted at in these popular books on mathematics. These became the early ”googologists”. This took on
something of a form of a hobby that still continues today, with amateurs writing papers claiming to have "invented
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Chapter 1. Exponentiation
the largest number ever." That being said, not everything produced is brilliant, nor is it all crank mathematics. There
is a variety of skill levels, and some of googology actually comes from professional mathematicians, not amateurs.
In particular, there seem to be three classes:
1. Googologisms that arise in professional mathematics as side-effects of more serious math problems, such as
Graham’s number and Skewes’ number.
2. Googologisms devised recreationally by professional mathematicians, such as chained arrow notation and
Steinhaus-Moser Notation.
3. Googologisms created solely by amateurs, such as array notation and Hyper-E notation.
During most of the 20th century, early googologists worked in isolation. Since the advent of the internet, however,
there has been a greater confluence of ideas, and several websites have sprung up to gather the loose bits of
information that form the body of knowledge, methodology, and conventions known as googology. Perhaps the
most important of these sites are Googology Wiki, Robert Munafo’s site, and One to Infinity.
3
1.1. Concept 1.1
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Furthermore, since 2002, a loosely knit community of large number enthusiasts, dubbing themselves googologists,
has emerged, building websites, sharing information, and developing a culture with a unique approach to one
particular challenge: "What is the largest number you can come up with?" Googologists generally avoid many
of the common responses such as "infinity," "anything you can come up with plus 1," "the largest number that can
be named in ten words," "the largest number imaginable," "a zillion," "a hundred billion trillion million googolplex"
or other indefinite, infinite, ill-defined, or inelegant responses. Rather googologists are interested in defining definite
numbers using efficient and far-reaching structural schemes, and don’t attempt to forestall the unending quest for
larger numbers, but rather encourage it. So perhaps a more accurate description of the challenge is: "What is the
largest number you can come up with using the simplest tools?"
As far as mathematical fields go, googology is an oddball. It precariously teeters on the edge of what we call
"science," becoming more of an art form as opposed to a mathematical study.
Although googology remains, and will probably always be, an obscure, esoteric, and impractical study, it at least
now has a name, a history, and a community.
List of Googologists
These people contributed significantly to Googological studies
TABLE 1.1:
Chris Bird
Jonathan Bowers
John Conway
Harvey Friedman
Nathan Ho
Lawrence Hollom
Andre Joyce
Donald Knuth
Agustin Rayo
Sbiis Saibian
Aarex Tiaokhiao
Nathan Richardson
Invented Bird’s array notation, helped investigate and
develop BEAF
Created BEAF, regarded the founding father of modern
googology
Invented chained arrow notation
Investigated combinatorial functions including n(k),
TREE(k), and SCG(k)
Founded Googology Wiki, republished BIG FOOT on
his site
Invented hyperfactorial array notation
Coined the word "googology" and set forth the first
large number naming systems
Invented arrow notation
Invented a famous function: one of the fastest-growing
functions known
Invented the Extensible-E System, and is writing a web
book about googology
Invented extensions to other notation
Created the course used to teach Googology, including
this textbook
Eras of Googology
There are three eras for the history of modern Googology. They are as follows, including the event that started each
era.
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Chapter 1. Exponentiation
TABLE 1.2:
Name of Era
Preinternet Era
Netcommunity Era
Worldwide War Era
Starting Event
Tetration, by Rucker, in 1901
Large Number Site, by Robert Munafo, in 1996
Googology Wiki, by Nathan Ho, in 2008
5
1.1. Concept 1.1
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Saibian’s Introduction to Googology
Large numbers have inspired both fear and awe in the minds of men since their dawn. In ancient times they were
the domain of the cosmos and the divine. Today they are treated as little more than a mathematical curiosity fit for
attracting the layman to mathematics and little else. None the less, large numbers have plagued the imaginations of
many a professional and amateur alike. They are ominous, austere, vast beyond description, godly, transcendent,
and often difficult or impossible to fully grasp. They are both a celebration of the human spirit and an affront to it.
For while we may revel in the vastness, get lost amongst their mindless yet tireless machinations, we are so forced
to confront out potential insignificance in a reality that stretches beyond all human comprehension!
The study of large numbers can be a deeply spiritual experience, for it taps into the very essence of what it means to
leave the profane behind in pursuit of the transcendent. While it is by no means a practical endeavor, most of these
numbers leaving our picayune existence far behind, it is at its core an expression of the unquenchable human spirit,
which always yearns for more. It is our attempt to grasp what is beyond ourselves; to understand what we may call
"the divine".
I entreat you, therefore, if my words have had an impact on you, to join me on a quest through the infinite multitudes
of the finite. A word of caution before we proceed: the traveling will only get more difficult as we proceed, and with
no end in sight. The venues large numbers can open up in your mind can be quite rewarding, but be warned that
madness lies this way. For we can no more imagine the end of numbers than we could wait for eternity. An earnest
study of large numbers will completely shift the way you think about infinity, and the finite, and your mind shall be
forever changed by it.
If these warnings have not turned you away, and you are still eager to learn the deep secrets of the numberscape,
then I invite you to read onwards, as casually or religiously as you’d like. Large numbers are not the domain of any
one man, or group, for there is a number for every man that ever lived, ever will live, or ever could live; ample real
estate for all humanity for all ages amongst the greatest of all heavens! What lengths are you prepared to go to try
and reach infinity even as you know the task is impossible? Why not see how "close you can get...
Review:
1.
2.
3.
4.
5.
6
What is the formal definition of Googology?
The earliest "googological" work can be dated back to ____.
According to modern (2002 - present) googologists, what is the purpose of googology?
Who wrote the notstion: BEAF?
The Googology Course was founded by ____, and its writing began in November, 2016: in the ____ era.
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Chapter 1. Exponentiation
1.2 Concept 1.2
Objective
Know the first 10 powers of 2, 3, 4, 5, and properties of these values
TABLE 1.3:
Power↓ Base→
2
3
4
5
6
7
8
9
10
2
4
8
16
32
64
128
256
512
1024
3
9
27
81
243
729
2187
6561
19683
59049
4
16
64
256
1024
4096
16384
65536
262144
1 048 576
5
25
125
625
3125
15625
78125
390625
1 953 125
9 765 625
You may notice that every other power of 2 corresponds to a power of 4. This is because 22 = 4. You need to multiply
a 2 by itself, (the number 2 is used two times), in order to get 4. Similarly, the powers of 8 correspond with every 3
power of 2 since 23 = 8.
In looking at the powers of 5, the last three digits have a pattern. If the exponent is an even number, the last three
digits will be 625 (not including exponents of 0 or 2). If the exponent is an odd number, the last three digits will be
125 (not including the 1st exponent).
No matter what the base is, there is always a modular recursion pertaining to the last digit(s) of the value. For the
powers of two, the last digit repeats in a cycle: 2, 4, 8, 6, 2, 4, 8, 6, and so on. The powers of 3 also have a pattern:
3, 9, 7, 1, 3, 9, 7, 1, etc. Whenever the base is a square, 4 for instance, there are only 2 numbers that go back in
forth; in this case, it is 4 and 6. The powers of 5 is a bit different, however. They all end in 5 as we mentioned in the
last paragraph. If the last digit in the square of a number ends in the original number, then it will always follow this
pattern. 62 is 36. 36 ends in 6, the original number. Therefore, all of the powers of 6 will end in 6. Most numbers
have 4 numbers that cycle.
How do you think we could compare two exponentiated numbers like 77 and 96 ? Which of the two is larger? For
this, you will need a log chart:
TABLE 1.4:
log(1)
log(2)
log(3)
log(4)
log(5)
log(6)
0
0.301
0.477
0.602
0.700
0.778
log(11)
log(12)
log(13)
log(14)
log(15)
log(16)
1.0414
1.0792
1.1139
1.1461
1.1761
1.2041
7
1.2. Concept 1.2
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TABLE 1.4: (continued)
log(7)
log(8)
log(9)
log(10)
0.845
0.903
0.954
1
log(17)
log(18)
log(19)
log(20)
1.2034
1.2553
1.2788
1.3010
To find out how many digits a number in the form of xy has, plug it into this equation: y*log(x) = n. Multiply the
exponent by the log of the base to get the number of digits. For example, to convert the base from 2 to 10 (it will
almost always be converted to 10) in 21024 , multiply 1024 by 0.301 and the answer is 308.224. Therefore, 21024 =
10308.224
Back to our earlier example, is 77 or 96 larger? To solve this, it would be nice to convert the base of EACH number
to 10. So we can restate the question to ask "is 7*log(7) or 6*log(9) larger? In computing this, our two values are
105.916 and 105.725 . We now know that 77 is larger than 99
Review:
1.
2.
3.
4.
5.
8
What are the last three digits of 51998 ?
315 = 27x (solve for x)
62x = mx (solve for m)
What is that last digit of 71001
Is 139 or 188 larger?
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Chapter 1. Exponentiation
1.3 Concept 1.3
Objective
Identify the hyperoperators, know the rules for exponentiation and be able to exponentiate with large bases.
The hyperoperator sequence is an infinite list of functions that is recursive. Each proceeding hyperoperator is a
repetition of the previous.
• Here are the first several hyperoperators
1.
2.
3.
4.
5.
6.
Repetitive counting is called addition (x+y)
Repetitive addition is called multiplication (x*y)
Repetitive multiplication is called exponentiation (xy )
Repetitive exponentiation is called tetration (y x)
Repetitive tetration is called pentation (y x)
Repetitive pentation is called hexation (xy )
• In the word, pentation, there is a prefix meaning 5. For hexation, there is a prefix meaning 6. The next
hyperoperators would be septation, octation, etc.
• Usually, hyperoperators are written in a specific notation where 83 = H3 (8, 3). The 3rd hyperoperator is used
on the numbers 8 and 3.
• The ’0th’ hyperoperator is called counting. To compute, take the second number, and add 1 to it.
TABLE 1.5: Here are some rules you need to know. m refers to any real number greater than 1
0^0
0^1
1^0
1^1
m^0
m^1
1
0
1
1
1
m
Review:
1.
2.
3.
4.
5.
Equate H0 (9, 11)
How many hyperoperators exist?
Equate H3 (2, 6)
Convert 4 3 to hyperoperator notation
What is (0^(0^(0^(0^(0)))))
9
1.3. Concept 1.3
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Rules of Exponentiation
Here are a few rules that will be helpful
1.
2.
3.
4.
5.
6.
(x+y)2 = (x+y)(x+y) = x2 +xy+y2
x^x^x ... x^x^x (x times) = x x = 2 x
Hm (2, 2) = 4
(ab)c = ac * bc
(ab )c = abc
ab * ac = ab+c
These rules can be used to solve exponentiation problems that have large bases. Let’s start with 86 .
86 {given} = (23 )6 {rule 5} =
218 {rule 5} = 29 *29 {rule 6} = 49 {rule 4} = 262144
Let’s try solving it another way:
86 {given} = 46 * 26 {rule 4} = 212 * 26 {rule 5} =
2
18
{rule 5} = 29 *29 {rule 6} = 49 {rule 4} =
262144
You may notice that the two highlighted sections of each problem are identical. Almost all exponentiation problems
end the same no matter what rules you use. Finding the right rules is up to you to figure out. Rules 1-3 will be used
less often, and only for specific cases.
Review:
1.
2.
3.
4.
5.
10
(x+2)2
412
85
124
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Chapter 2. Functions and Notations
C HAPTER
2
Functions and Notations
Chapter Outline
2.1
C ONCEPT 2.1
2.2
C ONCEPT 2.2
2.3
C ONCEPT 2.3
2.4
R EFERENCES
Now that you have completed Concept 1, actual "googological" material is now ready to be learned! This unit will
most likely be the most interesting since functions and notations are important in thinking of numbers beyond what
you may have been taught in the past. Get ready for a fun ride, since this is where numbers get big!
Concept 2 Objectives
1. Know scientific notation, up-arrow notation, down arrow notation, and Hyper-E notation
2. Be able to solve various factorials, and know basics of chained-arrow notation, BEAF, SGH, and FGH
3. Know the Hardy Hierarchy, and be able to compare different notations
An underline indicates that all rules for the notation or function are required to be known. Otherwise, only parts of
the notation or function are required to be known. More details are in each section of this concept.
Notation
A way to write a number that is inconvenient to write out decimally.
Function
Something that manipulates a number to represent a new value.
11
2.1. Concept 2.1
2.1 Concept 2.1
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Chapter 2. Functions and Notations
2.2 Concept 2.2
Objective
Be able to solve factorials, know chained arrow notation, BEAF, SGH, and FGH
Factorials and Variations
There are many different types of factorials:
TABLE 2.1:
Type of Factorial
Factorial
Expofactorial
Rule
n! = n * (n-1) * (n-2) ... (1)
an = n (n−1)
Example
5! = 5*4*3*2*1 = 720
a5 = 5^4^3^2^1 (Has almost
200,000 digits)
Double Factorial
n!! = n * (n-2) * (n-4) ... (1 or 2)
7!! = 7*5*3*1 = 105
Review:
1.
2.
3.
4.
5.
fdsa
fdsa
fdsa
fdsa
fdsa
Chained Arrow Notation
There are some basic rules for Chained Arrow Notation, but not all are listed here.
•
•
•
•
•
•
x→y = xy
x→1 = x
x→y→1 = x→y = xy
x→y→z = x↑z y = x↑↑↑... z ↑’s ... ↑↑↑y
Example: 2→4→3 = 2↑↑↑4 = 2^2^2 ... 65536 2’s ... 2^2^2
More rules will be explained in Googology II
Review:
1. Write 34 in chained arrow notation.
13
2.2. Concept 2.2
2.
3.
4.
5.
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Write 4256 in chained arrow notation. There should be 2 arrows in your answer
Convert 65536 into simplified chained arrow notation
Compute 2→2→x
Convert x→2→4 into up arrow notation. There should not be any 2’s in your answer.
Array Notation is one of the most difficult, and powerful notations you must know this year. First, start with an
empty set: {} this equals 1. Adding a number inside the set changes the value to the number inside: {a}=a. You can
add numbers outside the array to get this: a{c}b which equals a ↑c b. Just like arrow notation, we can expand arrays
like this:
a{5}3 = a{4}a{4}a. Subtract the inside of the array by 1, and then repeat ’a’ the number of times ’b’ represents.
Now to the next step of array notation: a{{c}}b. Having 2 brackets is called expansion. a{{c}}4 =
Here are some terms that would be helpful:
TABLE 2.2:
a{{1}}b
a{{2}}b
a{{3}}b
a{{{1}}}b
a{{{2}}}b
a{{{3}}}b
a{{{{1}}}}b
a{{{{2}}}}b
a{{{{{1}}}}}b
a{1}6 b
•
•
•
•
•
•
•
•
Expansion
Mutiexpansion
Powerexpansion
Explosion
Multiexplosion
Powerexplosion
Detonation
Multidetonation
Pentanation
Hexanation
{} = 1
{a} = a
{a, b} = a→b = ab
a {c} b = a ↑c b
{a, b, c} = a→b→c = a ↑c b
After 3 elements, you can not simply put every number into chained arrow notation
{a, b, c, d, ..., 1} = {a, b, c, d, ...}
{a, 1, b, c, d ...} = a
– This means if a 1 represents the 2nd or last element, it can be removed along with all elements after it
(WARNING: this rule does not apply if a 1 does not represent wither the 2nd or last element!)
• Array Notation will be explained more in-depth in Googology II
Review:
1. Write googol (10100 ) in BEAF
2. Write 636 in BEAF. There may be an exponent in your answer
3. Convert 10↑↑↑10↑↑↑10↑↑↑ ... ↑↑↑10 (10 10s) into BEAF
The most basic of functions is the Slow-growing Hierarchy. These are the rules you are required to know this year:
• gm(n)=m. Always!! If there is a constant following "g", that number is always the answer.
• gω(n)=n. If there is a lowercase omega after the g, whatever is in parenthesis is the answer. (Omega represents
the number in parenthesis).
14
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•
•
•
•
•
•
•
Chapter 2. Functions and Notations
gωω(n)=nn since Omega represents n.
... this pattern continues
gε0(n)=n↑↑n
gε1(n)=n↑↑2n
... this pattern continues
gεω(n)=n↑↑(n2 )
gεωω(n)=n↑↑(n2 )
Other rules will be explained in Googology II.
Review:
1. What is g4 (59)
2. What is gω(72)
3. What is g 2 (65)
4. What is g7ω(2)Justi f yyouranswer f ornumber4
15
2.3. Concept 2.3
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2.3 Concept 2.3
Learning Objectives
Vocabulary
Concept 2.1
5.
Objective
Know scientific notation, up-arrow notation, down arrow notation, and Hyper-E notation
Scientific Notation
You should already be familiar with this one, but as a review, here is an example:
52000 = 5.2 x 104
The exponent always determines the number of digits + the number of digits before the decimal place in
the coefficient. In the example above, the exponent after the 10 is 4 and in 5.2, there is 1 digit before the decimal
place. 4+1=5, therefore, we have proved that 54000 has 5 digits. It is "more correct" to exclude a single digit before
the decimal place. It is not necessary to be exact in scientific notation. For example, 7654321 can be written as 7.654
x 106 . In googology, scientific notation is most common for smaller numbers like 1060 and will not always have a
coefficient.
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Chapter 2. Functions and Notations
Review:
1.
2.
3.
4.
5.
Write 123 in scientific notation
Write 93700 in scientific notation
Write 7.43 x 103 in decimal form
3
Is 1 x 103 x 10 larger than a googol?
Justify your answer for number 4
Up-arrow Notation
Up-arrow Notation is the most basic, yet common notation used in Googology. It compares to the hyperoperator hierarchy. The definition of Up-arrow Notation starts with xy = x↑y. The 3rd hyperoperator is represented with 1 up arrow. The next rule is for tetration: y x = x↑↑y. The 4th hyperoperator is represented with
2 up arrows. To find the hyperoperator when given the Up-arrow Notation, simply add 2 to the number of arrows. Similarly, when given the hyperoperator notation, subtract 2 from the subscript to get the number of up
arrows in Up-arrow notation. It is important to know that x x which equals x↑↑x also equals x↑↑↑2. Therefore,
Hm+2 (x, y) = x↑m y. Also, Hm+2 (x, x) = x↑mx = x↑m+1 x.
TABLE 2.3:
Note:
x↑m y simply means x↑↑...↑↑y where the number of up arrows equals m. Raising the values of x, y, and m
all increase the value, but m is the strongest, followed by y. Increasing the value of x does change the value
dramatically, but not as much as m and y do.
Expanding something in Up-arrow notation is done like this: 4↑↑↑3 = 4↑↑4↑↑4 (reduce one arrow and repeat the first
number by the value of the second number). 4↑↑4↑↑4 = 4↑↑4↑4↑4↑4 (like exponentiation, powers are done from the
top to the bottom, so we must expand the numbers to the right first). 4↑↑4↑4↑4↑4 = 4↑↑4↑4↑256 = 4↑↑4↑(1.34 x
10154 ). We can now start to see how fast this notation can grow. However, this is a very small and simple notation.
There are more powerful ones to come...
FIGURE 2.1
Here is the official definition of Up-arrow Notation
To conclude, here are the rules you should keep in memory:
•
•
•
•
•
xy = x↑y
xx = 2 x = x↑x = x↑↑2
x
xx = x↑x↑x = x↑↑3
x x = x↑↑x = x↑↑↑2
Hm+2 (x, y) = x↑m y. Also, Hm+2 (x, x) = x↑mx = x↑m+1 x.
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2.3. Concept 2.3
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Review:
1.
2.
3.
4.
5.
Write 34 in up-arrow notation
Write 636 in up-arrow notation. There should be 2 arrows in your answer
Convert 99 into H4 (x, y)
Compute 2↑↑↑↑2
What other rule does number 4 correspond to?
Down-arrow Notation
This notation is nearly identical to Up-arrow notation. You should recall that after expanding an expression in
Up-arrow Notation, you are supposed to solve from right to left. This is where the one difference comes into play.
In Down-arrow Notation, however, you solve from left to right. Here is an example: 3↓↓4 = 3↓3↓3 = ((3↓3)↓3)↓3 =
(27↓3)↓3 = etc.
Solving with a single ↓ is no different from with a ↑.
FIGURE 2.2
Here is the official definition of Down-arrow Notation
Review:
1.
2.
3.
4.
5.
Compute 3↓↓2
Compute 2↓↓↓2
Convert 4↓↓4 to up-arrow notation
Convert 5↓↓↓2 to a single exponent
Convert (100↓2)*(2↓3) to scientific notation
Hyper-E Notation
There have been many extensions of this notation. In Googology 1, you are only required to know the original one:
Hyper-E Notation. All of these were written by Sbiis Saibian
TABLE 2.4: Sub-notations of Extensible-E Notation
Name
Hyper-E Notation
Extended Hyper-E Notation
Cascading-E Notation
Extended Cascading-E Notation
Hyper-Extended Cascading-E Notation
Abbreviation
E#
xE#
E^
xE^
#xE^
This notation is similar to scientific notation where ’E’ represents ’times 10 to the’
2E4 = 2 * 104
E9 = 109
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Chapter 2. Functions and Notations
You can represent a tower of exponents by repeating the ’E’
4
2EE4 = 2 * 1010
(
9)
EEE9 = 1010 10
However, not all chains are based on 10. Like logarithms, assume the base to be 10 unless otherwise noted. Here are
some examples where the base is not 10.
E(6)7 = 67
5
EE(2)5 = 22
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2.3. Concept 2.3
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Eventually, writing all of the E’s becomes cumbersome. Simply add #m at the end of the expression to define the
number of E’s
E8#5 = EEEEE8 = 10^10^10^10^10^8
E100#100 = EEE...EEE100 (100 E’s) = grangol
1
E1#3 = 1010 0 = Trialogue (instead of writing E#3, place a 1 to make E1#3)
We may even do 3 entry expressions as followed!
E100#100#2 = E100#(E100#100) = EEE...EEE100 (grangol E’s) = grangoldex
E100#100#3 = E100#(E100#100#2) = E100#(E100#(E100#100) = EEE...EEE100 (grandoldex E’s) = grangoldudex
E100#100#100#100 = E100#100#(E100#100#100#99) = E100#100(E100#100#(E100#100#100#98) = ... = gigangol
We can see how powerful adding extra entries can be. Also, we can stack the #’s together like so:
E100##100 = E100#100#100 ... #100#100#100 with 100 repetitions of 100# = gugold
E100###100 = E100##100## ... ##100 with 100 repetitions of 100## = throogol
E100####100 = E100###100 ... ###100 with 100 repetitions of 100### = teroogol
When we have two hyperions (#) stacked together (##), we call it a deutero-hyperion. Three of them would make a
trito-hyperion, and so on. There is one last rule before we complete this notation and here it is!
E100#####...#####100 with 100 hyperion marks = E100#100 100
Sets of 100 hyperion marks decompose into 99s, 99s, decompose into 98s, etc. Also note that the superscript 100
means that there are 100 #’s, and should not be confused with E100#(100 100) which involves tetration.
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Chapter 2. Functions and Notations
FIGURE 2.3
Here is the original definition of Hyper-E
Notation
FIGURE 2.4
Here is the newer, extended definition of
Hyper-E Notation
Review:
1.
2.
3.
4.
5.
Solve 3E(2)3
What is E1#2?
Expand E3###3 into deutero-hyperions.
Expand E5#2 3
Convert E2##4 to up arrow notation.
21
2.3. Concept 2.3
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Concept 2.2
Objective
Be able to solve factorials, know chained arrow notation, BEAF, SGH, and FGH
Factorials and Variations
There are many different types of factorials:
TABLE 2.5:
Type of Factorial
Factorial
Expofactorial
Rule
n! = n * (n-1) * (n-2) ... (1)
an = n (n−1)
Example
5! = 5*4*3*2*1 = 720
a5 = 5^4^3^2^1 (Has almost
200,000 digits)
Double Factorial
n!! = n * (n-2) * (n-4) ... (1 or 2)
7!! = 7*5*3*1 = 105
Review:
1.
2.
3.
4.
5.
fdsa
fdsa
fdsa
fdsa
fdsa
Chained Arrow Notation
There are some basic rules for Chained Arrow Notation, but not all are listed here.
•
•
•
•
•
•
x→y = xy
x→1 = x
x→y→1 = x→y = xy
x→y→z = x↑z y = x↑↑↑... z ↑’s ... ↑↑↑y
Example: 2→4→3 = 2↑↑↑4 = 2^2^2 ... 65536 2’s ... 2^2^2
More rules will be explained in Googology II
Review:
1.
2.
3.
4.
5.
22
Write 34 in chained arrow notation.
Write 4256 in chained arrow notation. There should be 2 arrows in your answer
Convert 65536 into simplified chained arrow notation
Compute 2→2→x
Convert x→2→4 into up arrow notation. There should not be any 2’s in your answer.
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Chapter 2. Functions and Notations
Array Notation
Array Notation is one of the most difficult, and powerful notations you must know this year. First, start with an
empty set: {} this equals 1. Adding a number inside the set changes the value to the number inside: {a}=a. You can
add numbers outside the array to get this: a{c}b which equals a ↑c b. Just like arrow notation, we can expand arrays
like this:
a{5}3 = a{4}a{4}a. Subtract the inside of the array by 1, and then repeat ’a’ the number of times ’b’ represents.
Now to the next step of array notation: a{{c}}b. Having 2 brackets is called expansion. a{{c}}4 =
Here are some terms that would be helpful:
TABLE 2.6:
a{{1}}b
a{{2}}b
a{{3}}b
a{{{1}}}b
a{{{2}}}b
a{{{3}}}b
a{{{{1}}}}b
a{{{{2}}}}b
a{{{{{1}}}}}b
a{1}6 b
•
•
•
•
•
•
•
•
Expansion
Mutiexpansion
Powerexpansion
Explosion
Multiexplosion
Powerexplosion
Detonation
Multidetonation
Pentanation
Hexanation
{} = 1
{a} = a
{a, b} = a→b = ab
a {c} b = a ↑c b
{a, b, c} = a→b→c = a ↑c b
After 3 elements, you can not simply put every number into chained arrow notation
{a, b, c, d, ..., 1} = {a, b, c, d, ...}
{a, 1, b, c, d ...} = a
– This means if a 1 represents the 2nd or last element, it can be removed along with all elements after it
(WARNING: this rule does not apply if a 1 does not represent wither the 2nd or last element!)
• Array Notation will be explained more in-depth in Googology II
Review:
1. Write googol (10100 ) in BEAF
2. Write 636 in BEAF. There may be an exponent in your answer
3. Convert 10↑↑↑10↑↑↑10↑↑↑ ... ↑↑↑10 (10 10s) into BEAF
The most basic of functions is the Slow-growing Hierarchy. These are the rules you are required to know this year:
• gm(n)=m. Always!! If there is a constant following "g", that number is always the answer.
• gω(n)=n. If there is a lowercase omega after the g, whatever is in parenthesis is the answer. (Omega represents
the number in parenthesis).
• gωω(n)=nn since Omega represents n.
• ... this pattern continues
• gε0(n)=n↑↑n
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2.3. Concept 2.3
•
•
•
•
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gε1(n)=n↑↑2n
... this pattern continues
gεω(n)=n↑↑(n2 )
gεωω(n)=n↑↑(n2 )
Other rules will be explained in Googology II.
Review:
1. What is g4 (59)
2. What is gω(72)
3. What is g 2 (65)
4. What is g7ω(2)Justi f yyouranswer f ornumber4
Concept 2.3
5.
Objective
Know MGH, be able to solve googological charts, and be able to compare different notations.
MGH
fdsa
Review:
1.
2.
3.
4.
5.
fdsa
fdsa
fdsa
fdsa
fdsa
Googological Charts
fdsa
Review:
1. fdsa
24
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2.
3.
4.
5.
Chapter 2. Functions and Notations
fdsa
fdsa
fdsa
fdsa
Comparing Notations
fdsa
log2 (3) 1.585 log2 (5) 2.322 log2 (6) 2.585 log2 (10) 3.322 log3 (5) 1.465 log3 (10) 2.096 log5 (10) 1.431 log10 (25)
1.398 log10 (50) 1.699 log25 (10) 1.215
Review:
1.
2.
3.
4.
5.
fdsa
fdsa
fdsa
fdsa
fdsa
Objective Know MGH, be able to solve googological charts, and be able to compare different notations.
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2.4. References
2.4 References
1.
2.
3.
4.
26
.
.
.
.
Up-arrow Notation.
Down-arrow Notation.
Hyper-E Notation.
New Hyper-E Notation.
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Chapter 3. Numbers
C HAPTER
3
Numbers
Chapter Outline
3.1
C ONCEPT 3.1
3.2
C ONCEPT 3.2
3.3
C ONCEPT 3.3
Concept 3 Objectives
1. Be able to classify numbers, know the Class 0 and Class 1 numbers, and be familiar with -illions
2. Know the common prefixes used in Googology and know basic large numbers
3. Identify and compare two or more googolisms.
27
3.1. Concept 3.1
3.1 Concept 3.1
Learning Objectives
Vocabulary
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Chapter 3. Numbers
3.2 Concept 3.2
Learning Objectives
Vocabulary
29
3.3. Concept 3.3
3.3 Concept 3.3
Learning Objectives
Vocabulary
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Chapter 3. Numbers
31