Huge Numbers
Tanja Rakovic
Background/History
Even though large numbers appear to be the same all over the world, different cultures have various ways of
naming and condensing them. The Ancient Indians began to use extremely large numbers because they were
related to their religious thoughts and practices. In ancient Vedic literature, there is an individual Sanskrit
name for each of the powers of 10 up to 1,000,000,000,000. The Ancient Indians also introduced the notion
of infinity many times. They noted that if one subtracts purna (Ancient Indian word for infinity, meaning
fullness) from purna, then one is left with purna. This is consistent with modern mathematics today, where
if you subtract infinity from infinity one is still left with infinity. Moreover, one of the largest numbers in the
Ancient Indian system is called jyotiba = 1080000 infinities. Something that is noticeably different from their
system compared to the US system is the Hindu units of time, calculated on a logarithmic scale, where 100 =
1 second as we know it today, and one year is approximately in the middle of 107 and 108 . A paramanu is in
between 100 and 101 , while an aayan is smaller than the “year” that we know of. This is important to note
in order to understand the difference between traditional US and Ancient Indian units of time. Even though
they used it mostly for astrological calculations and religious rituals, their system has evolved an impacted the
whole world to this day.
Theoretically, there is no largest finite number because there will always be a number closer to infinity that is
bigger than the one before it. Nevertheless, we still work with large numbers that we are more familiar with.
The most common way of naming these familiar large numbers in the English language is by using the words
million, billion, trillion, quadrillion, quintillion, sextillion, septillion, and octillion.
When a number is too large, then we write it using mathematical notation like symbols and equations. When
people consider huge numbers, usually the number of digits which can be written in scientific notation is
brought up. For example 10000= 1 × 104 . This simple notation allows people to condense large numbers
into ”smaller numbers” that actually represent the same number. It is used extensively in chemistry for more
precise measurements.
A famous large number, the googol, is written as 10100 which has 101 digits. Basically, it is defined as the
number one followed by one hundred zeros. This term was coined by Milton Sirotta, who was the nephew of
the mathematician named Edward Kasner. He asked his nephew to think of a name for a large number, and
after a short amount of time, Milton replied “googol.” It is the number of grains of sand that could fit in the
universe times ten billion. To get a sense of how big this number is, imagine flying a plane through this sand
at full speed for trillions of years. One would never pass through all the sand because there is just so much of
it. Moreover, if one were to take a grain of sand and examine it under a microscope, he/she would notice that
there are actually 10 billion smaller grains that make up that one grain. If this were the case for every grain
of sand then the total number of those smaller grains would be equal to a googol. The number googol is even
larger than the amount of elementary particles in the universe, which is 1080 .
googol= 10100 = 10 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000
000 000 000 000 000 000 000 000 000 000 000 000 000 000 000
100
A googolplex is 1010 , and this has about googol digits. More simply, it is 10 to the googol power. Googol
is relatively small compared to another large number called Graham’s number, because we as humans cannot
concretely say how many digits it has due to the extremely large scale. It is hard to understand the number
googolplex, but a hypothetical example can demonstrate its complexity. If the universe is filled with sand,
then this only gets us a ten billionth of the way to googol. Therefore, if we fill the universe fill sand and write
ten billion zeros on each grain of sand in the universe, then that is the only way one would be able to write
googolplex. This is obviously extremely unrealistic, especially considering that in a human’s lifetime, he/she
would be able to write all those zeros on half a grain of sand. This is an extremely complex idea to consider.
Another practical application would be if one filled the entire volume of the observable universe with dust
particles that are about 1.5 micrometers in size, then the different number of combinations in which one could
arrange and number the particles would be equal to a googolplex. Moreover, we cannot even try to print out
a googolplex because according to Moore’s Law, the power of computer processors doubles every one to two
years. Therefore, it would only make sense to print out a googolplex 524 years later because any attempts
before this would be overtaken by a faster processor. Because a googolplex is so large, it would take years to
print out, and this effort would be futile at least for now.
Even though a googolplex seems extremely large, there is an even larger number called the googolplexian. This
10100
is the number with a one followed by a googolplex of zeros. It can be represented as 1010
. Not much is
known about this number, but we do know that the googolplexianth is even bigger than the googolplexian.
100
1010
The googolplexianth is written as 1010
. Also a googolplexianth is equal to
1
googolplexian .
When thinking about these large numbers as a part of the universe, first the Planck length must be defined. It
is the smallest measurement of length which is approximately equal to 1.6 × 10−35 m. A googol Plank lengths
is roughly an inch. However, a googolplexianth of the universe is too small to even consider.
Another important number is 1080 . It is the common estimate for the number of atoms in the universe. To
get to this number, one must multiply one trillion by one trillion five times and then once more by a hundred
million. It is believed to be extreme underestimate because we do not know exactly how large the universe is.
It is fascinating to think about huge numbers in more practical terms. We already know that there are many
atoms in everyday objects like our phones, but to think of the amount of atoms in the entire universe seems
unrealistic.
The concept of infinity is usually thought to be the largest ”number” because it is greater than any finite
number. It is used within calculus and the concept of limits. However beginning in the 19th century, mathematicians have also expanded beyond infinite numbers by studying transfinite numbers. These numbers are
also greater than any finite number, but unlike infinite numbers, transfinite numbers are also larger than infinity (this idea comes from set theory). It was Georg Cantor who proved that infinite numbers exists, and
brought up the idea that numbers larger than these existed. Through the Diagonal Theorem he showed that
the set of rational numbers can be put into one-to-one correspondence with natural numbers. Therefore, the
rational numbers have the same cardinal number as the set of natural numbers. The ”transfinite cardinal” of
this set is called the smallest transfinite numbers.
The largest of the transfinite numbers, if they exist, are the large cardinals. Large cardinals are bigger than
the least a such that a = ωa . The existence of these cardinals cannot be proved because much information
needs to be assumed, such as the existence of large cardinals themselves.
Greek and Latin Notation
The Ancient Greeks used a system based on the myriad (a large number representing ten thousand during their
time). Their largest number was a myriad myriad, equivalent to one hundred million. Archimedes created
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a system of naming large numbers up until 108×10 . Essentially he was naming powers of a myriad myriad.
This is the largest number because it ”equals a myriad myriad to the myriad myriadth power, all taken to
the myriad myriadth power.” This is important because it reveals problems that Archimedes faced regarding
notation. Some explanations for this are that he stopped at this number because he did not find any new
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ordinal numbers larger than the myriad myriadth. His overall goal was to name large powers of 10 to give
rough estimates for these numbers in a simple way. However, the numbers became too large, and therefore
a new system had to be implemented. This is where Apollonius of Perga came in and invented a newer,
easier system of naming large numbers. This included large numbers that were not powers of 10. Most of
the numbers he named were based on the powers of a myriad. Later, the mathematician Diophantus used a
similar technique to name large numbers.
The more modern version of naming large numbers stemmed from the Romans. For example, they expressed
the number 1,000,000 as decies centena milia which means ten hundred thousand. Then in the thirteenth
century, the French introduced the word million which is used today. Currently, the US system of naming
numbers differs slightly from the European system because the same words represent different numbers. For
example, one billion in the US system is equal to 109 , while in the European system it is equal to 1012 . The way
numbers are said and recorded is not standard all over the world due to historical and language differences.
P
Huge numbers are also represented by Greek letters. For example sigma ( ) is used to represent mathematical
addition or standard deviation. With addition, it condenses all the expressions one would have when adding
many terms into one simplified expression.
For example, instead of writing (3 × 1) + (3 × 2) + (3 × 3) + (3 × 4) =30, we can write
4
X
3n = 30
n=1
This notation is important especially when dealing with areas under curves. For example, when looking at
Z 2
x2 dx
0
P
P
the summation of the right hand sum under the curve can be written as nk=1 f (x∗k ) × ∆x = nk=1 f (x∗k )2 × ∆x
where x∗k = (2/n) × k. When n → ∞, we want to calculate the area under the curve with an infinite amount
of rectangles. This is important because using rectangles to calculate the area under a curve does not give an
exact answer; it only gives a close estimate. Therefore when an infinite amount of rectangles is used, this give
an area that is more accurate.
Graham’s number
Graham’s number is an extremely large number that is 0 percent as big as infinity. This is a strange fact,
considering that infinity extends forever and so even though Graham’s number is large, it is still not comparable
to the size of infinity. There will always be a number that is even bigger than Graham’s number; it just does
not have a name and may not be useful currently. Graham’s number is the proven “upper bound” on the
solution to a problem relating to Ramsey theory, which is the study of combinatorial objects. As the scale of
the objects becomes large, a certain degree of order must occur.
Furthermore, Goldbach’s conjecture states that every integer is the sum of two primes. Mathematicians have
not checked if the Goldbach conjecture is true for numbers up to Graham’s number because it is so large. To
get a sense of how large it is, Graham’s number minus 1018 is roughly equal to Graham’s number.
The only way to describe Graham’s number accurately is by using Knuth’s up-arrow notation. This type of
notation is similar to the notation using exponents, but it is more extensive and is used for extremely large
numbers. For example, a ↑ b represents the usual exponential notation for ab = a × a × a... for a total of ‘b’
copies of ‘a’ multiplied together. If a second arrow is added, such that, the notation becomes a ↑ ↑ b, which
3
4
means that ‘b’ copies of ‘a’ multiplied to multiple exponents. For example, 4 ↑ ↑ 3 = 44 . When three arrows
are added to become a ↑ ↑ ↑ b, this means to make ‘b’ copies copies of a ↑ ↑. For example, 4 ↑ ↑ ↑ 3 is equal to
4 ↑ ↑ (4 ↑ ↑ 4). This means that there are three copies of the number 4 that are separated by the ↑ ↑ notation.
44
Also, the number in the parenthesis (4 ↑ ↑ 4) is equal to 44 . Then it is necessary to raise 4 to the 4th power
that many times because of the 4 ↑ ↑ outside the parenthesis.
An important thing to notice is that numbers using up arrow notation grow quickly (much more than using
simple multiplication), and the more they are used, the bigger the number becomes. For example, 2 × 10 =
20, but 210 = 1024. Therefore, each level of up arrows grows more quickly than the level before.
3 ↑↑ ............... ↑ 3
{z
}
|
3 ↑↑ ............ ↑ 3
|
{z
}
.
G=
.
.
3 ↑↑ .... ↑ 3
| {z }
3 ↑↑↑↑ 3
The above is the closest we can give to a description of the Graham number. It has 64 layers!
History of Graham’s Number
In 1977, Martin Gardner wrote that Graham created a bound “so vast”, represented by Graham’s number.
Graham was working on a problem related to combinatorics: ”Connect each pair of geometric vertices of an
n-dimensional hypercube to obtain a complete graph on 2n vertices. Color each of the edges of this graph either
red or blue. What is the smallest value of n for which every such coloring contains at least one single-colored
complete subgraph on four coplanar vertices? His published proof does not actually use Graham’s number, but
he did use a smaller number. The idea for Graham’s number did however stem from this problem. Ronald
Graham came up with it because it was simpler to explain than “his actual upper bound”. Since it was an
even bigger number, it was still considered an upper bound.
Graham’s number is extremely larger than most other numbers such as Skewes’ number and Moser’s number.
It is so large that even the observable universe is too small to show Graham’s number digitally. Even though
c
..
.
this number uses power towers (i.e: ab ), this is still not enough to show the magnitude of Graham’s number.
This is why Knuth’s up-arrow notation is important, as it is the best way to describe and understand the
number. Because of this, mathematicians were able to derive the last 12 digits, which are 262464195387.
Description of Knuth’s up-arrow notation
The mathematician and computer scientist Donald Knuth created up-arrow notation in order to continue “the
compounding nature of the better-known arithmetic operations”. We already know that ab = a × a × ... × a=
ab for b copies of a. As numbers started getting larger, Knuth developed a system that allowed this process
to continue, but for an infinite amount of arithmetic operations. A single up-arrow ↑ is used to represent the
represent repeated multiplication in a more condensed way. A simple example is 2 ↑ 3 = 23 = 8. When the
number gets even larger then two up-arrows ↑ ↑ need to be used. This repeated exponentiation is called a
2
power tower of a’s that is b levels high. For example 2 ↑ ↑ 3= 2 ↑ (2 ↑ 2) = 22 = 24 = 16.
a ↑ ↑ b = a ↑ (a ↑ (...(a ↑ a))) = aa
4
..
.
a
This process can be repeated an infinite amount of times, which allows people to use more mathematical
operations; thus creating larger and larger numbers. These numbers would not be able to be represented
concisely with regular powers, which is why Knuth’s up-arrow notation is important. Moreover, numbers
written with the upward arrows tend to grow very quickly. Therefore the arrows allow one to understand to
magnitude of these numbers without writing too many digits. For example, 2 ↑ ↑ 4= 65,536. However, 2 ↑ ↑ ↑
4 = 2 ↑ ↑ (2 ↑ ↑ (2↑ ↑ 2))= 2 ↑ ↑ (2↑ ↑ 4) = 2 ↑ ↑ 65,536. This number represents a power tower of 2’s that is
65,536 levels high. With the seemingly simple addition of another arrow, the number becomes extremely large.
Another reason why Knuth’s up-arrow notation is important is because it gives mathematicians the ability to
precisely define large numbers, such as Graham’s number. The numbers that use this notation are so large
that there is not enough room in the universe to write them. Therefore, with this notation, one can write the
same number but in a condensed form.
Another type of number that uses Knuth’s up-arrow notation is called the Ackermann Number. It is a number
in the form n↑...↑n
n . The first few of these numbers are 1 ↑ 1 = 1, and 2 ↑ ↑ 2 = 4. These numbers also grow
very quickly, and are especially useful for defining complex algorithms. Therefore, they are sometimes used a
base for programming language.
Skewes’ Numbers
The first Skewes number, represented by Sk1 was proved to exist in by John Littlewood in 1912. A South
e79
1034
African mathematician named Stanley Skewes found the upper bound, Sk1 = ee
≈ 1010
of a problem
(What is the smallest natural number x such that π(x) > li(x), where π(x) is the prime counting function and
li(x) is the logarithmic integral function?) who’s answer has not been found yet. This number has about
8.85 decillion digits, and falls between a googolplex and a googolduplex, and quickly broke the record for the
largest number used in mathematics at the time. Even though he did assume that the Riemann Hypothesis,
it is relevant because it differentiates between the first and second Skewes’ number, and other large numbers.
Furthermore, Littlewood also proved that there are infinite numbers that exist where li(x) is less than π(x).
In addition, π(x) can be used to approximate many large numbers as well. For example, π(1000)= 168,
and π(1024 )= 18435599767349200867866. To approximate π(x), it is first important to understand the prime
number theorem, which states that
lim
x→∞
where π(x) is not close to
x
log(x)
π(x)
x
log(x)
=1
because the numbers are so large.
x
x
π(x) will generally be larger than log(x)
. For example, when x=100 then π(x)=25 and log(x)
= 21.7147. If we
x
24
22
22
take a larger number, such as x=10 , then π(x)= 1.84355 × 10 , and log(x) = 1.80956 × 10 . It is noticeable
x
x
that log(x)
always has a slightly smaller value than π(x). However as x becomes larger, the ratio between log(x)
and π(x) becomes closer to 1.
The discovery of the Skewes’ numbers began with the endeavor to find the smallest value of x such that li(x)
is less than π(x). Skewes himself started by finding the upper bound to the solution described earlier.
The Second Skewes’ Number was discovered in 1955 when Skewes tried to find an upper bound for when π(x)
is greater than li(x). This time he did not assume that the Riemann hypothesis was true, which made it more
difficult for him to solve the problem. Nevertheless, he discovered the upper-bound which was bigger than the
ee
7.705
first Skewes number. It is equal to ee
. This number is bigger than a googolduplex and has about 3.3 ×
10963 number of digits. This number broke the record that the first Skewes number set.
After the discovery of the Skewes’ numbers, mathematicians began to find more accurate upper-bounds for
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the same problems Skewes was working on. For example, R. Sherman Lehman found the upper bound to be
1.65 × 101165 , which is a drastic decrease from the upper bound represented by both Skewes’ numbers. The
most recent improvement to the upper bound value was made by Stefanie Zegowitz, who reduced the upper
bound to e727.951332973 , which equals approximately 1.39716 ×10316 .
Even though the Skewes’ numbers were no longer recognized as the upper bounds, they were still identified
as important because of their magnitude. Skewes’ numbers were practically obsolete with the introduction of
Graham’s number in 1971.
Even today, we still do not know the solution to Skewes’ problems. The closest answer is the most recent one
proposed by Zegowitz, but this still cannot be 100 percent confirmed as the solution. There is also a lower
bound for the solution to Skewes’ problem, which is 1014 , as proven by Tadej Kotnik in 2008. Therefore, a
general, the most current theoretical solution to Skewes’ problem can be written as
1014 < N < 1.39716 ×10316
Steinhaus-Moser notation
Steinhaus-Moser notation is likewise used in mathematics to express large numbers. It is an extension of
Steinhaus’s polygon notation. A number n inscribed in a triangle is defined as n n . A number n in a square
equals “the number n inside n triangles”. Finally, a number n in a pentagon is equal to “the number n inside
n squares”. Steinhaus also defined “mega” as the number equivalent to 2 in a circle. Moreover, a “megistron”
is the number equivalent to 10 in a circle.
Moser’s number is defined as the the number equal to “2 in a megagon”, where a megagon is a polygon with
“mega” amount of sides. This number has also been proven to be: moser < 3 → 3 → 4 → 2 in Conway chained
arrow notation. This notation is another way of expressing large numbers. It is always a series of finite integers
separated by rightward arrows. Quite simply, for example, p → q represents pq . The same number can also
be represented by using Knuth’s up-arrow notation. Therefore moser < f 3 (4) = f (f (f (4))), where f(n) = 3
↑n 3. Therefore Moser’s number is extremely small compared to Graham’s number. This can be shown with
the notation: moser 3 → 3 → 64 → 2 < f 64 = Graham’s number.
Other forms of notation within the Steinhaus-Moser notation includes the functions “square(x)” and “triangle(x)”. There are also certain rules that apply if M(n,m,p) is the number represented by the number n in m
“nested p-sided polygons”. They are M(n,1,3)= nn , M(n,1,p+1)= M(n,n,p) and M(n,m+1,p)=M(M(n,1,p),m,p).
There are also rules when defining mega, megistron, and moser such that mega equals M(2,1,5), megistron
equals M(10,1,5) and moser equals M(2,1,M(2,1,5)).
It is necessary to get into more detail regarding “mega”. It is already known as a very large number that
equals M(2,1,5)= M(256,256,3). If we bring in the function f (x) = xx , mega can be written as f 256 (256)=
256
257
f 258 (2) where the superscripts represents a functional power. Therefore M(256,2,3)= (256256 )256 = 256256 .
This can be repeated for other values of mega such as M(256,5,3) which equals
256
256256
256256
257
.
Therefore it can be concluded that when mega = M(256, 256, 3) ≈ (256 ↑)256 257. (256 ↑)256 represents a
functional power of the function f(n)= 256n . By using Knuth’s arrow notation, this also equals 256 ↑ ↑ 257.
It is apparent that this is a big number, but there still exist even larger ones.
Practical Approaches
When applying huge numbers to the real world, we see that they are mostly used in astronomy and cosmology.
For example, if one goes by the Big Bang Theory, the universe is estimated to be 4.3 ×1017 seconds old. The
observable universe is 8.8 ×1026 meters and within it, there are 5×1022 stars.
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Huge numbers also play a role in computing. For example, if a hacker were to attempt to break into a computer
with a 40 character password, it would take him 2 ×1087 seconds or approximately 6×1079 years to try all the
possible combinations. To get a sense of how large this number is, 6×1079 years is older than the universe,
which is about 13.7 billion years old.
We usually do not think of how huge numbers play a role in our every day lives, but they are seen in computers,
the human body, etc. For example, the numbers of bits in a computer 500-1000 GB hard disk is estimated to
be about 1013 . When we talk about the human body, we can say that there are about 1014 cells in the human
body, and about 1014 neuronal connections in the human brain.
An interesting example of the practical application of huge numbers is when discussing the numbers of possible
games in chess. As soon as both players move, 400 possible ways to setup the board exist. After each player
makes a move the second time, then there are 197,742 possible games. After three moves, there are 121 million
possible games. Each time a player makes a move, there are so many more possibilities that exist. Some have
estimated that the number of chess games possible is about 10120 .
The Shannon number, named after Claude Shannon is based on an average of 103 possibilities for a pair of
moves first by White and then by Black, where a game usually lasts through forty moves by each color. He
also estimated the possible number of positions which is approximately equal to 1043 . By taking into account
possible illegal positions, for example when both kings are in check, an upper bound: 5 × 1052 was created
for the number of positions. The true number is considered to be about 1050 . However, more recent results
conclude that the upper bound is only 2155 . This shows that there is always a possibility to refine a solution
in order for it to be more accurate, because it is easy to be disorganized when dealing with large numbers.
Furthermore, large numbers have practical applications when dealing with lottery combinations. To begin, the
probability of winning the lottery is equal to
the number of winning lottery numbers
the total number of possible lottery numbers
In order to find the total number of possible lottery numbers, the formula
n!
(nr ) = r!(n−r)!
is needed. n! represents n × (n-1) × (n-2) × ... × 2 × 1. For example, if we pick two items from a set of 5
5∗4∗3∗2∗1
items, the formula is used where: (52 ) = 3∗2∗1∗2∗1
= 120
12 = 10. Therefore if a set has five items represented by
the notation: {1,2,3,4,5} then we can pick two items represented by ten different groups: {1,2}, {1,3}, {1,4},
{1,5}, {2,3}, {2,4}, {2,5}, {3,4}, {3,5}, and {4,5}. In order to apply this to the lottery, it is important to know
that we can pick 6 possible winning numbers out of 49 total different numbers. This can be represented in the
equation such that:
(49
6 )= 13,983,816
Therefore the probability of winning the lottery is
1
13,983,816 .
Even as the lottery continues to change with the addition of new games, more possible lottery numbers, and
even bonus numbers depending on each state, this is the standard probability for any lottery game with a total
number of possible lottery numbers equal to 49.
Conclusion
When we have knowledge on large numbers, this enables people to think about and understand the universe
in a totally different way. We are able to understand problems using large quantities, which allows us to
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understand the past and the future more accurately. Even though some numbers are too large to express, we
still have unimaginable ways of expressing them into more understandable terms. Therefore, in the future of
mathematics, it is likely that even larger numbers with meaning will be discovered and put into good use.
References
1. http://mathworld.wolfram.com/LargeNumber.html
2. http://www.mrob.com/pub/math/largenum.html
3. https://blogs.scientificamerican.com/roots-of-unity/
4. http://mathworld.wolfram.com/SkewesNumber.html
5. http://www.transfinite.com/content/about5
6. https://en.wikipedia.org/wiki/Transfinite_number
7. https://en.wikipedia.org/wiki/History_of_large_numbers
8. http://mathworld.wolfram.com/RamseyTheory.html
9. https://en.wikipedia.org/wiki/Steinhaus?Moser_notation
10. http://mathworld.wolfram.com/Megistron.html
11. https://plus.maths.org/content/writing-unwritable-arrow-notation
12. https://sites.google.com/site/pointlesslargenumberstuff/home/1/skewes
13. http://garsia.math.yorku.ca/~zabrocki/math5020f03/lot649/lot649v3.pdf
14. http://www.googolplexian.com
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