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Home , Billion, Googol, Googolplex, Large numbers, Trillion

Huge Tanja Rakovic

Background/History

Even though 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 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 , 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 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, , , 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 , 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 . 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 times ten billion. To get a sense of how big this number is, imagine flying a plane through this sand at full speed for 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

A is 1010100 , 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 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 100 is the number with a one followed by a googolplex of zeros. It can be represented as 101010 . Not much is known about this number, but we do know that the googolplexianth is even bigger than the googolplexian. 10100 1010 1 The googolplexianth is written as 10 . Also a googolplexianth is equal to 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 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, math- ematicians 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 in- finity (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 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 (a large number representing ten thousand during their time). Their largest number was a myriad myriad, equivalent to one hundred million. created a system of naming large numbers up until 108×1016 . 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

2 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.

Huge numbers are also represented by Greek letters. For example sigma (P) is used to represent mathematical 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 x2dx 0 Pn ∗ Pn ∗ 2 the summation of the right hand sum under the curve can be written as k=1 f(xk)×∆x = k=1 f(xk) ×∆x ∗ where xk= (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 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 means that ‘b’ copies of ‘a’ multiplied to multiple exponents. For example, 4 ↑ ↑ 3 = 444 . 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. 4 Also, the number in the parenthesis (4 ↑ ↑ 4) is equal to 444 . 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 ), 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 : ”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. 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 . . . this number uses power towers (i.e: abc ), 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 is called a power tower of a’s that is b levels high. For example 2 ↑ ↑ 3= 2 ↑ (2 ↑ 2) = 222 = 24 = 16. a . . . a ↑ ↑ b = a ↑ (a ↑ (...(a ↑ a))) = aa

4 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 n↑...↑n in the form 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 79 34 ee 1010 African mathematician named Stanley Skewes found the upper bound, Sk1= e ≈ 10 of a problem (What is the smallest 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

π(x) lim x =1 x→∞ log(x)

x where π(x) is not close to log(x) 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 24 22 x 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 e7.705 first Skewes number. It is equal to eee . 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

5 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 nn . 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 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 “trian- gle(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)= f 258(2) where the superscripts represents a functional power. Therefore M(256,2,3)= (256256)256256 = 256256257 . 257 256256 This can be repeated for other values of mega such as M(256,5,3) which equals 256256256 .

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 and . For example, if one goes by the 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.

6 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 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 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 n! (r ) = 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 5∗4∗3∗2∗1 120 items, the formula is used where: (2) = 3∗2∗1∗2∗1 = 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

1 Therefore the probability of winning the lottery is 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

7 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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