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Huge Numbers by Tanja Rakovic

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Huge Numbers by Tanja Rakovic 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 A googolplex 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 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 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 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, 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 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 = !a. 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 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.
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