Monday, January 23, 2012

M.I.T.'s Large Number Contest

Boston, 2007

There are 100,000,000,000,000 atoms in a single cell of a human cheek and a single hydrogen atom is 0.00000000024 meters. When talking about atoms and cells and a number of other things, numbers get very, very small and very, very large, requiring the use of way too many zeros.

Rather than writing that many noughts, mathematicians and scientists use a shorthand notation that replaces all of those zeros. Taking advantage of our numbering system that is based on the number 10, it’s easy to see that 1,000 is the same as 10 x 100 which is the same as 10 x 10 x 10. The notation they developed modified this from 10 x 10 x 10 to 103, pronounced ten to the third power or simply, ten to the third. That’s not all that useful for a thousand or even a hundred thousand. But when you’re dealing with the number of atoms in a cheek cell, it’s much easier to write 1014 than 100,000,000,000,000.
Mathematicians as far back as Nicole Oresme (c. 1323-1382) have been using exponential notation, though its use became much more widespread in the twentieth century in the 1940s and 1950s as computers were being introduced that could handle these kinds of numbers.
Scientists, looking at atoms and stars, needed a convenient way of noting these numbers and in the post-WWII era, a specific exponential notation, one in which specific numbers were multiplied by exponents of ten, became the standard and the term scientific notation was born. Rather than write 3,400,000,000,000,000, scientists would write 3.4 x 1015.
So there are 1014 cells in a human cheek cell and an average sized person has approximately 7 x 1027 atoms in their body. Numbers this large are beyond the ability of the human mind to fully grasp, but if we keep going, there are 7,000,000,000 people on Earth, so 4.9 x 1037 atoms make up the human race.
One step further and it is estimated that the entire planet is made up of 1.33 x 1050 atoms. That doesn’t sound like much more that the number used on the human race, but keep in mind that 1050 is 1037 multiplied by 10,000,000,000,000. Remember, each time you increase the exponent by one, you increase the number by a factor of ten.
Earth, the other planets, asteroids and other bodies in our solar system combined, make up less than two tenths of a percent (0.02%) of the solar system, the sun being the overwhelming mass. The number of atoms in the sun is 1.2 x 1057, or more than nine million times the number in the earth.
A typical galaxy has about 400 billion stars and it is estimated that there are 1069 atoms in an average galaxy. Finally, the universe, 80 billion galaxies large, contains around 1080 atoms total. That’s 100,000,000,000,000,000,000, 000,000,000,000,000, 000,000,000,000,000, 000,000,000,000,000, 000,000,000,000,000 atoms.
Think that’s a large number? In a minute, you’ll think that’s insignificant.
A few of years ago, a couple of professors, Dr. Evil (Adam N. Elga) from Princeton and the Mexican Multiplier (Agustin Rayo) from MIT, not their real names, decided to have a Large Number Duel. The goal was to write the largest specific number ever written on a standard chalkboard. All mathematical notations were permissible with two rules. First, nothing cheap like “your number plus one” and it must be finite, simply meaning not using infinity.
The match began like a boxing match with Dr. Evil in the red corner and the Mexican Multiplier in the blue. Dr. Evil started simple: he wrote the numeral one. The Multiplier countered by writing a string of ones across the chalkboard. Evil came back by erasing a line through half of the ones, turning them into exclamation points. In mathematics this is the symbol for factorials.
A factorial is the product of a number multiplied by every counting number less that it. For instance, 9! is 9x8x7x6x5x4x3x2x1 which is 362,880. 9!! is 362,880 x 362,879 x 362, 878 and so on. Remember, it’s estimated that the number of atoms in the universe is somewhere between, 1078 and 1082. The total number of atoms is closer to 9! than 9!!. Imagine how big a number would be with 9!!! or 9!!!! So think about how big a number that Dr. Evil wrote when he took half a chalkboard worth of ones and applied another half-chalkboard of factorials.
From there, the numbers got crazy big and most of the observers were unable to follow the complex mathematical calculations they were witnessing. One student even asked Evil if the number written on the board as a formula was even computable. Evil sheepishly said no.
Back and forth and back and forth until Dr. Evil finally fell to his knees acknowledging, “I’ve been crushed.” 


If you Google “Profs Duke It Out in Big Number Duel,” you can read the MIT school newspaper’s report on the contest.

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