aka...exponents and arrows and the math nerds who love them.
Is 4000 a big number? How about 1.2 million? 632, 415, 906, 255? What about 1015? As this is obviously a trick question, even the non-mathematic are likely guessing that the last of these is the biggest. Ten to the fifteenth power is a sixteen-digit number (as compared to the 12 digits of the next largest number mentioned above). Knowing that still doesn’t give a fair representation, though. Which seems larger: 1,000,000,000,000,000 or 1015?
Of course, neither really means anything to us.
Multiple exponents can make even larger numbers seem small. How big is (103)x(103)? Or (103)3? How about 10^33 (the ^ means to the power of, so 10^3=103 and 10^33 just means raise 10 to the 33 power)? How do these two number compare to the numbers at the end of the last paragraph?
First, 1,000,000,000,000,000 and 1015 are the same; anytime you raise 10 to a power, the result is a one with that number (the power) of zeroes after it. When multiplying a number to a power by the same number to a power, add the exponents ((103)x(103)=106 or 1,000,000). When raising a number to a power and then raising the result to a power, such as (103)3, you multiply the exponents. So (103)3 yields 109 or 1,000,000,000. Finally, raising a number to a power that is raised to a power, the power in the exponent must be done first (following order of operations), so 10^33 is 1027 or 1,000,000,000,000,000,000,000,000,000. So minor changes in the use of these little numbers (the exponents) or movements of the parentheses can make a huge difference in the scale of the result. We can also represent unwieldy numbers in very small notations through careful use of exponents. In addition to the notations used above, the up arrow, ↑, indicates an exponential use of powers, such as 10^(3↑↑3) (this is Knuth’s up-arrow notation, for those who care) means 10^(3^(3^3)), which is 10^(3^27) or 10^7625597484987. This number is a one with over 7 trillion zeroes after it, which I represented with four digits, three arrows and a pair of parentheses (and only then because I wanted to keep using 10 to a power and used threes to keep the final exponent small enough to show here. The addition of a third arrow does not mean a third cubing of the 7 trillion digit number; instead, it means to do the double arrow operation three times: 3↑↑↑3= (3↑↑(3↑↑3), which is 3 cubed 7625597484987 times (a number far too large to represent in any form other than that already used here). Multiple arrows, even with small numbers, will yield numbers that have minimal, if any, use.
Taking our extreme number game a step further, Graham’s number is a number that makes creative use of the up-arrow notation. First, calculate 3↑↑↑↑3; this will be called g(1). To calculate g(2), calculate 3↑↑….↑↑3, with number of arrows between the 3’s being equal to g(1). Of course, remembering what 3 up arrows did to the threes in the previous example and the fact that an additional arrow means to do the triple arrow operation 3 times (3↑↑↑(3↑↑↑3), which is unwieldy when reduced to the already impressive double arrow notation. So, if g(2) is a pair of 3’s with g(1) arrows, this number is not only large beyond imagination, we cannot even comprehend how large the result is in comparison to g(1). Moving on, Graham’s number is g(64)….that’s right, a mind blowing ratio between mind blowing numbers 64 TIMES! Ridiculous!
So, The Math Factor (broadcast on KUAF, an NPR associate, and available on podcast) had a contest to see who could express the largest number on the fly. The contestants (AgustÃn Rayo against Adam Elga) took turns one-upping each other until Rayo introduced us to the number now known as Rayo’s Number. Simply put (sort of), Rayo’s Number is the smallest number larger than the largest finite number that can be expressed in the language of first order set theory (common math and logic symbolism) using no more than a googol (one with one hundred zeros after it or 10^100) symbols. A bit of a cheat, but the winner of the contest.
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