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clashofpawns 13-Feb-18, 00:13 |
 A little more arithmeticAt the basic level we have "succession" where a natural number succeeds to the next natural number. As in 1 to 2, 2 to 3, or 11 to 12. Addition is a higher order process. When we compute 7 + 6, what we're really doing is taking the succession of 7, six times. 7->8->9->10->11->12->13 It's a shorthand for computing many successions and we could say it's a higher order operator. Multiplication is to addition, as addition is to succession. Exponentiation comes next, then tetration. Then something you might call hyper-tetration, then hyper-hyper-tetration, etc. Donald Knuth, one of my favorite mathematicians and one of my favorite computer scientists, and certainly my favorite mathematician+computer scientist, invented a notation called "Knuth's Up Arrow Notation" that generalizes this operator progression. Example: 3↑4 = 3 to the 4th power. In other words, one arrow represents exponentiation. Two arrows would represent iterated exponentiation. Or the iteration of what one arrow represents. Example: 2↑↑4 = 2^(2^(2^2)) = 2^2^4 = 2^16 = 65536 As you can see, already with tetration, numbers grow very quickly. Something like 2↑↑↑4 is well beyond what I can easily express here. If we try, it goes something like this: 2↑↑(2↑↑(2↑↑2)) = 2↑↑(2↑↑4) = 2↑↑65536 = 2^2^2^2^2^...(a tower of exponents 65536 2's tall)...^2 So this notation gives us a convenient way to represent numbers whose logarithm is inexpressible without higher order arithmetic operators. In other words a number, even the number of digits of which, is inexpressible with mere exponentiation. What about a number that has so many digits that even the number that represents the number of digits is itself a number with more digits than could be expressed. And so on, arbitrarily far. Each additional '↑' gives rise to a whole new universe of complexity. This brings us to Graham's Number. There are larger numbers. In fact, I will speak to the fast growing hierarchy and infinite ordinals at a later time and when I do I'll create a notation that allows us to exceed Graham's Number with tiny seeds to elementary functions seeded with low level infinite ordinals. But for now let us bask in the glow of a finitude that seemingly exceeds the infinite. Consider the sequence 'g' defined thusly: g_0 = 4 g_(n+1) = 3(↑^(g_n))3 In other words: g = {4, 3↑↑↑↑3, 3↑↑↑↑↑.....(how many ↑'s you ask? exactly 3↑↑↑↑3 ↑'s!)...↑↑↑3, ...} Graham's number is g_64. g_1 is already a number that belies any comprehension whatsoever and g_2 is literally the application of an operator on an order that is equivalent to g_1. The usefulness, or more accurately, the arrival of this number is attributed to mathematician Ronald Graham. en.wikipedia.org Finally, I'll mention that, though the fast growing hierarchy dominates another notation invented by another favorite mathematician of mine, John Conway, namely Conway's Chained Arrow Notation, it's fascinating how much more powerful Chained Arrow Notation actually is. 3→3→64→2 < g_64 < 3→3→65→2 Though Chained Arrow Notation is far more powerful, it is much less wieldly. Much less intuitive. Up Arrow Notation appeals instantly. Chained Arrow Notation is much subtler and as such, I prefer to skip over it indefinitely. Though I proved an interesting result about the relationship between Chained Arrow Notation and patterns in the base-17 representations of mersenne primes. That is alas beyond the scope of this post |
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“So mathematicians, do you have an answer to that tricky problem?” “Sort of” “What do you mean, sort of?” “Well, we know it’s definitely bigger than 13” “That’s great!” “And smaller than a number so absurdly large there aren’t enough particles in the universe to represent the number of digits it has” “Oh” |
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clashofpawns 14-Feb-18, 01:52 |
I view that as a failing on the part of the Universe, not Graham's Number. Stupid thing needs to eat it's just increasingly thin skin and increasingly frigid bones |
