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Conway's chained-arrow notation





One of various methods that have been devised recently for representing extremely large numbers. Developed by John Conway, it is based on Knuth's up-arrow notation but is even more powerful. The two systems are related thus:

     a right-arrow b right-arrow 1 = a up-arrow b
     a right-arrow b right arrow 2 = a up arrowup-arrow b
     a right-arrow b right-arrow 3 = a up-arrowup-arrowup-arrow b
     a right-arrow b right-arrow c = a up-arrowup-arrow...up-arrowup-arrow b (c up arrows)

Longer chains are evaluated by the following general rules:

     a right-arrow ... right-arrow b right-arrow c right-arrow 1 = a right-arrow ... right-arrow b right-arrow c
     a right-arrow ... right-arrow b right-arrow 1 right-arrow d + 1 = a right-arrow ... right-arrow b, and
     a right-arrow ... right-arrow b right-arrow c + 1 right-arrow d + 1 = a right-arrow ... right-arrow b right-arrow (a right-arrow ... right-arrow b right-arrow c right-arrow d + 1) right-arrow d

It's important to recognize that the Conway arrow isn't an ordinary dyadic operator. Where three or more numbers are joined by arrows, the arrows don't act separately but rather the whole chain has to be considered as a unit. The chain might be thought of as a function with a variable number of arguments, or as a function whose single argument is an ordered list or vector. The Ackermann function is equivalent to a three-element chain: A(m, n) = ( 2 right-arrow (n + 3) right-arrow (m - 2)) - 3. It can also be shown that Graham's number is bigger than 3 right-arrow 3 right-arrow 64 right-arrow 2 and smaller than 3 right-arrow 3 right-arrow 65 right-arrow 2.


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   • SYMBOLS AND NOTATION