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Have You Heard of Tetration? b + n , b + 1 + 1 + ··· + 1 | {z } n b × n , b + b + ··· + b | {z } n bn , b × b × · · · × b | {z } n b : : b: nb , b | {z } n Exponentiation associates to the right, so for example 4b means bb b bb b , not bb . Why? Which is bigger? Have you heard of tetration? It’s the fourth hyperoperation. 1 b × n , b + b + ··· + b | {z } n bn , b × b × · · · × b | {z } n b : : b: nb , b | {z } n Exponentiation associates to the right, so for example 4b means bb b bb b , not bb . Why? Which is bigger? Have you heard of tetration? It’s the fourth hyperoperation. b + n , b + 1 + 1 + ··· + 1 | {z } n 1 bn , b × b × · · · × b | {z } n b : : b: nb , b | {z } n Exponentiation associates to the right, so for example 4b means bb b bb b , not bb . Why? Which is bigger? Have you heard of tetration? It’s the fourth hyperoperation. b + n , b + 1 + 1 + ··· + 1 | {z } n b × n , b + b + ··· + b | {z } n 1 b : : b: nb , b | {z } n Exponentiation associates to the right, so for example 4b means bb b bb b , not bb . Why? Which is bigger? Have you heard of tetration? It’s the fourth hyperoperation. b + n , b + 1 + 1 + ··· + 1 | {z } n b × n , b + b + ··· + b | {z } n bn , b × b × · · · × b | {z } n 1 Exponentiation associates to the right, so for example 4b means bb b bb b , not bb . Why? Which is bigger? Have you heard of tetration? It’s the fourth hyperoperation. b + n , b + 1 + 1 + ··· + 1 | {z } n b × n , b + b + ··· + b | {z } n bn , b × b × · · · × b | {z } n b : : b: nb , b | {z } n 1 Why? Which is bigger? Have you heard of tetration? It’s the fourth hyperoperation. b + n , b + 1 + 1 + ··· + 1 | {z } n b × n , b + b + ··· + b | {z } n bn , b × b × · · · × b | {z } n b : : b: nb , b | {z } n Exponentiation associates to the right, so for example 4b means bb b bb b , not bb . 1 Which is bigger? Have you heard of tetration? It’s the fourth hyperoperation. b + n , b + 1 + 1 + ··· + 1 | {z } n b × n , b + b + ··· + b | {z } n bn , b × b × · · · × b | {z } n b : : b: nb , b | {z } n Exponentiation associates to the right, so for example 4b means bb b bb b , not bb . Why? 1 Have you heard of tetration? It’s the fourth hyperoperation. b + n , b + 1 + 1 + ··· + 1 | {z } n b × n , b + b + ··· + b | {z } n bn , b × b × · · · × b | {z } n b : : b: nb , b | {z } n Exponentiation associates to the right, so for example 4b means bb b bb b , not bb . Why? Which is bigger? 1 b = a − n n = a − b b = ap/n n = a/b = n = b p a n logb a n ? b = a4 n = logb a n2 1 2 4 16 65;536 265;536 ≈22:0035×1019;728 A: Real fast. Q: Does it have an inverse? A: Yeah, 2—which do you want? Q: Is one better than the other? A: Maybe, you decide: If ::: then ::: , and also ::: . b + n = a b × n = a bn = a nb = a Q: Why should we care? A: Its inverse grows as slow as its self grows fast, and tetration grows real fast: Tetration FAQ Q: How fast does it grow? 2 b = a − n n = a − b b = ap/n n = a/b = n = b p a n logb a n ? b = a4 n = logb a n2 1 2 4 16 65;536 265;536 ≈22:0035×1019;728 Q: Does it have an inverse? A: Yeah, 2—which do you want? Q: Is one better than the other? A: Maybe, you decide: If ::: then ::: , and also ::: . b + n = a b × n = a bn = a nb = a Q: Why should we care? A: Its inverse grows as slow as its self grows fast, and tetration grows real fast: Tetration FAQ Q: How fast does it grow? A: Real fast. 2 b = a − n n = a − b b = ap/n n = a/b = n = b p a n logb a n ? b = a4 n = logb a n2 1 2 4 16 65;536 265;536 ≈22:0035×1019;728 A: Yeah, 2—which do you want? Q: Is one better than the other? A: Maybe, you decide: If ::: then ::: , and also ::: . b + n = a b × n = a bn = a nb = a Q: Why should we care? A: Its inverse grows as slow as its self grows fast, and tetration grows real fast: Tetration FAQ Q: How fast does it grow? A: Real fast. Q: Does it have an inverse? 2 b = a − n n = a − b b = ap/n n = a/b = n = b p a n logb a n ? b = a4 n = logb a n2 1 2 4 16 65;536 265;536 ≈22:0035×1019;728 Q: Is one better than the other? A: Maybe, you decide: If ::: then ::: , and also ::: . b + n = a b × n = a bn = a nb = a Q: Why should we care? A: Its inverse grows as slow as its self grows fast, and tetration grows real fast: Tetration FAQ Q: How fast does it grow? A: Real fast. Q: Does it have an inverse? A: Yeah, 2—which do you want? 2 b = a − n n = a − b b = ap/n n = a/b = n = b p a n logb a n ? b = a4 n = logb a n2 1 2 4 16 65;536 265;536 ≈22:0035×1019;728 A: Maybe, you decide: If ::: then ::: , and also ::: . b + n = a b × n = a bn = a nb = a Q: Why should we care? A: Its inverse grows as slow as its self grows fast, and tetration grows real fast: Tetration FAQ Q: How fast does it grow? A: Real fast. Q: Does it have an inverse? A: Yeah, 2—which do you want? Q: Is one better than the other? 2 n2 1 2 4 16 65;536 265;536 ≈22:0035×1019;728 b = a − n n = a − b b = ap/n n = a/b = n = b p a n logb a n ? b = a4 n = logb a Q: Why should we care? A: Its inverse grows as slow as its self grows fast, and tetration grows real fast: Tetration FAQ Q: How fast does it grow? A: Real fast. Q: Does it have an inverse? A: Yeah, 2—which do you want? Q: Is one better than the other? A: Maybe, you decide: If ::: then ::: , and also ::: . b + n = a b × n = a bn = a nb = a 2 n2 1 2 4 16 65;536 265;536 ≈22:0035×1019;728 b = ap/n n = a/b = n = b p a n logb a n ? b = a4 n = logb a Q: Why should we care? A: Its inverse grows as slow as its self grows fast, and tetration grows real fast: Tetration FAQ Q: How fast does it grow? A: Real fast. Q: Does it have an inverse? A: Yeah, 2—which do you want? Q: Is one better than the other? A: Maybe, you decide: If ::: then ::: , and also ::: . b + n = a b = a − n n = a − b b × n = a bn = a nb = a 2 n2 1 2 4 16 65;536 265;536 ≈22:0035×1019;728 p = n = b p a n logb a n ? b = a4 n = logb a Q: Why should we care? A: Its inverse grows as slow as its self grows fast, and tetration grows real fast: Tetration FAQ Q: How fast does it grow? A: Real fast. Q: Does it have an inverse? A: Yeah, 2—which do you want? Q: Is one better than the other? A: Maybe, you decide: If ::: then ::: , and also ::: . b + n = a b = a − n n = a − b b × n = a b = a/n n = a/b bn = a nb = a 2 n2 1 2 4 16 65;536 265;536 ≈22:0035×1019;728 p n ? b = a4 n = logb a Q: Why should we care? A: Its inverse grows as slow as its self grows fast, and tetration grows real fast: Tetration FAQ Q: How fast does it grow? A: Real fast. Q: Does it have an inverse? A: Yeah, 2—which do you want? Q: Is one better than the other? A: Maybe, you decide: If ::: then ::: , and also ::: . b + n = a b = a − n n = a − b b × n = a b = ap/n n = a/b n n b = a b = a n = logb a nb = a 2 n2 1 2 4 16 65;536 265;536 ≈22:0035×1019;728 ? n = logb a Q: Why should we care? A: Its inverse grows as slow as its self grows fast, and tetration grows real fast: Tetration FAQ Q: How fast does it grow? A: Real fast. Q: Does it have an inverse? A: Yeah, 2—which do you want? Q: Is one better than the other? A: Maybe, you decide: If ::: then ::: , and also ::: . b + n = a b = a − n n = a − b b × n = a b = ap/n n = a/b n = = n = b a b p a n logb a n n b = a b = a4 2 n2 1 2 4 16 65;536 265;536 ≈22:0035×1019;728 Q: Why should we care? A: Its inverse grows as slow as its self grows fast, and tetration grows real fast: Tetration FAQ Q: How fast does it grow? A: Real fast.
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