4 ms·

And it's even smaller if compared to Graham's number. Every time I try to imagine that one it feels like I'm going to mental asylum.

amelius 10y ago
Related only to large numbers, but is there some theory about generalizing and extending our usual mathematical operators +, ×, and ^ (power)? + applied N times becomes ×N × applied N times becomes ^N ^ applied N times becomes ...? etcetera And would such a theory have any practical use?
- abengoam 10y ago
  https://en.wikipedia.org/wiki/Hyperoperation https://en.wikipedia.org/wiki/Hyperoperation
- NegativeK 10y ago
  Knuth's up-arrow notation is one of a few ways to address the addition->multiplication->exponentiation->tetration->... extensions: https://en.wikipedia.org/wiki/Knuth%27s_up-arrow_notation https://en.wikipedia.org/wiki/Knuth%27s_up-arrow_notation As for practicalities, the mathematician in me will let the scientists deal with that.
  - dack 10y ago
    I like the description of graham's number (by graham) using the up-arrow notation https://youtu.be/GuigptwlVHo?t=31 https://youtu.be/GuigptwlVHo?t=31
- drjesusphd 10y ago
  Graham's number comes from such an extension, and is about 64 such operations deeper. It's useful in the sense that it provides an upper bound for some proof that hasn't yet been proven for ALL numbers. http://waitbutwhy.com/2014/11/1000000-grahams-number.html http://waitbutwhy.com/2014/11/1000000-grahams-number.html
- maze-le 10y ago
  > And would such a theory have any practical use? Yes, sureley. In Computablilty Theory you have the famous Ackermann-function[1]. It is actually an operator-extension, like you just described. It is important, because it grows overexponentially, but is still computable (unlike e.g. the busy-beaver-function). [1]: https://en.wikipedia.org/wiki/Ackermann_function https://en.wikipedia.org/wiki/Ackermann_function
  - infogulch 10y ago
    Given that Graham's Number is an (over-)extension of the Ackermann-function that means that it's still computable, correct? Given the series leading up to Graham's Number, G = g_64 (g_1, g_2, ...), BB(n) will outgrow g_n, right? If so what's the smallest n such that BB(n) > g_n?
    - infogulch 10y ago
      After a bit of research I found that there is a proof [0][1] that BB(12) > g_1. But that's still a ways from g_2, let alone g_64. [0]: "A Lower Bound on Rado's Sigma Function for Binary Turing Machines" by Milton Green (1964) [1]: https://en.wikipedia.org/wiki/Busy_beaver#Known_values_for_.CE.A3_and_S https://en.wikipedia.org/wiki/Busy_beaver#Known_values_for_....
deleted 10y ago
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amelius 10y ago

Related only to large numbers, but is there some theory about generalizing and extending our usual mathematical operators +, ×, and ^ (power)? + applied N times becomes ×N × applied N times becomes ^N ^ applied N times becomes ...? etcetera And would such a theory have any practical use?

abengoam 10y ago
https://en.wikipedia.org/wiki/Hyperoperation https://en.wikipedia.org/wiki/Hyperoperation
NegativeK 10y ago
Knuth's up-arrow notation is one of a few ways to address the addition->multiplication->exponentiation->tetration->... extensions: https://en.wikipedia.org/wiki/Knuth%27s_up-arrow_notation https://en.wikipedia.org/wiki/Knuth%27s_up-arrow_notation As for practicalities, the mathematician in me will let the scientists deal with that.
- dack 10y ago
  I like the description of graham's number (by graham) using the up-arrow notation https://youtu.be/GuigptwlVHo?t=31 https://youtu.be/GuigptwlVHo?t=31
drjesusphd 10y ago
Graham's number comes from such an extension, and is about 64 such operations deeper. It's useful in the sense that it provides an upper bound for some proof that hasn't yet been proven for ALL numbers. http://waitbutwhy.com/2014/11/1000000-grahams-number.html http://waitbutwhy.com/2014/11/1000000-grahams-number.html
maze-le 10y ago
> And would such a theory have any practical use? Yes, sureley. In Computablilty Theory you have the famous Ackermann-function[1]. It is actually an operator-extension, like you just described. It is important, because it grows overexponentially, but is still computable (unlike e.g. the busy-beaver-function). [1]: https://en.wikipedia.org/wiki/Ackermann_function https://en.wikipedia.org/wiki/Ackermann_function
- infogulch 10y ago
  Given that Graham's Number is an (over-)extension of the Ackermann-function that means that it's still computable, correct? Given the series leading up to Graham's Number, G = g_64 (g_1, g_2, ...), BB(n) will outgrow g_n, right? If so what's the smallest n such that BB(n) > g_n?
  - infogulch 10y ago
    After a bit of research I found that there is a proof [0][1] that BB(12) > g_1. But that's still a ways from g_2, let alone g_64. [0]: "A Lower Bound on Rado's Sigma Function for Binary Turing Machines" by Milton Green (1964) [1]: https://en.wikipedia.org/wiki/Busy_beaver#Known_values_for_.CE.A3_and_S https://en.wikipedia.org/wiki/Busy_beaver#Known_values_for_....

deleted 10y ago

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