Measuring large numbers - Revision history

Sligocki: Add Guides category (and remove from Stubs, it seems quite a full article)

2026-11-10T01:53:00Z

Add Guides category (and remove from Stubs, it seems quite a full article)

← Older revision		Revision as of 01:53, 10 November 2026
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	~~{{Stub}}~~
	The busy beaver functions are known to grow faster than any computable function, as do many of their variants. Because of this, values of the function, or lower bounds on the function can be very, very large, so it is tempting to find ways of representing these numbers; this article will cover these methods.		The busy beaver functions are known to grow faster than any computable function, as do many of their variants. Because of this, values of the function, or lower bounds on the function can be very, very large, so it is tempting to find ways of representing these numbers; this article will cover these methods.

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	== References ==		== References ==
			<references />
	[[Category:~~Stub~~]]		[[Category:Guides]]

Loader3229: /* Comparing FGH and BB */

2026-11-09T11:50:05Z

Comparing FGH and BB

← Older revision		Revision as of 11:50, 9 November 2026
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	\| <math>\omega+2</math>\|\| 10 \|\| 12 \|\| 12<ref>Jeff Ketchersid, "[https://github.com/jeff87654/busy-beaver/blob/main/BB12/1RB1RA_1RI1RC_0LD1LG_0RE1LD_1LK1RF_1RG0RA_1RA0LH_1LG1LH_1RE0LJ_1LH1LI_1RZ0LL_0LH1LK.md M12: a 12-state machine at level f_(ω+2)]". Github file, October 2026. Accessed 27 October 2026.</ref>		\| <math>\omega+2</math>\|\| 10 \|\| 12 \|\| 12<ref>Jeff Ketchersid, "[https://github.com/jeff87654/busy-beaver/blob/main/BB12/1RB1RA_1RI1RC_0LD1LG_0RE1LD_1LK1RF_1RG0RA_1RA0LH_1LG1LH_1RE0LJ_1LH1LI_1RZ0LL_0LH1LK.md M12: a 12-state machine at level f_(ω+2)]". Github file, October 2026. Accessed 27 October 2026.</ref>
	\|-		\|-
	\| <math>\omega\times 2</math>\|\| 10 \|\| 14 \|\| 20		\| <math>\omega\times 2</math>\|\| 10 \|\| 14 \|\| 15
	\|-		\|-
	\| <math>\omega^2</math>\|\| 10 \|\| 20 \|\| 21		\| <math>\omega^2</math>\|\| 10 \|\| 20 \|\| 21

Loader3229: /* Comparing FGH and BB */

2026-11-07T03:33:48Z

Comparing FGH and BB

← Older revision		Revision as of 03:33, 7 November 2026
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	\| <math>\psi_0(\varepsilon_{\Omega_\omega+1})</math>\|\| \|\| \|\|		\| <math>\psi_0(\varepsilon_{\Omega_\omega+1})</math>\|\| \|\| \|\|
	\|-		\|-
	\| <math>\psi_0(\Omega_{\Omega_{\Omega_{...}}})</math>\|\| \|\| \|\|		\| <math>\psi_0(\Omega_{\Omega_{\Omega_{...}}})</math> (Extended Buchholz's psi function)\|\| \|\| \|\|
	\|-		\|-
	\| <math>lim(BMS)</math> (Bashicu matrix system)\|\| \|\| \|\| 150		\| <math>lim(BMS)</math> (Bashicu matrix system)\|\| \|\| \|\| 150

Loader3229: /* Comparing FGH and BB */

2026-11-07T03:33:14Z

Comparing FGH and BB

← Older revision		Revision as of 03:33, 7 November 2026
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	\|-		\|-
	\| <math>\varepsilon_0</math> (Epsilon ordinals)\|\| 10 \|\| 31 \|\| 48<ref>https://discord.com/channels/960643023006490684/1331570843829932063/1567585091172040726</ref>		\| <math>\varepsilon_0</math> (Epsilon ordinals)\|\| 10 \|\| 31 \|\| 48<ref>https://discord.com/channels/960643023006490684/1331570843829932063/1567585091172040726</ref>
			\|-
			\| <math>\varepsilon_1</math>\|\| 48 \|\| 48 \|\|
			\|-
			\| <math>\zeta_0</math>\|\| \|\| \|\|
	\|-		\|-
	\| <math>\varphi(\omega , 0)</math> (Veblen's phi function)\|\| \|\| \|\|		\| <math>\varphi(\omega , 0)</math> (Veblen's phi function)\|\| \|\| \|\|

Polygon: /* Comparing FGH and BB */ 51--> 48 for f_e_0(10)

2026-11-05T11:43:48Z

Comparing FGH and BB: 51--> 48 for f_e_0(10)

← Older revision		Revision as of 11:43, 5 November 2026
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	\| <math>\omega^\omega+1</math>\|\| 31 \|\| 31 \|\| 31		\| <math>\omega^\omega+1</math>\|\| 31 \|\| 31 \|\| 31
	\|-		\|-
	\| <math>\varepsilon_0</math> (Epsilon ordinals)\|\| 10 \|\| 31 \|\| 51<ref>https://discord.com/channels/960643023006490684/~~1026577255754903572~~/~~1276881449685094495</ref><ref>Sligocki, "[[Talk:Champions#BB(51)_epsilon_0_champion\|BB(51) epsilon_0 champion]]". bbchallenge wiki talk page edit, September 2024. Accessed 7 October 2026.~~</ref>		\| <math>\varepsilon_0</math> (Epsilon ordinals)\|\| 10 \|\| 31 \|\| 48<ref>https://discord.com/channels/960643023006490684/1331570843829932063/1567585091172040726</ref>
	\|-		\|-
	\| <math>\varphi(\omega , 0)</math> (Veblen's phi function)\|\| \|\| \|\|		\| <math>\varphi(\omega , 0)</math> (Veblen's phi function)\|\| \|\| \|\|

Sligocki: /* Fast-growing hierarchy */ Tidy up fast-growing hierarchy

2026-11-05T06:53:05Z

Fast-growing hierarchy: Tidy up fast-growing hierarchy

← Older revision		Revision as of 06:53, 5 November 2026
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	== Fast-growing hierarchy ==		== Fast-growing hierarchy ==
	The fast-growing hierarchy uses ~~the fundamental sequences of ordinals to define diagonalizations~~ across previous functions in the ~~hierarchy~~. Functions in the fast-growing hierarchy are written in the form <math>f_\alpha(n)</math>, and are calculated according to three rules:		The [[fast-growing hierarchy]] uses function iteration and diagonalization across previous functions to extend far beyond the Ackermann function. Functions in the fast-growing hierarchy are written in the form <math>f_\alpha(n)</math>, and are calculated according to three rules:

	<math>f_0(n) = n + 1</math>		<math>f_0(n) = n + 1</math>

Sligocki: /* Knuth's up-arrow notation */ Tidy up knuth up-arrow section

2026-11-05T06:48:39Z

Knuth's up-arrow notation: Tidy up knuth up-arrow section

← Older revision		Revision as of 06:48, 5 November 2026
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	<math>a \uparrow \uparrow b =\begin{cases}		<math>a \uparrow \uparrow b =\begin{cases}
	b + 1, & \text{if } b \leq 0,		b + 1, & \text{if } -1 < b \leq 0,
	\\ a^{a \uparrow \uparrow (b - 1)}, & \text{otherwise.}		\\ a^{a \uparrow \uparrow (b - 1)}, & \text{otherwise.}
	\end{cases}</math>		\end{cases}</math>
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	Note that when using this method, the function <math>10 \uparrow\uparrow x</math>, and other functions similar to it, are not analytic functions; however, that requirement is unnecessary when using this method to represent large numbers. Additionally, methods of obtaining analytic hyperoperators are complex, typically involving Abel functions, Riemann mappings, or Jabotinsky matrices; this method forgoes this, allowing for the ease of calculation.		Note that when using this method, the function <math>10 \uparrow\uparrow x</math>, and other functions similar to it, are not analytic functions; however, that requirement is unnecessary when using this method to represent large numbers. Additionally, methods of obtaining analytic hyperoperators are complex, typically involving Abel functions, Riemann mappings, or Jabotinsky matrices; this method forgoes this, allowing for the ease of calculation.

	In this format, the lower bound for <math>S(2,5)</math> can ~~instead~~ be written as <math>S(2,5) > 10 \uparrow\uparrow 4.8142</math>.		In this format, the lower bound for <math>S(2,5)</math> can be written more precisely as <math>S(2,5) > 10 \uparrow\uparrow 4.8142</math>.


	~~In popular mathematics<!-- This is an oxymoron, as mathematics is not popular; please find a better way to state this. -->,~~ Graham's number (<math>= g_{64}</math>) is ~~represented~~ using ~~the following recursive function involving~~ Knuth's up-arrow notation:		[[Graham's number]] (<math>g_{64}</math>) is a famously large number that is defined using Knuth's up-arrow notation:

	<math>g_n = \begin{cases}		<math>g_n = \begin{cases}
	3 \uparrow \uparrow \uparrow \uparrow 3, & \text{if }n = 1		3 \uparrow \uparrow \uparrow \uparrow 3, & \text{if }n = 1
	\\ 3 \uparrow^{g_{n - 1}} 3, & \text{~~otherwise.~~}		\\ 3 \uparrow^{g_{n - 1}} 3, & \text{if } n > 1
	\end{cases}</math>		\end{cases}</math>

Sligocki: /* Scientific notation */ Tidy up sci notation section

2026-11-05T06:42:25Z

Scientific notation: Tidy up sci notation section

← Older revision		Revision as of 06:42, 5 November 2026
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	== Scientific notation ==		== Scientific notation ==
	Scientific notation is one of the most common ways of representing ~~exceedingly~~ large numbers; numbers written in standard scientific notation symbolically consist of a multiplicand (taking a value between 1 and the base, though is sometimes omitted), a base (commonly 10) and an ~~index~~ (usually an integer, though sometimes takes non-integer values if the multiplicand is omitted).		Scientific notation is one of the most common ways of representing large numbers; numbers written in standard scientific notation symbolically consist of a multiplicand (taking a value between 1 and the base, though is sometimes omitted), a base (commonly 10) and an exponent (usually an integer, though sometimes takes non-integer values if the multiplicand is omitted).

	For example, <math>8.68 \cdot 10^6</math> represents the number 8,680,000, whereas <math>6.022 \cdot 10^{23}</math> represents the number 602,200,000,000,000,000,000,000.		For example, <math>8.68 \cdot 10^6</math> represents the number 8,680,000, whereas <math>6.022 \cdot 10^{23}</math> represents the number 602,200,000,000,000,000,000,000. These numbers are often approximations (to some number of digits) or lower/upper bounds on the precise integer.

	~~In cases where the index is not an integer, the representation usually represents an irrational number, though sometimes it can represent a rational number, or even an~~ integer.		Non-integer exponents are also often used to approximate. For example, <math>10^{7.67}</math> representing a lower bound for the number 46,773,515.

	For example, <math>10^{7.67}</math> ~~represents~~ the number 46,773,~~514.1287..., an irrational number, whereas <math>16^{6.75}</math> represents the number 134,217,728, an integer~~.


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	When representing ~~certain exceedingly large~~ numbers using scientific notation, the ~~index~~ itself may also be represented using scientific notation.		When representing even larger numbers using scientific notation, the exponent itself may also be represented using scientific notation. For example:

	<math>S(2,5) > 10^{10^{10^{3,314,360}}}</math>		<math>S(2,5) > 10^{10^{10^{3,314,360}}}</math>

Polygon: /* Comparing FGH and BB */ Updated with Jacobzhengs new BB(20) champion

2026-11-04T14:54:44Z

Comparing FGH and BB: Updated with Jacobzhengs new BB(20) champion

← Older revision		Revision as of 14:54, 4 November 2026
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	\| <math>\omega+2</math>\|\| 10 \|\| 12 \|\| 12<ref>Jeff Ketchersid, "[https://github.com/jeff87654/busy-beaver/blob/main/BB12/1RB1RA_1RI1RC_0LD1LG_0RE1LD_1LK1RF_1RG0RA_1RA0LH_1LG1LH_1RE0LJ_1LH1LI_1RZ0LL_0LH1LK.md M12: a 12-state machine at level f_(ω+2)]". Github file, October 2026. Accessed 27 October 2026.</ref>		\| <math>\omega+2</math>\|\| 10 \|\| 12 \|\| 12<ref>Jeff Ketchersid, "[https://github.com/jeff87654/busy-beaver/blob/main/BB12/1RB1RA_1RI1RC_0LD1LG_0RE1LD_1LK1RF_1RG0RA_1RA0LH_1LG1LH_1RE0LJ_1LH1LI_1RZ0LL_0LH1LK.md M12: a 12-state machine at level f_(ω+2)]". Github file, October 2026. Accessed 27 October 2026.</ref>
	\|-		\|-
	\| <math>\omega\times 2</math>\|\| 10 \|\| 14 \|\| 21		\| <math>\omega\times 2</math>\|\| 10 \|\| 14 \|\| 20
	\|-		\|-
	\| <math>\omega^2</math>\|\| 10 \|\| 21 \|\| 21		\| <math>\omega^2</math>\|\| 10 \|\| 20 \|\| 21
	\|-		\|-
	\| <math>\omega^\omega</math>\|\| 10 \|\| 29 \|\| 31		\| <math>\omega^\omega</math>\|\| 10 \|\| 29 \|\| 31

Loader3229: /* Comparing FGH and BB */

2026-10-30T00:48:17Z

Comparing FGH and BB

← Older revision		Revision as of 00:48, 30 October 2026
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	\| <math>\omega^2</math>\|\| 10 \|\| 21 \|\| 21		\| <math>\omega^2</math>\|\| 10 \|\| 21 \|\| 21
	\|-		\|-
	\| <math>\omega^\omega</math>\|\| 10 \|\| 40 \|\| 40		\| <math>\omega^\omega</math>\|\| 10 \|\| 29 \|\| 31
	\|-		\|-
	\| <math>\omega^\omega+1</math>\|\| 41 \|\| 41 \|\| 51		\| <math>\omega^\omega+1</math>\|\| 31 \|\| 31 \|\| 31
	\|-		\|-
	\| <math>\varepsilon_0</math> (Epsilon ordinals)\|\| 10 \|\| 51 \|\| 51<ref>https://discord.com/channels/960643023006490684/1026577255754903572/1276881449685094495</ref><ref>Sligocki, "[[Talk:Champions#BB(51)_epsilon_0_champion\|BB(51) epsilon_0 champion]]". bbchallenge wiki talk page edit, September 2024. Accessed 7 October 2026.</ref>		\| <math>\varepsilon_0</math> (Epsilon ordinals)\|\| 10 \|\| 31 \|\| 51<ref>https://discord.com/channels/960643023006490684/1026577255754903572/1276881449685094495</ref><ref>Sligocki, "[[Talk:Champions#BB(51)_epsilon_0_champion\|BB(51) epsilon_0 champion]]". bbchallenge wiki talk page edit, September 2024. Accessed 7 October 2026.</ref>
	\|-		\|-
	\| <math>\varphi(\omega , 0)</math> (Veblen's phi function)\|\| \|\| \|\|		\| <math>\varphi(\omega , 0)</math> (Veblen's phi function)\|\| \|\| \|\|