Ballium's number is a number coined in a parody YouTube video by Meerkats Anonymous, in which a fictional physicist Samuel Ballium claims it to be "the largest number."[1][2]According to the video, Ballium's number is exactly

\[(794{,}843{,}294{,}078{,}147{,}843{,}293.7 + 1/30) \cdot e^{\pi^{e^\pi}}\]

In the video, a spoof on newscasts about scientific findings, Ballium explains how he accidentally added "Hamlet" to an equation he was working on and stumbled upon this number. The video also goes on to say that the "Microsoft calculator struggles to produce an output, even when set to scientific mode."

The number is in fact smaller than a googolplex. This can be shown easily enough, by rounding \(e\) and \(\pi\) up to the next nearest integer and replacing the first component with \(10^{21}\):

\[\text{Ballium's number} < 10^{21} \cdot 3^{4^{3^4}} = 10^{21} \cdot 3^{4^{81}} = 10^{21} \cdot 3^{2^{162}} = 10^{21} \cdot 3^{10^{162 \log_{10} 2}} < 10^{21} \cdot 3^{10^{162 \cdot 0.4}} = \]

\[10^{21} \cdot 3^{10^{64.8}} < 10^{21} \cdot 10^{10^{64.8}} = 10^{21 + 10^{64.8}} < 10^{10^{64.8} + 10^{64.8}} < 10^{10^{65.8}} < 10^{10^{66}}\]

This is less than \(10^{10^{100}}\), so Ballium's number is smaller than googolplex.

In fact, \(10^{10^{66}}\) is a very generous upper bound. The actual value is closer to \(10^{10^{11}}\). It can be shown that:

\[10^{10^{11}} < \text{Ballium's number} < 10^{10^{12}}\]

Ballium's number contains roughly 138 billion digits before the decimal point, storage of which is possible on today's computers. However, the process of computing those digits exactly may be impractically long. The exact number of digits before the decimal point is 138,732,019,350.

Ballium's number is a typical example of common attempts to name a very large number. Its form seems to be largely inspired by Skewes' number, but it fails to be very large, mainly because the topmost exponent of the second component is too small.

Ballium's number is also an example of ultrafinitism.


Notation Lower bound Upper bound
Scientific notation \(2\times10^{138\,732\,019\,349}\) \(3\times10^{138\,732\,019\,349}\)
Arrow notation \(2\,571\uparrow40\,682\,658\,827\) \(245\uparrow58\,067\,130\,727\)
Down-arrow notation \(477\downarrow\downarrow5\) \(478\downarrow\downarrow5\)
Steinhaus-Moser Notation 10[3][3] 11[3][3]
Copy notation 1[1[12]] 2[2[12]]
H* function H(46H(2)) H(47H(2))
Taro's multivariable Ackermann function A(3,A(3,35)) A(3,A(3,36))
Pound-Star Notation #*((1))*(4)*7 #*((1))*(34)*3
BEAF {2571,40682658827} {245,58067130727}
Hyper-E notation 2E138,732,019,349 3E138,732,019,349
Bashicu matrix system (0)(1)[5] (0)(1)[6]
Hyperfactorial array notation (13!)! (14!)!
Fast-growing hierarchy \(f_2(f_2(33))\) \(f_2(f_2(34))\)
Hardy hierarchy \(H_{\omega^22}(33)\) \(H_{\omega^22}(34)\)
Slow-growing hierarchy \(g_{\omega^{\omega^{\omega+1}+\omega^\omega\times 3}}(10)\) \(g_{\omega^{\omega^{\omega+1}+\omega^\omega\times 4}}(10)\)

Notes and references

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