A coogol is equal to \(10^{100}\) using hypermathematics.[1] The term was coined by googology101 on 2009/02/12. Hypermathematics is the same as normal mathematics except for one thing: addition is replaced by concatenation: 2+2=22 and 34+75=3475 for example.
A coogol is \((10^{2 \cdot 10^{99}+1} - 10)/99\) in normal notation and approximately E99.301#2 in Hyper-E Notation.
Calculation[]
Hyper-E notation is used. ＋ is a normal notation. From \(m^n\) = \(m\cdot(E[m](n-1))\),[2]
\(10^n\) = \(10 \cdot E(n-1)\) = \(\Sigma_{k=1}^{E(n-1)}E(k+k-1)\) = \((E(E(n-1)+E(n-1)+1) - 10)/99\)
Therefore \(10^n\) in hypermathematics is equal to \((10^{2 \cdot 10^{n-1}+1} - 10)/99\) in normal mathematics, and can be approximated by \(10^{2 \cdot 10^{n-1}-1}\).
From this result, a coogol is \((10^{2 \cdot 10^{99}+1} - 10)/99\) in normal notation and approximated as \(10^{2 \cdot 10^{99}-1} \approx 10^{10^{99.301}} = E99.301\#2 \).
Approximation in other notations[]
| Notation | Approximation |
|---|---|
| Arrow notation | \(\begin{align*}&10\uparrow\uparrow3.3\\\approx&56\uparrow\uparrow3\end{align*}\) |
| Down-arrow notation | \(80\downarrow\downarrow53\) |
| Bowers' Exploding Array Function | \(\{56,3,2\}\) |
| Chained arrow notation | \(56\rightarrow3\rightarrow2\) |
| Hyper-E notation | E99.301#2 |
| Graham Array Notation | \([56,3,2]\) |
| Hyperfactorial array notation | \((68!)!\) |
| Scientific notation | \(18^{55^{57}}\) |
| Steinhaus-Moser Notation | \(56[3][3]\) |
| Strong array notation | \(s(56,3,2)\) |
| Fast-growing hierarchy | \(f_{2}^2(325)\) |
| Hardy hierarchy | \(H_{\omega^{2}\times2}(325)\) |
| Slow-growing hierarchy | \(g_{\omega^{\omega^{57}}}(55)\) |
Note[]
Based on wrong calculation,[3] coogol was approximated as E20#99. It was corrected to E99.301#2 as described in the calculation section.
Sources[]
- ↑ Googol is a tiny dot, Hypermathematics 2009/02/12. retrieved 2022/03/10
- ↑ Hypermathematics#Exponentiation
- ↑ Hypermathematics#Wrong calculation
See also[]
Hypermathematics: bigoogol · trigoogol · quadrigoogol · coogol(plex)
Hyperlicious: wakoogol(plex) · wakamoogol(plex) · wonkapoogol(plex) · ultron
Numbers with a W: woogol · wiggol · waggol · weegol · wigol · woggol · wagol · bwoogol · bwiggol · bwaggol · bweegol · bwigol · bwoggol · bwagol
Primes: Gooprol(plex) · Booprol · Trooprol · Quadrooprol
Other: Bentley's Number · Pigol · Egol · Phigol · gongol(plex) · kaboodol(plex) · gaz(illion)