Universal Number

The Universal Number

(As of current knowledge) every possibly measurable quantity can be represented with $10^{134}$ bytes (or bits).

Why?

First, I would like to make some assumptions! Let’s same you make a measurement to distinguish the physical states of the world.

For example, a potato could of many lengths! We use rulers to distinguish which universe¹ we are in such that the potato is so long. Effectively, the smallest length increment is the Planck length. This is about $1.6\times10^{-35}\mathrm{m}$ The diameter of the observable universe ² is $8.8\times10^{26}\mathrm m$. If we made a ruler this long which every decrement link one Planck length apart, the number of possible potato lengths is $8.8\times10^{26}\mathrm m/ (1.6\times10^{-35}\mathrm{m}) \approx 14\times 10^{61}$ Big number but some we can handle 😉. The number of bytes to represent such number of states is $n$ such that $\left(2^8\right)^n = 10^{ 63 }$. ³ This is because a byte is $8$ bits so $2^8$ is the number of states one byte could represent. So $n = \log_{2^8}\left(10^{ 63 }\right) = 63\log_{256}(10) \approx 25.99 \approx 26$ So, $26$ bytes for all possible potatoes.

In order to find the number of possible measurable quantitites, the idea is to find the number of states possible and then take the $\log_{256}$ of it.

Let’s say the maximum number if states you can contain in a given surface area is bounded by the entropy of a black hole . That is $S_{BH}/A = k_B/(4\ell^2_P)$ The number of microstates (which is the number of possible states) is given by $$\Omega = \exp(S_{BH}/k_{B}) = \exp( {1.253×10^{ 80 }/\mathrm m^2 \times A} ).$$ Given that the observable universe has a surface area of $2.43\times54\mathrm m^2$ and so ${\ln\Omega\approx3.04\times10^{134}}$. $\Omega$ is big number. $\ln \Omega$ can be seen as the number of natural bits, but we want in terms of bytes. So, we just need to change bases to find $\log_{256} \Omega$. This is just $\log_{256} \Omega = \ln\Omega/\ln 256 \approx 10^{134}$. This number is so big because it’s the number of every possible* measurement.