# nat (unit)

The natural unit of information (symbol: nat),[1] sometimes also nit or nepit, is a unit of information or entropy, based on natural logarithms and powers of e, rather than the powers of 2 and base 2 logarithms, which define the bit. This unit is also known by its unit symbol, the nat. The nat is the coherent unit for information entropy. The International System of Units, by assigning the same units (joule per kelvin) both to heat capacity and to thermodynamic entropy implicitly treats information entropy as a quantity of dimension one, with 1 nat = 1. Physical systems of natural units that normalize the Boltzmann constant to 1 are effectively measuring thermodynamic entropy in nats.

When the Shannon entropy is written using a natural logarithm,

${\displaystyle \mathrm {H} =-\sum _{i}p_{i}\ln p_{i}}$

it is implicitly giving a number measured in nats.

One nat is equal to 1/ln 2 shannons (or bits) ≈ 1.44 Sh or, equivalently, 1/ln 10 hartleys ≈ 0.434 Hart.[1] The factors 1.44 and 0.434 arise from the relationships

${\displaystyle \left(2^{x}=e^{1}\Rightarrow x={\tfrac {1}{\ln 2}}\right)}$, and
${\displaystyle \left(10^{x}=e^{1}\Rightarrow x={\tfrac {1}{\ln 10}}\right)}$.

One nat is the information content of the uniform distribution on the interval [0,e].

## History

Boulton and Wallace used the term nit in conjunction with minimum message length[2] which was subsequently changed by the minimum description length community to nat to avoid confusion with the nit used as a unit of luminance.[3]

Alan Turing used the natural ban.[4]

## References

1. ^ a b "IEC 80000-13:2008". International Electrotechnical Commission. Retrieved 21 July 2013.
2. ^ Boulton, D. M.; Wallace, C. S. (1970). "A program for numerical classification". Computer Journal. 13 (1): 63–69.
3. ^ Comley, J. W. & Dowe, D. L. (2005). "Minimum Message Length, MDL and Generalised Bayesian Networks with Asymmetric Languages". In Grünwald, P.; Myung, I. J. & Pitt, M. A. (eds.). Advances in Minimum Description Length: Theory and Applications. Cambridge: MIT Press. sec. 11.4.1, p271. ISBN 0-262-07262-9.
4. ^