# nat (unit)

Short description: Unit of information

The natural unit of information (symbol: nat),[1] sometimes also nit or nepit, is a unit of information or information entropy, based on natural logarithms and powers of e, rather than the powers of 2 and base 2 logarithms, which define the shannon. This unit is also known by its unit symbol, the nat. One nat is the information content of an event when the probability of that event occurring is 1/e.

One nat is equal to 1/ln 2 shannons ≈ 1.44 Sh or, equivalently, 1/ln 10 hartleys ≈ 0.434 Hart.[1]

## 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]

## Entropy

Shannon entropy (information entropy), being the expected value of the information of an event, is inherently a quantity of the same type and with a unit of information. The International System of Units, by assigning the same unit (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.[lower-alpha 1] Systems of natural units that normalize the Boltzmann constant to 1 are effectively measuring thermodynamic entropy with the nat as unit.

When the shannon entropy is written using a natural logarithm, $\displaystyle{ \Eta = - \sum_i p_i \ln p_i }$ it is implicitly giving a number measured in nats.

## Notes

1. This implicitly also makes the nat the coherent unit of information in the SI.

## References

1. Boulton, D. M.; Wallace, C. S. (1970). "A program for numerical classification". Computer Journal 13 (1): 63–69. doi:10.1093/comjnl/13.1.63.
2. 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.. Advances in Minimum Description Length: Theory and Applications. Cambridge: MIT Press. sec. 11.4.1, p271. ISBN 0-262-07262-9.
3. Hodges, Andrew (1983). Alan Turing: The Enigma. New York City: Simon & Schuster. ISBN 0-671-49207-1. OCLC 10020685.