### Nuclear Benford

July 2, 2012 Surprisingly, I've never written an article about one of the most fascinating natural laws, Benford's law. The law says that in most number lists (see exceptions, below), the digit, one, will appear with a frequency of about 30%, and not the "expected" 11%. This first digit law was discovered by the astronomer, Simon Newcomb, in 1881. In those days before ubiquitous computing, logarithmic tables were often used in multiplication and division. Newcomb found that such books were most worn at the beginning, where the ones were. Newcomb may have discovered this law, but it's named after optical physicist, Frank Benford. Benford published a thorough examination of this effect in 1938. A table of his findings appears below.[1]*Table I from "The law of anomalous numbers," by Frank Benford, Proceedings of the American Philosophical Society, vol. 78, no. 4 (March, 1938), p. 553. (Via Google Books))*

Benford's law contains predictions for the probability that other numbers will take first place. The percentages, based on an idealized statement of the law, P = log

_{10}(1 + (1/n)), appear in the following table and graph.

Digit | % | Digit | % | |

1 | 30.1 | 6 | 6.7 | |

2 | 17.6 | 7 | 5.8 | |

3 | 12.5 | 8 | 5.1 | |

4 | 9.7 | 9 | 4.6 | |

5 | 7.9 |

*Probability that a particular digit will occupy the first place in a number.(Graph rendered with Gnumeric)*

It's easy to see how this law is possible. If you start counting (1, 2, 3, 4, 5... etc.), you start with a 100% probability of one being the first digit. This decreases to 50% when you reach two, and 11.1% when you reach nine. Then, the probability increases to 52.6% up to nineteen. This behavior continues with larger numbers. The best explanation of Benford's law is scale invariance. We measure things in nature with a particular yardstick, but the yardstick can be divided into inches, centimeters, or some other unit. No matter what the unit is, the distribution of our data points won't change. The only thing that changes is how we look at them. The only distribution that has such a property is the logarithmic distribution expressed by Benford's law. Some datasets do not follow Benford's law. It's easy to see that normally distributed data, such as IQ scores, will not. In IQ scores, one as a first digit should occur about half the time, with 9 and 8 having a much higher probability than their Benford values, and 2, 3 and 4 having a first digit probability of about zero. One interesting use of Benford's law is for forensic accounting. If someone is "cooking the books," their data entries would likely be random, and they won't obey Benford's law. However, in this modern age, a competent crook would likely program a spreadsheet to generate numbers with a Benford distribution. Not surprisingly, mathematically-generated number lists, such as the powers of numbers, the Fibonacci numbers, and factorials, fit Benford's law quite exactly. One interesting thing about the table in Benford's paper is the entry for "X-ray Volts," which is presumably the voltage needed to excite the various X-ray lines in the elements (his text doesn't explain this). It can be seen that this quantum process follows Benford's law quite closely. Physicists have found that many nuclear processes follow Benford's law.[2-6] It was found that the half-life of 2059 β-decays demonstrate Benford's law.[2] An analysis of the half-life of 2298 nuclear decays by all modes, including α-decay, follows the law quite closely, as shown in the figure.[3-4] The authors of the study state, however, that this is just an interesting result that gives no insight into the mechanism of nuclear decay.

*Benford's law for 2298 nuclear decay half-lives, using data from refs. 3-4.The columns show the data, and the error bars correspond to what would be expected from Benford's law for the sample size.(Graph rendered with Gnumeric)*

### References:

- Frank Benford, "The law of anomalous numbers". Proceedings of the American Philosophical Society, vol. 78, no. 4 (March, 1938), pp. 551–572 (via Google Books).
- Dong-Dong Ni, Lai Wei and Zhong-Zhou Ren, "Nuclear Physics: Benford's Law and β-Decay Half-Lives," Communications in Theoretical Physics, vol. 51, no. 4 (April, 2009), pp. 713-716.
- Janos Farkas and Gyorgy Gyurky, "The significance of using the Newcomb-Benford law as a test of nuclear half-life calculations," arXiv Preprint Server, June 18, 2010.
- János Farkas and György Gyürky, "The Significance Of Using The Newcomb–Benford Law As A Test Of Nuclear Half-Life Calculations," Acta Physica Polonica B, vol. 41, no. 6 (June, 2010), pp. 1213-1221.
- B. Buck, A.C. Merchant and S.M. Perez, "An Illustration of Benford's first digit law using alpha decay half lives, Eur. J. Phys., vol. 14, no. 2 (March, 1993), pp. 59-63); PDF copy available, here.
- Charles G. Wohl, "Benford's Law"

*Permanent Link to this article*

Linked Keywords: Physical law; natural law; Benford's law; number; digit; frequency; astronomer; Simon Newcomb; ubiquitous computing; logarithmic tables; multiplication; division; optical physicist; Frank Benford; Google Books; probability; Gnumeric; counting; scale invariance; yardstick; inch; centimeter; distribution; normal distribution; normally distributed; intelligence quotient; IQ score; forensic accounting; cooking the books; randomness; random; criminal; crook; spreadsheet; mathematics; mathematically-generatedvexponentiation; powers of numbers; Fibonacci numbers; factorial; voltage; X-ray spectroscopy; X-ray lines; chemical element; quantum mechanics; physicist; nuclear; half-life; β-decay; α-decay; error bar.

### Google Search

Free Downloads:

STEM artwork

for your holiday gifts

Latest Books by Dev Gualtieri

Thanks to Cory Doctorow of BoingBoing for his favorable review of Secret Codes!

Other Books

- 3D-printing Stainless Steel - December 11, 2017

- Fingerprints - December 4, 2017

- Collision Avoidance - November 27, 2017

- Ultra-pure Green Light - November 20, 2017

- High Energy Cosmic Rays - November 13, 2017

- Advanced Aluminum Alloys - November 6, 2017

- Joseph Polchinski - October 30, 2017

- Our Magnetic Universe - October 23, 2017

- Cavitation - October 16, 2017

- Pell Numbers - October 9, 2017

- Miniature Antennas - October 2, 2017

- Fizzy Graphene - September 25, 2017

- The First Angiosperm - September 18, 2017

- Noise Thermometry and the Boltzmann Constant - September 11, 2017

- Walking in the Rain - September 4, 2017

- Agitated Atoms - August 28, 2017

- Partial Solar Eclipse at New Jersey - August 24, 2017

- Magnetocapacitive Tunnel Junctions - August 21, 2017

- Tardigrades - August 14, 2017

- Roman Concrete - August 7, 2017

- Solar Spicules - July 31, 2017

- Schroeder Diffuser - July 24, 2017

- Rough Microparticles - July 17, 2017

- Robot Musicians - July 10, 2017

- Walter Noll (1925-2017) - July 6, 2017

- cosmogony - July 3, 2017

- Crystal Prototypes - June 29, 2017

- Voice Synthesis - June 26, 2017

- Refining Germanium - June 22, 2017

- Granular Capillarity - June 19, 2017

- Kirchhoff–Plateau Problem - June 15, 2017

- Self-Assembly - June 12, 2017

- Physics, Math, and Sociology - June 8, 2017

- Graphene from Ethylene - June 5, 2017

- Crystal Alignment Forces - June 1, 2017

- Martian Brickwork - May 29, 2017

- Carbon Nanotube Textile - May 25, 2017

- The Scent of Books - May 22, 2017

- Patterns from Randomness - May 18, 2017

- Terpene - May 15, 2017

- The Physics of Inequality - May 11, 2017

- Asteroid 2015 BZ509 - May 8, 2017

- Fuzzy Fibers - May 4, 2017

- The Sofa Problem - May 1, 2017

### Deep Archive

Deep Archive 2006-2008

**Blog Article Directory on a Single Page**