Chin. Phys. Lett.  2019, Vol. 36 Issue (7): 070201    DOI: 10.1088/0256-307X/36/7/070201
A Proof of First Digit Law from Laplace Transform
Mingshu Cong1, Bo-Qiang Ma1,2**
1School of Physics and State Key Laboratory of Nuclear Physics and Technology, Peking University, Beijing 100871
2Center for High Energy Physics, Peking University, Beijing 100871
Cite this article:   
Mingshu Cong, Bo-Qiang Ma 2019 Chin. Phys. Lett. 36 070201
Download: PDF(484KB)   PDF(mobile)(481KB)   HTML
Export: BibTeX | EndNote | Reference Manager | ProCite | RefWorks
Abstract The first digit law, also known as Benford's law or the significant digit law, is an empirical phenomenon that the leading digit of numbers from real world sources favors small ones in a form $\log(1+{1}/{d})$, where $d=1, 2,\ldots, 9$. Such a law has been elusive for over 100 years because it has been obscure whether this law is due to the logical consequence of the number system or some mysterious mechanism of nature. We provide a simple and elegant proof of this law from the application of the Laplace transform, which is an important tool of mathematical methods in physics. It is revealed that the first digit law originates from the basic property of the number system, thus it should be attributed as a basic mathematical knowledge for wide applications.
Received: 22 March 2019      Published: 20 June 2019
PACS:  02.30.Uu (Integral transforms)  
  02.50.-r (Probability theory, stochastic processes, and statistics)  
  02.50.Cw (Probability theory)  
Fund: Supported by the National Natural Science Foundation of China under Grant No 11475006.
URL:       OR
E-mail this article
E-mail Alert
Articles by authors
Mingshu Cong
Bo-Qiang Ma
[1]Newcomb S 1881 Am. J. Math. 4 39
[2]Benford F 1938 Proc. Am. Philos. Soc. 78 551
[3]Burke J and Kincanon E 1991 Am. J. Phys. 59 952
[4]Ley E 1996 Am. Stat. 50 311
[5]Torres J et al 2007 Eur. J. Phys. 28 L17
[6]Leemis L M et al 2000 Am. Stat. 54 236
[7]Brown R J C 2005 Analyst 130 1280
[8]Costas E et al 2008 Aquat. Bot. 89 341
[9]Nigrini M and Miller S J 2007 Math. Geol. 39 469
[10]Shao L and Ma B Q 2010 Astropart. Phys. 33 255
[11]Shao L and Ma B Q 2010 Physica A 389 3109
[12]Shao L and Ma B Q 2010 Phys. Rev. E 82 041110
[13]Buck B et al 1993 Eur. J. Phys. 14 59
[14]Ni D and Ren Z 2008 Eur. Phys. J. A 38 251
[15]Ni D D et al 2009 Commun. Theor. Phys. 51 713
[16]Liu X J et al 2011 Eur. Phys. J. A 47 78
[17]Jiang H et al 2011 Chin. Phys. Lett. 28 032101
[18]Shao L and Ma B Q 2009 Mod. Phys. Lett. A 24 3275
[19]Tolle C R et al 2000 Chaos 10 331
[20]Berger A et al 2005 Trans. Am. Math. Soc. 357 197
[21]Berger A 2005 Discrete Contin. Dyn. Syst. A 13 219
[22]Nigrini M J 1996 J. Am. Tax. Assoc. 18 72
[23]Nigrini M J 1999 Intern. Auditor 56 21
[24]Rose A M and Rose J M 2003 J. Accountancy 196 58
[25]Diekmann A 2007 J. Appl. Stat. 34 321
[26]Pinkham R S 1961 Ann. Math. Stat. 32 1223
[27]Berger A et al 2008 J. Theor. Probab. 21 97
[28]Hill T P 1995 Am. Math. Mon. 102 322
[29]Hill T P 1995 Proc. Am. Math. Soc. 123 887
[30]Hill T P 1995 Stat. Sci. 10 354
[31]Berger A et al 2009 Benford Online Bibliography
[32]Cong M et al 2019 Phys. Lett. A 383 1836
[33]Marsden J and Hoffman M 1999 Basic Complex Anal. 3rd edn (New York: W. H. Freeman) p 471
[34]Engel H A and Leuenberger C 2003 Stat. Probab. Lett. 63 361
Related articles from Frontiers Journals
[1] Bin Cheng, Ya-Ming Chen, Xiao-Gang Deng. Solution to the Fokker–Planck Equation with Piecewise-Constant Drift *[J]. Chin. Phys. Lett., 0, (): 070201
[2] Bin Cheng, Ya-Ming Chen, Xiao-Gang Deng. Solution to the Fokker–Planck Equation with Piecewise-Constant Drift[J]. Chin. Phys. Lett., 2020, 37(6): 070201
[3] Jun Song, Rui He, Hao Yuan, Jun Zhou, Hong-Yi Fan. Joint Wavelet–Fractional Fourier Transform[J]. Chin. Phys. Lett., 2016, 33(11): 070201
[4] CAO Jian-Zhu, FANG Chao**, SUN Li-Feng . Analytical Solution of Fick's Law of the TRISO-Coated Fuel Particles and Fuel Elements in Pebble-Bed High Temperature Gas-Cooled Reactors[J]. Chin. Phys. Lett., 2011, 28(5): 070201
[5] YANG Jun, SHA Kan, GAN Woon-Seng, YAN Yong-Hong, TIAN Jing. A Simplified Algorithm for Impedance Calculation of Arbitrarily Shaped Radiators[J]. Chin. Phys. Lett., 2005, 22(10): 070201
Full text