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### Benford’s Law

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He conjectured a simple formula: nature seems to have a tendency to arrange numbers so that the proportion starting with the digit D is equal to “log10 of 1 + (1/D).”

Newcomb`s observations were then virtually ignored until 57 years later when Frank Benford published his paper. (Benford, F. “The Law of Anomalous Numbers.” Proc. Amer. Phil. Soc. 78, 551-572, 1938). He rediscovered the phenomena and came up with the same law as Newcomb. Conducting a monumental research, he analyzed 20229 set of numbers gathered from everything from listings of the areas of rivers to physical constants and death rates, he showed that they all adhere to the same law: around 30.1 per cent began with the digit 1, 17.6 per cent with 2, 12.5 per cent with 3, 9.7 per cent with 4, 7.9 percent with 5, 6.7 percent with 6, 5.8 per cent with 7, 5.1 percent with 8 and 4.6 percent with 9.

Benford’s law is scale-invariant (the distribution of digits is unaffected by changes of units) and base-invariant. In fact in 1995, 114 years after Newcomb’s discovery, Theodore Hill, proved that any universal law of digit distributions that is base invariant has to take the form of Benford’s law (“Base invariance implies Benford’s law”, Proceedings of the American Math. Society, vol 123, p 887).

In applying Benford’s law three rules should be observed: first the sample size should be big enough to give the predicted proportions a chance to show themselves so you will not find Benford’s law in the ages of your family of 5 people. Second, the numbers should be free of artificial limits so

obviously you cannot expect the telephone numbers in your neighborhood follow Benford’s law. Third, you don’t want numbers that are truly random. By definition, in a random number, every digit from 0 to 9 has an equal chance of appearing in any position in that number.

### An excellent fraud-buster

This fascinating mathematical theorem is a powerful and relatively

 GROUP 1 30 suras whose no.of verses start with 1 Sura No. No. of Verses 1 4 176 2 5 120 3 6 165 4 9 127 5 10 109 6 11 123 7 12 111 8 16 128 9 17 111 10 18 110 11 20 135 12 21 112 13 23 118 14 37 182 15 49 18 16 60 13 17 61 14 18 62 11 19 63 11 20 64 18 21 65 12 22 66 12 23 82 19 24 86 17 25 87 19 26 91 13 27 93 11 28 96 19 29 100 11 30 101 11

simple tool for pointing suspicion at frauds, embezzlers, tax evaders and sloppy accountants.

The income tax agencies of several nations and several states, have started using detection software based on Benford's Law to detect fabrication of data in financial documents and income tax returns.

The idea is that if the numbers in a set of data like sales figures, buying and selling prices, insurance claim costs and expenses claims, more or less match the frequencies and ratios predicted by Benford's Law, the data are probably honest. But if a graph of such numbers is markedly different from the one predicted by Benford's Law, it arouses suspicion of fraud.

### Application to the Quran

The Quran is divided into chapters of unequal length, each of which is called a sura.
The shortest of the suras has ten words, and the longest sura, which is placed second in the text, has over 6000 words. From the second sura onward, the suras gradually get shorter, although this is not a hard and fast rule. The last sixty suras take up about as much space as the second sura. This unconventional structure does not follow people’s expectations as to what a book should be. However it appears to be a deliberate design on part of the author of the Quran. Let’s verify the evidence:

Quran consists of 114 suras. Each sura is composed of certain number of verses, for example sura 1 has 7 verses and sura 96 (the first sura revealed to prophet Muhammad ) has 19 verses. So we have a set of 114 data to which we can apply the Benford’s law. The first table showing Group 1 with 30 suras is shown on the left:

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