Notis: Det följande är en reproduktion av artiklen "Benford's Law and the Numerical Structure of the Quran" i Juli-upplagan 2000 av Submitters Perspective, den månatliga bulletinen av United Submitters International.

You shall not accept any information, unless you verify it for yourself. I have given you the hearing, the eyesight, and the brain, and you are responsible for using them. [Quran 17:36]

Henri Poincare, a famous French mathematician of late 19th century, once said “If God speaks to man, He undoubtedly uses the language of mathematics.”

The Quran is intended to be an eternal miracle. A highly sophisticated mathematical system based on prime number 19 was embedded into the fabric of the Quran (decoded between 1969-1974 and onwards with the aid of computers). This system provided verifiable PHYSICAL evidence that “The Book is, without a doubt, a revelation from the Lord of the universe” (32:2), and incontrovertibly ruled out the possibility that it could be the product of a man living in the ignorant Arabian society of the 7th century. It also proved that no falsehood could enter into the Quran, as promised by God.

To ascertain that they fully delivered their Lord's messages, He protectively enveloped what He entrusted them with and He counted the numbers of all things. 72:28 (7+2+2+8=19)

Furthermore the mathematical miracle of the Quran shed new light on the exceptional style and structure of the book. Here, we will look into one of these aspects through Digital Analysis based on a modern mathematical theorem known as Benford’s Law which has proved strikingly effective in detecting frauds.

Benford’s Law

According to Benford’s discovery, if you count any collection of objects – whether it be pebbles on the beach, the number of words in a magazine article or dollars in your bank account – then the number you end up with is more likely to start with a “1” than any other digit. Somehow, nature has a soft spot for digit “one.” Frank Benford, a physicist with the General Electric Company, was not the first who made this astonishing observation. 19 years before the end of 19th century, the American astronomer and mathematician Simon Newcomb noticed that the pages of heavily used books of logarithms were much more worn and smudged at the beginning than at the end, suggesting that for some reason, people did more calculations involving numbers starting with 1 than 8 and 9. (Newcomb, S. "Note on the Frequency of the Use of Digits in Natural Numbers." Amer. J. Math 4, 39-40, 1881)

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 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:

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

There are nine tables showing the groups. In these tables Group X includes all the suras containing a number of verses starting with the digit X

Thus, there are 30 suras in the Quran whose number of verses start with digit “1”, 17 suras with digit “2” and so on.  Looking into these tables immediately reveals that the distribution of digits almost accurately adheres to Benford’s law.(graph below).this pattern also conforms to mathematical miracle of the Quran.

30*1+17*2+3*12+4*11+5*14+6*7+7*8+8*10+9*5=437=19*23

By: Majeed Motahari

GROUP 2 17 suras whose no. of verses start with 2   Sura No. No. of Verses 1 2 286 2 3 200 3 7 206 4 26 227 5 48 29 6 57 29 7 58 22 8 59 24 9 71 28 10 72 28 11 73 20 12 81 29 13 84 25 14 85 22 15 88 26 16 90 20 17 92 21	GROUP 3 12 suras whose no. of verses start with 3   Sura No. No. of Verses 1 2 286 2 3 200 3 7 206 4 26 227 5 48 29 6 57 29 7 58 22 8 59 24 9 71 28 10 72 28 11 73 20 12 81 29	GROUP 4 11 suras whose no. of verses start with 4   Sura No. No. of Verses 1 2 286 2 3 200 3 7 206 4 26 227 5 48 29 6 57 29 7 58 22 8 59 24 9 71 28 10 72 28 11 73 20
GROUP 5 14 suras whose no.of verses start with 5   Sura No. No. of Verses 1 2 286 2 3 200 3 7 206 4 26 227 5 48 29 6 57 29 7 58 22 8 59 24 9 71 28 10 72 28 11 73 20 12 81 29 13 84 25 14 85 22	GROUP 6 7 suras whose no.of verses start with 7   Sura No. No. of Verses 1 2 286 2 3 200 3 7 206 4 26 227 5 48 29 6 57 29 7 58 22	GROUP 7 8 suras whose no.of verses start with 7   Sura No. No. of Verses 1 2 286 2 3 200 3 7 206 4 26 227 5 48 29 6 57 29 7 58 22 8 59 24
GROUP 8 10 suras whose no.of verses start with 8   Sura No. No. of Verses 1 2 286 2 3 200 3 7 206 4 26 227 5 48 29 6 57 29 7 58 22 8 59 24 9 71 28 10 72 28