### Bedford's Law & Quran

Submitters Perspective
c
ourtesy of www.masjidtucson.org

## BEDFORD'S LAW & THE QURAN

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 "1". Benford was not the first to make 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, a physicist with the General Electric Company, published his paper. (Bedford, F. "The Law of Anomalous Numbers." Proc. Amer. Phil. Soc. 78, 551-572, 1938). He rediscovered the phenomenon and came up with the same law as Newcomb. Conducting a monumental research, he analyzed 20229 set of numbers gathered from everywhere 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 distribution that is base invariant has to take the form of Benford's law ("Base invariance implies Benford's law", Proceedings of the American Mathematical 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 to 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 expense 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 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 . 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 the part of the author of the Quran.

Let's verify the evidence:

Quran consists of 114 suras. Each sura is composed of a 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 Benford's law. The result is shown in the following tables:

Group X includes All the suras containing a number of verses starting with the digit X

 Group one. These are  30 Suras Sura Number 4 5 6 9 10 11 12 16 17 18 Number of verses 176 120 165 127 123 111 128 111 110

 Group one Sura Number 20 21 23 37 49 60 61 62 63 64 Number of verses 135 112 118 182 18 13 14 11 11 18 Group one Sura Number 65 66 82 86 87 91 93 96 100 101 Number of verses 12 12 19 17 19 15 11 19 11 11

 Group two . These are 17 Suras Sura Number 2 3 7 26 48 57 58 59 71 72 Number of verses 286 200 206 227 29 29 22 24 28 28

 Group two Sura Number 73 81 84 85 88 90 92 Number of verses 20 29 25 22 26 20 21

 Group three . These are 12 suras. Sura Number 31 32 45 46 47 67 76 83 89 103 108 110 Number of verses 34 30 37 35 38 30 31 36 30 3 3 3

 Group four These are 11 Suras Sura Number 13 35 50 52 70 75 78 79 80 106 112 Number of verses 43 45 45 49 44 40 40 46 42 4 4

 Group five These are 14 Suras. Sura Number 14 34 41 42 44 54 68 69 74 77 97 105 111 113 Number of verses 52 54 54 53 59 55 52 52 56 50 5 5 5 5

 Group six These are 7 Suras. Sura Number 24 29 30 51 53 109 114 Number of verses 64 69 60 60 62 6 6

 Group seven These are 8 Suras. Sura Number 1 8 22 25 33 39 55 107 Number of verses 7 75 78 77 73 75 78 7

 Group eight These are 10 Suras. Sura Number 28 36 38 40 43 94 95 98 99 102 Number of verses 88 83 88 85 89 8 8 8 8 8

 Group nine These are 5 Suras. Sura Number 15 19 27 56 104 Number of verses 99 98 93 96 9

Thus, there are 30 suras in the Quran containing a number of verses starting with digit "1", 17 suras with digit "2", 12 suras with digit"3", 11 suras with digit "4", 14 suras with digit "5", 7 suras with digit "6", 8 suras with digit "7", 10 suras with digit "8" and 5 suras with digit "9". As it is seen on the graph, this digital distribution is remarkably close to Benford’s prediction.

This data also conforms to the Quran’s code:

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

### Is It A Mere Coincidence?

We observed that Group one contains 30 suras. Remember that number 30 is the 19th composite number. Number 30 appears to have a crucial role in Quran’s mathematical system. The only time that number 19 is mentioned in the Quran is verse 30 (sura 74). Also note that the number of suras (114=19*6) is immediately preceded with 30th prime number (113). Furthermore, the 19th prime is 67 and sura 67 happens to have 30 verses (Group three). Also see Editor’s Note#2 in The End of the World coded in the Quran.

Another fascinating feature of Group one reveals itself when we arrange the suras in the chronological order of revelation; Sura 82 with 19 verses fits into the 19th place.

 Sura Number 96 87 93 100 91 101 86 20 17 10 11 12 6 37 18 Number of verses 19 19 11 11 15 11 17 135 111 109 123 111 165 182 110 Chronological order of revelation 1 8 11 14 26 30 36 45 50 51 52 53 55 56 69

 Sura Number 16 21 23 82 60 4 65 63 49 66 64 61 62 5 9 Number of verses 128 112 118 19 13 176 12 11 18 12 18 14 11 120 127 Chronological order of revelation 70 73 74 82 91 92 99 104 106 107 108 109 110 112 113

Henri Poincare:* Mathematician, born in Nancy, France. He studied at Paris, where he became professor in 1881 (=19 x 99) . He was eminent in physics, mechanics, and astronomy, and contributed to many fields of mathematics. He created the theory of automorphic functions, using new ideas from group theory, non-Euclidean geometry, and complex function theory. The origins of the theory of chaos are in a famous paper of 1889 on real differential equations and celestial mechanics. Many of the basic ideas in modern topology, triangulation, and homology are due to him. He gave influential lecture courses on such topics as thermodynamics, and almost anticipated Einstein's theory of special relativity, showing that the Lorentz transformations form a group. In his last years he published several books on the philosophy of science and scientific method, and was also well known for his popular expositions of science.

### BEDFORD’S LAW & The Quran

"If God speaks to man, he undoubtedly uses the language of mathematics."   Henri Poincare*

"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)

The Quran is intended to be an eternal miracle. The highly sophisticated mathematical system based on prime number 19 embedded into the fabric of the Quran ( decoded between 1969-1974 with the aid of computers), 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)

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.