Some data examples in the above picture come from the Spanish National Bureau of Statistics, and the data are counted according to Ben Ford's logarithmic law. However, the data obtained according to the lottery is random and unified.
Does your residential address number begin with 1 According to a strange mathematical law, about one third of the residential numbers take 1 as their first number. Many other areas with almost no connectivity have the same situation: for example, the historical data of Dow Jones index, the order of file sizes stored in personal computers, the length of major rivers in the world, the numbers on the front page of newspapers and many other things.
This law is named Benford's law after its second founder, Frank Benford. Ben Ford, a physicist at General Electric Company, discovered this law in 1935. This law tells people how often each number (from 1 to 9) appears in various databases as the first important Arabic number.
Except that the number 1 always occupies about one third of the frequency, the frequency of the number 2 is 17.6%, the frequency of the number 3 is 12.5%, and the frequency of the number 9 is 4.6%. In mathematical terms, the formula of this logarithmic law is f (d) = log [1+(1/d)], where f stands for frequency and d stands for the number to be verified. This phenomenon makes people feel strange, and so do scientists at the University of Có rdoba, such as Jerez Torres, Sansolis Ferozt, Antonio Camino and Antonio Sola. Scientists published an article in the European Journal of Physics entitled How Numbers Began? (the first number law) ",and gives a brief historical review of this law. Their paper also expounds the effective application of the first number law and explains why no one can reasonably explain the frequency phenomenon of this number.
Torres, an expert in plasma physics, said, "Bendford's law has always been one of my interests since I learned it. In statistical physics class, I have always used this law as an amazing example to stimulate students' curiosity. "Torres explained that before Bendford, there was a respected astronomer named simon newcomb who discovered this law in 188 1 year. Newcomb's contemporary scientists did not pay enough attention to his scientific discoveries. Both Bendford and newcomb were puzzled by this rule: when browsing the logarithmic table, they noticed that the beginning of the book was much dirtier than the end. In other words, when their colleagues go to the library and choose books on various subjects, they all choose the first page to start reading.
Ben Ford's observation on this issue is more profound than newcomb's. He began to investigate other numbers and found that the phenomenon of "first number law" appeared in completely different data, such as population, mortality, physical and chemical constants, baseball statistics, half-life radioisotopes, answers in physics books, prime numbers and Fibonacci numbers. In other words, as long as the data obtained by the unit of measurement system conforms to this law.
On the other hand, data obtained and restricted at will usually do not conform to this Ford law. For example, lottery numbers, telephone numbers, gasoline prices, dates, and weight or height data of a group of people are relatively randomly or arbitrarily specified, and are not obtained by the measurement system.
As explained by Torres and his colleagues, scientists have been closely following Ben Ford's study of this number phenomenon for decades, but they have hardly found the first number law itself except for more examples. However, scientists have found some strange phenomena. For example, when examining the second important number in the database, the law still works, but the importance of the second important number has decreased. Similarly, the characteristics displayed by the third and fourth important numbers begin to become the same, and the frequency of the fifth important number is 10%, which is just the average. The second strange phenomenon aroused more scientific interest:
In the published article, scientists wrote, "In 196 1 year, Pickhan discovered the first conventional related conclusion, which showed that Bendford's law was a scale invariant principle and the only law that put forward the principle of digital scale invariance. In other words, because the length of the world's rivers is expressed in kilometers, it satisfies this Ford's law, and it also satisfies this law of miles, light years, microns or other length units. "
Torres also explained that in the late 20th century, some important forecasting theories (cardinal constancy and uniqueness, etc. ) was confirmed by Ted Hill and other mathematicians. Although some examples have been explained (for example, almost all residential address numbers start with the number 1, and the low-order digits always appear before the high-order digits), there are still no available criteria to explain various examples. At the same time, scientists also explained that there is no priority standard that can tell us when we should or should not follow this law to set numbers. Torres said, "Now many theoretical achievements have been made in the study of this law, but some theoretical achievements are still unclear. Why do some digital settings, such as general physical constants, conform to this law perfectly? We should not only understand the mathematical reason of this rule, but also grasp the characteristics of this group of experimental data. For example, what is their connection point? Where are they from? Obviously, they are quite independent. I hope to find the overall inevitability and sufficient conditions of this law in the future. Many people are interested in this law, especially economists. But I also know that this law may never be possible. "
However, scientists have applied this law to many practical applications. For example, a company's annual account data should meet this law, and economists can find out forged data according to this law. Because it is difficult to satisfy this rule by falsifying data. (Interestingly, scientists found that the numbers 5 and 6, rather than 1, were the most popular numbers, indicating that counterfeiters tried to "hide" data in the middle of accounts. )
Ben Ford's law is also used to detect fraud in election voting. According to this law, scientists discovered voting fraud in Florida, Venezuela and Mexico in 2004 and 2006 respectively.
Torres said, "The first law of numbers is that stories discovered through dirty pages are completely unbelievable. There is no denying that Ben Ford's law has been applied. When this rule is discovered, it is not clear what benefits it can bring. To me, this seems to be just a digital singularity. This is an example of a typical simplicity that may contain unexpected magic. "