Essay - Zipf's Law and Benford's Law This Paper Will Address Both...

Zipf's Law and Benford's *****

***** paper will address both Zipf's Law and ***** *****, in order for the reader to clearly understand ***** of the laws and *****ir significance to mathematics. "Zipf's law, named after the Harvard l*****guistic professor George Kingsley Zipf (1902-1950), is the observation that frequency of occurrence of some event (P), as a function ***** the rank (i) when the rank *****s determined by ***** above frequency of occurrence, is a power-***** function Pi ~ 1/ia with the exponent a close to unity" (Li, n.d.).

Benford's L*****w, also called the first-digit law, states that in l*****ts ***** numbers from many (but not all) real-life sources of data, the leading digit occurs much more ********** than ***** others (namely about 30% of the time" (Wikipedia, 2006). Both of ********** statements, however, must be clarified in order to truly understand what these *****s mean and what they are about.

***** Benford's Law is concerned, the larger ***** the digit is, the less the likelihood that it will be the lead*****g digit where any num*****r ***** concerned (Bogomolny, n.d.). This applies to any kind of figures, from those that have social significance to those that ***** more closely tied to the natural world. These can include ***** taken ***** newspaper articles, stock prices, electricity bills, population numbers, are***** and/or lengths for rivers, death rates, both mathematical and physical constants, ***** ***** processes ***** are described ***** the 'power laws,' which are seen as being very common within nature (Bogomolny, n.d.).

***** order to explain this, it is important to ***** that ***** first digits have a certain select distribution, and this distribution must be completely independent from the me*****uring system ***** is used. To be more specific, ***** indicates that, if an individual would convert from feet ***** meters, ***** example, the distribution would ***** ***** changed (*****, n.d.). It is what ***** termed as 'scale invariant,' a***** therefore it is also logarithmic (Bogomolny, n.d.). When me*****uring either the distance or ***** length of something, the ***** digit, which is non-zero, should ***** a distribution that is the same regard***** of what the unit of measure is. Th***** unit ***** measure could be inches, yards, meters, *****, miles, light years, ***** virtually any other type of *****ment (Hill, 1995).

It is important to ***** aw*****, *****, when considering the ***** of feet and yards, that *****re are three feet in every yard, so one must consider ***** there is probability regarding the first digit ***** this ***** and that there is equal probability of th***** digit being 1 (in yards) or 3, 4, ***** 5 (***** feet) (Bogomolny, n.d.). By applying *****is idea to all of the possible scales ***** could be used for measurement, one would get a ***** that is *****. Combining that with ***** idea that log 10(1) is equal to zero a***** log 10(10) is equal to one, Benford's L***** is seen (Bogomolny, n.d.). In other words, ***** there ***** a distribution of the *****