. T hroughout the history of mathematics we encounter many kinds of numbers: nat
ID: 3195147 • Letter: #
Question
. T hroughout the history of mathematics we encounter many kinds of numbers: natural, prime, constructible, perfect, algebraic, irrational, transcendental, and so on. In this question you will investigate yet another kind of number for a real number to be "normal in base 10" (b) Is the number normal in base 10? Do some on-line exploring and then write out a few sentences explaining your opinion on this question. If you happen to believe that the answer is "yes", what surprising consequence concerning blocks of consecutive zeros would be true about ?Explanation / Answer
(a)The real numbers include all the rational numbers, such as the integer 5 and the fraction 4/3, and all the irrational numbers, such as 2 (1.41421356..., the square root of 2, an irrational algebraic number). Included within the irrationals are the transcendental numbers, such as (3.14159265...). Real numbers can be thought of as points on an infinitely long line called the number line or real line, where the points corresponding to integers are equally spaced. Any real number can be determined by a possibly infinite decimal representation, such as that of 8.632, where each consecutive digit is measured in units one tenth the size of the previous one. (B) A real number x is normal in base b if in its representation in base b all digits occur, in an asymptotic sense, equally often. In addition, for each m, the b^m different m-strings must occur equally often. In other words, lim n->infinity N(s,n)/n = b^-m for each m-string s, where N(s,n) is the number of occurrences of s in the first n base-b digits of x. A number that is normal in all bases is called normal.
The apparent randomness of Pi's digits had been observed prior to the precise definition of normality. De Morgan, for example, pointed out that one would expect the digits to occur equally often, but yet the number of 7's in the first 608 digits is 44, much lower than expected. However, it turned out that his count was based on inaccurate data.
There are lots of normal numbers - Borel proved that the set of non-normal numbers has measure zero - but it is difficult to provide concrete examples. While an undergraduate at Cambridge University, D. Champernowne proved that 0.12345678910111213 ... is normal in base 10, but an explicit example of a normal number is still lacking.
The question of Pi's normality only scratches the surface of the deeper question whether the digits of Pi are "random". That normality is not sufficient follows from the observation that a truly random sequence of digits ought to be normal when only digits in positions corresponding to perfect squares are examined. But if all such positions in a normal number are set to 0, the number is still normal. On the other hand, more rigorous definitions of "random" exclude Pi because Pi's decimal expansion is a recursive sequence.
Thus deeper questions are lurking, but so little is known about Pi's decimal expansion that it is reasonable to focus on whether Pi is normal to base ten. To put our ignorance in perspective, note that it is not even known that all digits appear infinitely often: perhaps
Pi = 3.1415926.....01001000100001000001...
In order to gather evidence for Pi's normality one would like to examine as many digits as possible. Those who have pursued the remote digits of Pi have often been pejoratively referred to as "digit hunters", but certain recent developments have added some glamor to the centuries-old hunt.
Related Questions
Navigate
Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.