Normal Numbers
A real number is said to be simply normal to the base b if every possible digit appears in its fractional part with a frequency of occurrence of 1/b, where b is the base of the expansion. By this we understand that if a number is expressed to the base b (as, for example, we regularly express numbers to the base 10) and if for each digit, d (0 d b-1), we count the occurrence of d in the first N positions after the decimal point, divide this count by N and then let N increase without bound; and if the ratio of this count approaches the limit 1/b we say that the number is simply normal to the base b.
In, say, the expansion of (which has been expanded, at last count, to 5.4 billion digits), we would naturally expect any particular digit, say 3, to appear no more nor less than any of the other nine digits, that is, one in 10 times. And this is what would happen if it turned out that were normal. Each digit should have an equal distribution or equal likelihood of occurrence.
Symbolically, taking A(d,b,N) to mean the number of occurrences of digit d when our number is expressed to the base b in the first N digits, then lim A(d,b,N)/N = 1/b is our standard for simple normality base b, where we take the limit as N grows toward infinity.
If in addition, is simply normal to base bn, for all positive integers n, then we say that is entirely normal to the base b.
And if, moreover, is entirely normal to every base b greater than 1, we say that is absolutely normal, or sometimes just normal.
It can be shown that a number simply normal to the base bn is also simply normal to base b; however, the converse is not true. For example, the rational number 0.0123456789 ... with this block of numbers repeated indefinitely is simply normal to the base 10, as any of the digits 0 through 9 appear with a frequency of 1/10; but it is not normal to the base 1010, since it misses 1010.- 1 digits. That is, 0123456789 is a single digit base 1010, and it is the only one that appears.
What is signal about normal numbers is Borel's normal number theorem, which asserts that almost all numbers are normal. The phrase "almost all" must be understood in the context of Lebesgue measure.
We say that a subset of the real line has Lebesgue measure zero if it can be covered by countably many intervals whose total length is less than any preselected positive quantity, however small. Borel's normal number theorem is then equivalent to the claim that the set of numbers that are not normal has Lebesgue measure zero Thus, for example, on the interval [0,1], the set of normal numbers will have measure 1 and its compliment; the set of numbers that are not normal will have measure 0.
It may also be demonstrated that the Borel normal number theorem is equivalent, in probability theory, to the strong law of large numbers. In this setting we exploit the concept of probability measure, which is nothing other than Lebesgue measure restricted to the [0,1] interval, so that events are identified with points or sets of points in [0,1], and their probability of occurrence is their Lebesgue measure (roughly, their total length), which is not less than 0 nor greater than 1.
All this may be clarified by considering the celebrated Cantor set, which is an example of a set that fails to be normal.
We begin by considering the [0,1] interval and represent the numbers between 0 and 1 in ternary expansion, that is their expansion to the base 3. The digits are all either 0, 1, or 2, and in the first position 0 will represent all numbers in the first third of the unit interval, 1 the numbers in the middle third of the unit interval, and 2 the final third of the interval. Each of these three intervals may again be subdivided into thirds and the second digit of the expansion, 0, 1, or 2 represents the first, second, or third trisection of each of the three initial divisions, and so on.
The Cantor set, then, may be represented by the set of numbers in [0,1] whose ternary expansion does not contain a 1. Geometrically, such representations designate, in the first stage, corresponding to the first digit, the first and third subinterval of [0,1] with the middle third deleted. The second digit, which is either 0 or 2, corresponds to the first and last third of each of the two thirds of [0,1] whose first digit is 0 or 2, and so on.
The Cantor set is not normal, for the digit 1 not only fails to occur with a frequency of 1/3, but fails to appear at all. According to Borel's normal number theorem, the Cantor set must have measure 0.
Probabilistically, we may interpret the Cantor set as representing random events in which one of the three possible outcomes never occurs. The probability of such a "Cantor-like" event occurring is zero.
The Cantor set is of interest because it possesses a variety of unusual properties, among them is the fact that it is an uncountable set whose measure is zero. This, though, is a property that it shares with the compliment of normal numbers, which is also uncountable.
Another property of the Cantor set is that it is possible to construct on the interval [0,1] a function that is singular, meaning that it has zero derivative on all but a set of measure zero, namely on all but the Cantor set, yet the function is non-constant. Analogously, it is possible to construct on [0,1] a singular function, strictly increasing, with zero derivative on all but a set of measure 0, namely on all but the compliment of the set of normal numbers.
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