Liouville Numbers
We define a number z to be algebraic if it satisfies a polynomial with integer coefficients. And we say that z is algebraic of degree n if n is the degree of the smallest polynomial satisfied by z. For example, the degree of 2 is 2, since it satisfies the polynomial x2 - 2 = 0 and no polynomial of smaller degree. Similarly, 2 + 3 is algebraic of degree 4, as we invite the reader to show by inventing a 4th degree polynomial which this number satisfies, and also convincing himself that there is no smaller.
This raises the question: Are there numbers which are not algebraic, that is, which do not satisfy any polynomial with coefficients in Z?
This question was answered in the affirmative in 1844 by Joseph Liouville (1809-1882). He first proved that if a real number z is algebraic of degree n we can say about it that there exists a constant M such that for all rational numbers p/q, with p,q contained in Z, the distance from z to p/q is greater than M/qn, where n is the degree of the minimal polynomial satisfied by z. He then defined a class of numbers, which we now call Liouville numbers, to be a set of numbers that, it turns out, fail to have this property. Specifically, a number z is a Liouville number if it is real and irrational and if for any positive integer n, there is a rational number p/q such that the distance from z to p/q is less than 1/qn.
By formally proving what is almost apparent, that a Liouville number cannot satisfy the property above characterizing an algebraic number, we prove that a Liouville number cannot be algebraic.
To demonstrate that the class of Liouville numbers is not vacuous, Liouville easily showed that 1/10n! is a Liouville number and thus is not algebraic. Here we take the sum from n = 1 to n = infinity. The number 10 may just as well be replaced by any other, and n! may be replaced by any sequence of numbers that diverges rapidly to infinity. A number that is not algebraic is said to be transcendental.
This was the first number shown to be transcendental and preceded Georg Cantor's non-constructive proof based on a cardinality argument by thirty years.
Other properties of Liouville numbers are as follows:
- 1) The Liouville numbers are dense in R.
- 2) The Liouville numbers are of second category.
- 3) The Liouville numbers comprise a subset of the real line having Lebesgue measure 0.
- 4) The Liouville numbers have s-dimensional (Hausdorff measure) equal to 0, for all positive s.
- 5) The real line can be partitioned into a set of Lebesgue measure 0 and a set of first Category by taking the Liouville numbers and their compliment.
These concepts will be explained but not proved.
We say that a subset E of R is dense in R if every interval in R contains a member of E. We say that a set F is nowhere dense if it is dense in no interval, that is, if every interval has a subinterval contained in the compliment of F. We say that a set is of first category if it is a countable union of nowhere dense sets. (A set is said to be countable if there is a one to one mapping from the set onto the integers.) And a set is of second category if it is not of first category. For example, Q, the rational numbers, is of first category since it may be represented as a countable union of singletons - {p/q}. Note however that Q is not itself nowhere dense. Indeed, Q is dense in R.
We define a set E to be of Lebesgue measure 0 if for any preassigned quantity, however small, we can cover E with a countable collection of intervals whose total length is less than this preassigned quantity. If there exists a positive real number s so that if we can raise the length of each individual interval to the power s, and if again their sum is arbitrarily small, then we say E has s-dimensional Hausdorff measure 0. Symbolically, |I| < ε, where ε is our arbitrarily small quantity, and EI. It can be shown that this property characterizes the Liouville numbers for all positive real numbers s, and thus the Liouville numbers have s-dimensional Hausdorff measure 0 for all positive real numbers s.
Calling the Liouville numbers E, then E, together with its compliment, is of course all of R, so the real line then can be partitioned in to a set of Lebesgue measure 0, namely E and a set of first category, namely the compliment of E.
This conclusion is quite remarkable, as a set of Lebesgue measure 0 is thought of, with some justice, as small or thin, and so is a set of first category, yet together they make up the entire real line.
Both the algebraic numbers and the Liouville numbers have Lebesgue measure 0. Cantor demonstrated that the set of algebraic numbers is a countable set, and we know that such a set must have Lebesgue measure 0. What remains--the set of transcendental numbers which are not Liouville numbers--must have positive measure, measure 1, say in [0,1]. Thus the Liouville numbers amounts only to the shoreline of the sea of transcendental numbers. It is however the only large general class of transcendental numbers with which we were initially acquainted and afforded the first techniques of creating specific transcendental numbers.
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