Elementary Properties
An equivalent definition to the one given above is that for any positive integer n, there exists an infinite number of pairs of integers (p,q) obeying the above inequality.
It is relatively easily proven that if x is a Liouville number, x is irrational. Assume otherwise; then there exist integers c, d with d > 0 and x = c/d. Let n be a positive integer such that 2n − 1 > d. Then if p and q are any integers such that q > 1 and p/q ≠ c/d, then
which contradicts the definition of Liouville number.
Read more about this topic: Liouville Number
Famous quotes containing the words elementary and/or properties:
“Listen. We converse as we live—by repeating, by combining and recombining a few elements over and over again just as nature does when of elementary particles it builds a world.”
—William Gass (b. 1924)
“The reason why men enter into society, is the preservation of their property; and the end why they choose and authorize a legislative, is, that there may be laws made, and rules set, as guards and fences to the properties of all the members of the society: to limit the power, and moderate the dominion, of every part and member of the society.”
—John Locke (1632–1704)