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)
“A drop of water has the properties of the sea, but cannot exhibit a storm. There is beauty of a concert, as well as of a flute; strength of a host, as well as of a hero.”
—Ralph Waldo Emerson (1803–1882)