Engel Expansions, Continued Fractions, and Fibonacci
Kraaikamp and Wu (2004) observe that an Engel expansion can also be written as an ascending variant of a continued fraction:
They claim that ascending continued fractions such as this have been studied as early as Fibonacci's Liber Abaci (1202). This claim appears to refer to Fibonacci's compound fraction notation in which a sequence of numerators and denominators sharing the same fraction bar represents an ascending continued fraction:
If such a notation has all numerators 0 or 1, as occurs in several instances in Liber Abaci, the result is an Engel expansion. However, Engel expansion as a general technique does not seem to be described by Fibonacci.
Read more about this topic: Engel Expansion
Famous quotes containing the words engel and/or continued:
“Now folks, I hereby declare the first church of Tombstone, which ain’t got no name yet or no preacher either, officially dedicated. Now I don’t pretend to be no preacher, but I’ve read the Good Book from cover to cover and back again, and I nary found one word agin dancin’. So we’ll commence by havin’ a dad blasted good dance.”
—Samuel G. Engel (1904–1984)
“There is not any present moment that is unconnected with some future one. The life of every man is a continued chain of incidents, each link of which hangs upon the former. The transition from cause to effect, from event to event, is often carried on by secret steps, which our foresight cannot divine, and our sagacity is unable to trace. Evil may at some future period bring forth good; and good may bring forth evil, both equally unexpected.”
—Joseph Addison (1672–1719)