Finite Continued Fractions
Every finite continued fraction represents a rational number, and every rational number can be represented in precisely two different ways as a finite continued fraction. These two representations agree except in their final terms. In the longer representation the final term in the continued fraction is 1; the shorter representation drops the final 1, but increases the new final term by 1. The final element in the short representation is therefore always greater than 1, if present. In symbols:
For example,
Read more about this topic: Continued Fraction
Famous quotes containing the words finite and/or continued:
“We know then the existence and nature of the finite, because we also are finite and have extension. We know the existence of the infinite and are ignorant of its nature, because it has extension like us, but not limits like us. But we know neither the existence nor the nature of God, because he has neither extension nor limits.”
—Blaise Pascal (16231662)
“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 (16721719)