Properties of Value of Finite Continued Fraction

Theorem

Value is Strictly Positive

Let $(a_0, \ldots, a_n)$ be a finite continued fraction in $\R$ of length $n \geq 0$.

Let all partial quotients $a_k>0$ be strictly positive.

Let $x = [a_0, a_1, \ldots, a_n]$ be its value.

Then $x>0$.

Value is at Least First Term

Let $(a_0, \ldots, a_n)$ be a finite continued fraction in $\R$ of length $n \geq 0$.

Let the partial quotients $a_k>0$ be strictly positive for $k>0$.

Let $x = [a_0, a_1, \ldots, a_n]$ be its value.

Then $x \geq a_0$, and $x>a_0$ if the length $n\geq 1$.

Floor of Simple Finite Continued Fraction

Let $\sequence {a_k}_{k \mathop \ge 0}$ be a simple finite continued fraction of length $n \ge 0$.

Let $x = [a_0, \ldots, a_n]$ be its value.

Then the floor of $x$ is the partial denominator $a_0$:

$\floor x = a_0$

unless $n = 1$ and $a_1 = 1$, in which case $x = \floor x = a_0 + 1$.

Also see

• Properties of Value of Infinite Continued Fraction