Properties of Value of Finite Continued Fraction
Theorem
Value is Strictly Positive
Let $\sequence {a_0, \ldots, a_n}$ be a finite continued fraction in $\R$ of length $n \ge 0$.
Let all partial denominators $a_k > 0$ be strictly positive.
Let $x = \sqbrk {a_0, a_1, \ldots, a_n}$ be its value.
Then $x > 0$.
Value is at Least First Term
Let $\sequence {a_0, \ldots, a_n}$ be a finite continued fraction in $\R$ of length $n \ge 0$.
Let the partial denominators $a_k > 0$ be strictly positive for $k>0$.
Let $x = [a_0, a_1, \ldots, a_n]$ be its value.
Then $x \ge a_0$, and $x > a_0$ if the length $n \ge 1$.
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 $x = \floor x = a_0 + 1$.