Finite Simple Continued Fraction has Rational Value

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Theorem

Let $n \ge 0$ be a natural number.

Let $\paren {a_0, \ldots, a_n}$ be a simple finite continued fraction‎ of length $n$.


Then its value $\sqbrk {a_0, \ldots, a_n}$ is a rational number.


Proof

This will be proved by induction on the number of partial quotients.


For all $n \in \N^*$, let $P \left({n}\right)$ be the proposition that the continued fraction $\sqbrk {a_0, a_1, \ldots, a_n}$ has a rational value.


Basis for the Induction

$P(0)$ is true, as $\sqbrk {a_0}$ is an integer and therefore rational.

This is our basis for the induction.


Induction Hypothesis

Now we need to show that, if $P \left({k}\right)$ is true, where $k \ge 1$, then it logically follows that $P \left({k+1}\right)$ is true.


So this is our induction hypothesis:

The continued fraction $\sqbrk {a_0, a_1, \ldots, a_k}$ has a rational value.


Then we need to show that the continued fraction $\sqbrk {a_0, a_1, \ldots, a_k, a_{k + 1} }$ also has a rational value.


Induction Step

Consider the continued fraction $\sqbrk {a_0, a_1, \ldots, a_k, a_{k + 1} }$.

By definition of value of a finite continued fraction:

$\sqbrk {a_0, a_1, \ldots, a_k, a_{k + 1} } = a_0 + \dfrac 1 {\sqbrk {a_1, a_2, \ldots, a_k, a_{k + 1} } }$


$a_0$ is an integer, as we have seen.

By the induction hypothesis, $\sqbrk {a_1, a_2, \ldots, a_k, a_{k + 1} }$ is a rational number, and so is its reciprocal.

Hence as $a_0 + \dfrac 1 {\sqbrk {a_1, a_2, \ldots, a_k, a_{k + 1} } }$ is the sum of two rational numbers, it is itself rational.


So $P \left({k}\right) \implies P \left({k+1}\right)$ and the result follows by the Principle of Mathematical Induction.


Therefore for all $n \in \N^*$, the continued fraction $\sqbrk {a_0, a_1, \ldots, a_n}$ has a rational value.

$\blacksquare$


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