Question

Recall that a real number x is rational if x = p/q for integers p, q with q = ̸= 0.

Prove that if x is rational then 1/(2x+1) is rational. Then prove that if 1/(2x+1) is rational then x is rational.

Answer 1

Answer on Question #85964 – Math – Discrete Mathematics

Question

Recall that a real number $x$ is rational if $x = p/q$ for integers $p, q$ with $q \neq 0$.

Prove that if $x$ is rational then $1/(2x + 1)$ is rational. Then prove that if $1/(2x + 1)$ is rational then $x$ is rational.

Solution

1) If $x$ is rational then $x = \frac{p}{q}$, where $p$ and $q$ are integers with $q \neq 0$.

So $\frac{1}{2x + 1} = \frac{1}{2^{\frac{p}{q}} + 1} = \frac{\frac{1}{2p + q}}{q} = \frac{q}{2p + q}$. Since $q$ and $2p + q$ are integers with $2p + q \neq 0$, then $\frac{q}{2p + q} = \frac{1}{2x + 1}$ is rational.

2) If $\frac{1}{2x + 1}$ is rational then $\frac{1}{2x + 1} = \frac{p}{q}$, where $p$ and $q$ are integers with $q \neq 0$.

In addition, $p = q * \frac{1}{2x + 1} \neq 0$. Therefore $2x + 1 = \frac{q}{p}$, and $2x = \frac{q}{p} - 1 = \frac{q - p}{p}$. So $x = \frac{q - p}{2p}$.

Since $q - p$ and $2p$ are integers with $2p \neq 0$, then $\frac{q - p}{2p} = x$ is rational.

Answer provided by https://www.AssignmentExpert.com