Question

is the number $\pi + 7$ rational or irrational? \
rational \
irrational

Explanation:

Step1: Recall properties of irrational numbers

An irrational number is a number that cannot be expressed as a fraction of two integers. $\pi$ is a well - known irrational number, and it has a non - repeating, non - terminating decimal expansion ($\pi\approx3.1415926535\cdots$).

Step2: Analyze the sum of a rational and an irrational number

A rational number can be written as $\frac{a}{b}$ where $a,b\in\mathbb{Z}$ and $b
eq0$. Let's assume that $\pi + 7$ is rational. Then we can write $\pi+7=\frac{p}{q}$, where $p,q\in\mathbb{Z}$ and $q
eq0$. Then we can rearrange this equation to get $\pi=\frac{p}{q}-7=\frac{p - 7q}{q}$. But $\frac{p - 7q}{q}$ is a rational number (since $p - 7q$ and $q$ are integers and $q
eq0$), and we know that $\pi$ is irrational. This is a contradiction, so our assumption that $\pi + 7$ is rational is wrong. Therefore, $\pi+7$ is irrational.

Answer:

irrational