Question

classify the numbers as rational or irrational.
π -7 π + -7
rational irrational

Explanation:

Step1: Recall definitions

A rational number can be expressed as $\frac{p}{q}$ where $p,q$ are integers and $q
eq0$. An irrational number cannot be expressed as such and has non - repeating, non - terminating decimal expansion.

Step2: Classify $-7$

$-7$ can be written as $\frac{-7}{1}$, where $-7$ and $1$ are integers and $1
eq0$. So, $-7$ is rational.

Step3: Classify $\pi$

$\pi = 3.1415926\cdots$ has a non - repeating, non - terminating decimal expansion. It cannot be written as a fraction of two integers. So, $\pi$ is irrational.

Step4: Classify $\pi + (-7)$

The sum of an irrational number ($\pi$) and a rational number ($-7$) is irrational. Because if $\pi+(-7)$ were rational, say $r=\pi - 7$ where $r$ is rational, then $\pi=r + 7$ would be rational (since the sum of two rationals is rational), which contradicts the fact that $\pi$ is irrational. So, $\pi+(-7)$ is irrational.

Answer:

rational: $-7$
irrational: $\pi$, $\pi + (-7)$