Question

is the number \\(\sqrt{37} + -2\\) rational or irrational? \
rational \
irrational

Explanation:

Step1: Analyze $\sqrt{37}$

A rational number can be expressed as $\frac{p}{q}$ where $p,q$ are integers and $q
eq0$. $\sqrt{37}$: since 37 is not a perfect square, $\sqrt{37}$ is irrational (by definition of irrational numbers: numbers that cannot be expressed as a fraction of two integers, and non - perfect square square roots are irrational).

Step2: Analyze the sum of irrational and rational

The number $- 2$ is a rational number (it can be written as $\frac{-2}{1}$). The sum of an irrational number and a rational number: Let the irrational number be $a$ (where $a$ cannot be written as $\frac{p}{q}$) and the rational number be $\frac{m}{n}$. Suppose $a+\frac{m}{n}=\frac{k}{l}$ (where $k,l$ are integers and $l
eq0$). Then $a=\frac{k}{l}-\frac{m}{n}=\frac{kn - ml}{ln}$, which would imply that $a$ is rational, a contradiction. So the sum of an irrational number and a rational number is irrational. Since $\sqrt{37}$ is irrational and $-2$ is rational, $\sqrt{37}+(-2)=\sqrt{37}-2$ is irrational.

Answer:

irrational