Question

classify (√3 + 1/3) as rational or irrational and explain your reasoning.
a suppose an irrational + rational = rational. then irrational = rational - rational. this means irrational = rational. thus the irrational number √3 plus the rational number 1/3 equals (√3 + 1/3) which is both rational and irrational.
b suppose an irrational + rational = rational. then irrational = rational - rational. this means irrational = rational, which is a contradiction. thus the irrational number √3 plus the rational number 1/3 equals the rational number (√3 + 1/3)
c suppose an irrational + rational = rational. then irrational = rational - rational. this means irrational = rational, which is a contradiction. thus the irrational number √3 plus the rational number 1/3 equals the irrational number (√3 + 1/3)
d suppose an irrational + rational = rational. then irrational = rational - rational. this means irrational = rational, which is a contradiction. thus the irrational number √3 plus the rational number 1/3 equals (√3 + 1/3) which is neither rational nor irrational.

Explanation:

Step1: Recall properties of rational and irrational numbers

The sum of a rational number $r$ and an irrational number $i$ cannot be rational. Suppose, for the sake of contradiction, that $i + r=q$ where $r$ is rational, $i$ is irrational and $q$ is rational. Then $i = q - r$. Since the difference of two rational numbers $q$ and $r$ is rational, this would mean that $i$ is rational, which is a contradiction. The number $\sqrt{3}$ is irrational (it cannot be written as a fraction $\frac{a}{b}$ where $a,b\in\mathbb{Z}$ and $b
eq0$) and $\frac{1}{3}$ is rational (it is of the form $\frac{a}{b}$ with $a = 1$ and $b=3$).

Step2: Analyze the given sum

The sum $\sqrt{3}+\frac{1}{3}$ is irrational because an irrational number ($\sqrt{3}$) plus a rational number ($\frac{1}{3}$) is irrational.

Answer:

C. The irrational number $\sqrt{3}$ plus the rational number $\frac{1}{3}$ equals the irrational number $(\sqrt{3}+\frac{1}{3})$