Question

rational and irrational number arithmetic
consider the numbers: $0, 3, \sqrt{3}, -\sqrt{3}, \frac{1}{\sqrt{3}}, \pi$

1. which number can be added to $\sqrt{3}$ to show that the sum of a rational number and an irrational number is not always an irrational number?

a $-\sqrt{3}$
b $\pi$
c $\sqrt{3}$
d $\frac{1}{\sqrt{3}}$

1. which number can be multiplied to $\sqrt{3}$ to show that the product of an irrational number and an irrational number is not always an irrational number?

Explanation:

Step1: Evaluate first question options

We need a number that when added to $\sqrt{3}$ (irrational) gives a rational sum.

• Option A: $\sqrt{3} + (-\sqrt{3}) = 0$, and 0 is rational.
• Option B: $\sqrt{3} + \pi$ is irrational.
• Option C: $\sqrt{3} + \sqrt{3} = 2\sqrt{3}$, irrational.
• Option D: $\sqrt{3} + \frac{1}{\sqrt{3}} = \frac{3 + 1}{\sqrt{3}} = \frac{4}{\sqrt{3}}$, irrational.

Step2: Evaluate second question options

We need a number that when multiplied by $\sqrt{3}$ (irrational) gives a rational product.

• Option A: $\sqrt{3} \times \pi = \pi\sqrt{3}$, irrational.
• Option B: $\sqrt{3} \times 1 = \sqrt{3}$, irrational.
• Option C: $\sqrt{3} \times 3 = 3\sqrt{3}$, irrational.
• Option D: $\sqrt{3} \times \frac{1}{\sqrt{3}} = 1$, and 1 is rational.

Answer:

First question: A. $-\sqrt{3}$
Second question: D. $\frac{1}{\sqrt{3}}$