Question

1. expressing \\(\sqrt{2}\\) as the ratio \\(\frac{\sqrt{2}}{1}\\) means the \\(\sqrt{2}\\) is a rational number.
2. the product of two irrational numbers is irrational.

Explanation:

Problem 17 (True/False, likely? Let's analyze)

Step1: Recall Rational Number Definition

A rational number is a number that can be expressed as $\frac{p}{q}$ where $p$ and $q$ are integers and $q
eq 0$. But for a number to be rational, $p$ and $q$ must be **integers**. In $\frac{\sqrt{2}}{1}$, the numerator $\sqrt{2}$ is not an integer (since $\sqrt{2}$ is irrational, approximately 1.414... and not a whole number or fraction of integers in simplest form).

Step2: Determine if $\sqrt{2}$ is Rational

Since $\sqrt{2}$ cannot be expressed as a ratio of two integers (this is a proven result: assume $\sqrt{2} = \frac{a}{b}$ in lowest terms, then $2b^2 = a^2$, so $a$ is even, let $a = 2k$, then $2b^2 = 4k^2 \implies b^2 = 2k^2$, so $b$ is even, contradicting lowest terms). So expressing it as $\frac{\sqrt{2}}{1}$ doesn't make it rational because the numerator isn't an integer. Thus the statement is false.

Problem 18 (True/False)

Step1: Recall Irrational Number Definition

An irrational number is a number that cannot be expressed as $\frac{p}{q}$ with $p, q$ integers, $q
eq 0$.

Step2: Find a Counterexample

Consider two irrational numbers: $\sqrt{2}$ and $\sqrt{2}$. Their product is $\sqrt{2} \times \sqrt{2} = 2$, which is a rational number (2 can be written as $\frac{2}{1}$). So the product of two irrational numbers is not always irrational. Thus the statement is false.

Answer:

s:

1. False
2. False