Understanding Rational Numbers

In the number system, numbers can be broadly classified into rational and irrational numbers. A rational number is any number that can be expressed in the form $\frac{p}{q}$, where $p$ and $q$ are integers, and $q$ is not equal to zero. Examples of rational numbers include integers (like 5, which is $\frac{5}{1}$), fractions (like $\frac{3}{4}$), and terminating or repeating decimals (like 0.5 or 0.333...).

On the other hand, an irrational number is a number that cannot be expressed as a simple fraction. Their decimal representations are non-terminating and non-repeating. Common examples include square roots of non-perfect squares, such as $\sqrt{2}$, $\sqrt{3}$, $\sqrt{5}$, and special numbers like $\pi$ (pi).

Analyzing the Given Options for Rationality

Let's examine each of the provided options to determine which one is a rational number.

• Option 1: $\sqrt{3}$

  The number 3 is not a perfect square. Therefore, $\sqrt{3}$ is an irrational number because its decimal representation is non-terminating and non-repeating. This number cannot be written in the form $\frac{p}{q}$.

• Option 2: $\sqrt{5}$

  The number 5 is not a perfect square. Similar to $\sqrt{3}$, $\sqrt{5}$ is an irrational number. Its decimal expansion is non-terminating and non-repeating, making it impossible to express as a fraction of two integers. This is a concept within the broader topic of the real numbers.

• Option 3: $\frac{2\sqrt{3}}{\sqrt{3}}$

  This expression can be simplified. We have $\sqrt{3}$ in both the numerator and the denominator. As long as $\sqrt{3}$ is not zero (which it isn't), we can cancel out the common term $\sqrt{3}$:

  $\frac{2\sqrt{3}}{\sqrt{3}} = 2 \times \frac{\sqrt{3}}{\sqrt{3}} = 2 \times 1 = 2$

  The result of the simplification is 2. The number 2 is an integer, and any integer can be written as a fraction with a denominator of 1, for example, $\frac{2}{1}$. Since 2 can be expressed in the form $\frac{p}{q}$ where $p=2$ and $q=1$ are integers and $q \neq 0$, the number 2 is a rational number. Simplifying the expression helped us identify the type of number.

• Option 4: $\sqrt{6}$

  The number 6 is not a perfect square ($2^2 = 4$, $3^2 = 9$). Therefore, $\sqrt{6}$ is an irrational number. Its decimal form is non-terminating and non-repeating, characteristic of irrational numbers within the set of real numbers.

Identifying the Rational Number

Based on our analysis of each option, the only number that can be expressed in the form $\frac{p}{q}$ where $p$ and $q$ are integers ($q \neq 0$) is the simplified form of option 3, which is 2. Thus, $\frac{2\sqrt{3}}{\sqrt{3}}$ represents a rational number.