Q1(iii):

Classify the following numbers as rational or irrational: (iii) $\frac{2\sqrt{7}}{7\sqrt{7}}$

Initial Setup & Given Expression

We are tasked with classifying the following mathematical expression as either a rational or an irrational number:

$\frac{2\sqrt{7}}{7\sqrt{7}}$

Step 1: Algebraic Simplification

Before classifying any real number, it must be reduced to its simplest form. We begin by analyzing the factors in both the numerator and the denominator.

• Numerator: $2\sqrt{7}$ (which is the product of the integer $2$ and the irrational number $\sqrt{7}$)
• Denominator: $7\sqrt{7}$ (which is the product of the integer $7$ and the irrational number $\sqrt{7}$)

[Per the fundamental properties of fractions and real numbers, any non-zero common factor present in both the numerator and the denominator can be divided out without changing the value of the expression]. Since $\sqrt{7} \approx 2.645$ and strictly $\sqrt{7} \neq 0$, we can safely cancel this common factor:

$\frac{2\sqrt{7}}{7\sqrt{7}} = \frac{2 \cdot \sqrt{7}}{7 \cdot \sqrt{7}} = \frac{2}{7} \cdot \frac{\sqrt{7}}{\sqrt{7}}$

$\frac{2}{7} \cdot 1 = \frac{2}{7}$

Step 2: Theoretical Classification

We must now evaluate the simplified expression, $\frac{2}{7}$, against the formal axioms of the real number system.

[By definition, a rational number ($\mathbb{Q}$) is any number that can be expressed in the form $\frac{p}{q}$, where $p$ and $q$ are integers ($\mathbb{Z}$), and the denominator $q$ is not equal to zero ($q \neq 0$)].

Analyzing our simplified fraction $\frac{2}{7}$:

• $p = 2$, which is an integer ($2 \in \mathbb{Z}$).
• $q = 7$, which is an integer ($7 \in \mathbb{Z}$) and strictly non-zero ($7 \neq 0$).

Because the simplified form perfectly satisfies the necessary and sufficient conditions of a rational number, the original expression is inherently rational.

Visual Representation of the Number System Classification

Final Solution: The given expression $\frac{2\sqrt{7}}{7\sqrt{7}}$ simplifies algebraically to $\frac{2}{7}$. Because it can be expressed as the ratio of two integers where the denominator is not zero, it is a rational number.