Q1(i):

Classify the following numbers as rational or irrational: (i) $2 - \sqrt{5}$

Step 1: Analyzing the Components of the Expression

We are given the mathematical expression $2 - \sqrt{5}$. To classify this number, we must first analyze the individual terms that constitute the expression:

• The number $2$: This is an integer. By definition, any integer can be expressed in the form $\frac{p}{q}$ where $p$ and $q$ are integers and $q \neq 0$ (e.g., $\frac{2}{1}$). Therefore, $2$ is a rational number.
• The number $\sqrt{5}$: The number $5$ is a prime number and not a perfect square. The square root of any non-perfect square yields a non-terminating, non-repeating decimal. Therefore, $\sqrt{5}$ is an irrational number.

Step 2: Applying the Properties of Real Numbers

In the real number system, specific closure properties govern the arithmetic operations between rational and irrational numbers. The relevant theorem states:

[Theorem: The sum or difference of a non-zero rational number and an irrational number is always an irrational number.]

Since we are subtracting an irrational number ($\sqrt{5}$) from a rational number ($2$), the resulting difference must be irrational.

Step 3: Formal Proof by Contradiction

To rigorously prove that $2 - \sqrt{5}$ is irrational, we utilize a proof by contradiction.

Assumption: Let us assume the contrary, that $2 - \sqrt{5}$ is a rational number. Let this rational number be represented by $r$.

$2 - \sqrt{5} = r$

Rearranging the equation to isolate $\sqrt{5}$:

$2 - r = \sqrt{5}$

Logical Deduction:

• We know that $2$ is a rational number.
• By our assumption, $r$ is a rational number.
• [Per the Closure Property of Rational Numbers under Subtraction], the difference between two rational numbers ($2 - r$) must also be a rational number.
• This implies that $\sqrt{5}$ is equal to a rational number.

Contradiction: This contradicts the universally established mathematical fact that $\sqrt{5}$ is an irrational number. Therefore, our initial assumption that $2 - \sqrt{5}$ is rational must be false.

Step 4: Decimal Expansion Analysis (Verification)

We can also verify this classification by examining the decimal expansion of the expression:

The decimal expansion of $\sqrt{5}$ is non-terminating and non-repeating:

$\sqrt{5} \approx 2.236067977...$

Substituting this into our expression:

$2 - \sqrt{5} = 2 - 2.236067977...$

$2 - \sqrt{5} = -0.236067977...$

The resulting decimal is also non-terminating and non-repeating, which is the definitive characteristic of an irrational number.

Visual Representation: Number Line Mapping

The following geometric representation demonstrates the subtraction of the irrational length $\sqrt{5}$ from the rational coordinate $2$, landing on the irrational coordinate $2 - \sqrt{5}$.

Final Solution: The expression $2 - \sqrt{5}$ is an irrational number.