Q1(ii):

Classify the following numbers as rational or irrational: (ii) $(3 + \sqrt{23}) - \sqrt{23}$

Initial Expression & Setup

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

$ (3 + \sqrt{23}) - \sqrt{23} $

To determine its classification, we must first simplify the expression to its most fundamental form.

Step 1: Algebraic Simplification

We begin by removing the parentheses. [Per the Associative Property of Addition, grouping symbols can be removed when only addition and subtraction are involved].

$ (3 + \sqrt{23}) - \sqrt{23} = 3 + \sqrt{23} - \sqrt{23} $

Step 2: Cancellation of Irrational Terms

Next, we group the like terms. The expression contains two terms involving the square root of 23: $+\sqrt{23}$ and $-\sqrt{23}$. These terms are additive inverses of each other. [Per the Additive Inverse Property, any number added to its negative equals zero, i.e., $x - x = 0$].

$ 3 + (\sqrt{23} - \sqrt{23}) = 3 + 0 $

$ = 3 $

Step 3: Theoretical Classification of the Result

The simplified result is the integer $3$. We must now evaluate this result against the formal definitions of rational and irrational numbers.

• Rational Number: A number that can be expressed in the form $\frac{p}{q}$, where $p$ and $q$ are integers, and $q \neq 0$.
• Irrational Number: A number that cannot be expressed as a simple fraction (its decimal expansion is non-terminating and non-repeating).

The integer $3$ can be rewritten as a fraction by placing it over a denominator of $1$:

$ 3 = \frac{3}{1} $

Here, $p = 3$ (an integer) and $q = 1$ (an integer where $1 \neq 0$). Because it strictly satisfies the condition $\frac{p}{q}$, the number is rational.

Visual Simplification Flowchart

Final Solution: The expression $(3 + \sqrt{23}) - \sqrt{23}$ simplifies to $3$, which is a rational number.