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Q1(ii):
Classify the following numbers as rational or irrational:
(ii) $(3 + \sqrt{23}) - \sqrt{23}$
Solution :
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.
More Questions from Class 9 Mathematics Number Systems EXERCISE 1.4
- Q1(i): Classify the following numbers as rational or irrational: (i) $2 - \sqrt{5}$
- Q1(iii): Classify the following numbers as rational or irrational: (iii) $\frac{2\sqrt{7}}{7\sqrt{7}}$
- Q1(iv): Classify the following numbers as rational or irrational: (iv) $\frac{1}{\sqrt{2}}$
- Q1(v): Classify the following numbers as rational or irrational: (v) $2\pi$
- Q2(i): Simplify each of the following expressions: (i) $(3 + \sqrt{3})(2 + \sqrt{2})$
- Q2(ii): Simplify each of the following expressions: (ii) $(3 + \sqrt{3})(3 - \sqrt{3})$
- Q2(iii): Simplify each of the following expressions: (iii) $(\sqrt{5} + \sqrt{2})^2$
- Q2(iv): Simplify each of the following expressions: (iv) $(\sqrt{5} - \sqrt{2})(\sqrt{5} + \sqrt{2})$
- Q3: Recall, $\pi$ is defined as the ratio of the circumference (say $c$) of a circle to its diameter (say $d$). That is, $\pi = \frac{c}{d}$. This seems to contradict the fact that $\pi$ is irrational. How will you resolve this contradiction?
- Q4: Represent $\sqrt{9.3}$ on the number line.
- Q5(i): Rationalise the denominators of the following: (i) $\frac{1}{\sqrt{7}}$
- Q5(ii): Rationalise the denominators of the following: (ii) $\frac{1}{\sqrt{7} - \sqrt{6}}$
- Q5(iii): Rationalise the denominators of the following: (iii) $\frac{1}{\sqrt{5} + \sqrt{2}}$
- Q5(iv): Rationalise the denominators of the following: (iv) $\frac{1}{\sqrt{7} - 2}$
CBSE Solutions for Class 9 Mathematics Number Systems
Chapters in CBSE - Class 9 Mathematics
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