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Q2:
Are the square roots of all positive integers irrational? If not, give an example of the square root of a number that is a rational number.
Solution :
Step 1: Theoretical Foundation & Definitions
To rigorously evaluate the proposition, we must first establish the mathematical definitions of the sets of numbers involved:
- Positive Integers ($\mathbb{Z}^+$): The set of whole numbers greater than zero, denoted as $\{1, 2, 3, 4, 5, \dots\}$.
- Rational Numbers ($\mathbb{Q}$): Any number that can be expressed in the form $\frac{p}{q}$, where $p$ and $q$ are integers ($\mathbb{Z}$) and $q \neq 0$. Their decimal expansions are either terminating or non-terminating but repeating.
- Irrational Numbers ($\mathbb{I}$): Real numbers that cannot be expressed as a simple fraction $\frac{p}{q}$. Their decimal expansions are non-terminating and non-repeating [Per the fundamental properties of real numbers].
Step 2: Evaluating the Proposition
The question poses a universal statement: "The square roots of all positive integers are irrational."
In formal logic, to disprove a universal statement ("for all $x$"), one only needs to provide a single valid counterexample ("there exists an $x$ such that..."). If we can find even one positive integer whose square root is a rational number, the entire proposition is proven false [By the principle of Proof by Counterexample].
Step 3: Constructing the Counterexample
Let us examine the positive integer $n = 4$.
Taking the principal square root of $4$:
$ \sqrt{4} = 2 $
Now, we must determine if the result, $2$, is a rational number. We apply the definition of a rational number ($\frac{p}{q}$):
$ 2 = \frac{2}{1} $
Here, $p = 2$ (which is an integer) and $q = 1$ (which is an integer and $1 \neq 0$). Because $2$ perfectly satisfies the conditions of a rational number, $\sqrt{4}$ is rational. This directly contradicts the claim that the square roots of all positive integers are irrational.
Step 4: Generalization of the Counterexample
This phenomenon is not isolated to the number $4$. It applies to all perfect squares. A perfect square is an integer that is the square of an integer.
Let $k$ be any positive integer. If we define a positive integer $n$ such that $n = k^2$, then:
$ \sqrt{n} = \sqrt{k^2} = k $
Since $k$ is an integer, it can always be written as $\frac{k}{1}$, making it a rational number. We can observe this pattern in the table below:
| Positive Integer ($n$) | Square Root ($\sqrt{n}$) | Fractional Form ($\frac{p}{q}$) | Classification |
|---|---|---|---|
| $1$ | $1$ | $\frac{1}{1}$ | Rational |
| $2$ | $\approx 1.414213\dots$ | Cannot be expressed | Irrational |
| $3$ | $\approx 1.732050\dots$ | Cannot be expressed | Irrational |
| $4$ | $2$ | $\frac{2}{1}$ | Rational |
| $9$ | $3$ | $\frac{3}{1}$ | Rational |
Step 5: Visualizing Rational vs. Irrational Square Roots
The geometric representation below maps the square roots of positive integers onto a real number line, demonstrating the precise locations where square roots yield rational integers versus irrational values.
Final Solution: No, the square roots of all positive integers are not irrational. A definitive counterexample is the positive integer $4$. The square root of $4$ is $2$, which is a rational number because it can be expressed exactly as the fraction $\frac{2}{1}$. This holds true for all perfect squares (e.g., $\sqrt{9}=3$, $\sqrt{16}=4$, $\sqrt{25}=5$).
More Questions from Class 9 Mathematics Number Systems EXERCISE 1.2
- Q1(i): State whether the following statements are true or false. Justify your answers. (i) Every irrational number is a real number.
- Q1(ii): State whether the following statements are true or false. Justify your answers. (ii) Every point on the number line is of the form $\sqrt{m}$, where $m$ is a natural number.
- Q1(iii): State whether the following statements are true or false. Justify your answers. (iii) Every real number is an irrational number.
- Q3: Show how $\sqrt{5}$ can be represented on the number line.
- Q4: Classroom activity (Constructing the ‘square root spiral’) : Take a large sheet of paper and construct the ‘square root spiral’ in the following fashion. Start with a point $O$ and draw a line segment $OP_1$ of unit length. Draw a line segment $P_1P_2$ perpendicular to $OP_1$ of unit length (see Fig. 1.9). Now draw a line segment $P_2P_3$ perpendicular to $OP_2$. Then draw a line segment $P_3P_4$ perpendicular to $OP_3$. Continuing in this manner, you can get the line segment $P_{n-1}P_n$ by drawing a line segment of unit length perpendicular to $OP_{n-1}$. In this manner, you will have created the points $P_2, P_3,...., P_n,...$ ., and joined them to create a beautiful spiral depicting $\sqrt{2}, \sqrt{3}, \sqrt{4}, ...$
CBSE Solutions for Class 9 Mathematics Number Systems
Chapters in CBSE - Class 9 Mathematics
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