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Q1(iii):
State whether the following statements are true or false. Justify your answers.
(iii) Every real number is an irrational number.
Solution :
Step 1: Theoretical Foundation of the Real Number System
To evaluate the statement, we must first establish the rigorous mathematical definition of real numbers. The set of real numbers, denoted by $\mathbb{R}$, is defined as the union of two mutually exclusive and exhaustive subsets: the set of rational numbers ($\mathbb{Q}$) and the set of irrational numbers. [Per the fundamental axioms of set theory and the real number system].
Mathematically, this relationship is expressed as:
$\mathbb{R} = \mathbb{Q} \cup (\mathbb{R} \setminus \mathbb{Q})$
Where:
- $\mathbb{Q}$ represents rational numbers (numbers that can be expressed as $\frac{p}{q}$, where $p$ and $q$ are integers and $q \neq 0$).
- $\mathbb{R} \setminus \mathbb{Q}$ represents irrational numbers (numbers that cannot be expressed as a simple fraction, possessing non-terminating and non-repeating decimal expansions).
Step 2: Logical Analysis of the Statement
The given statement asserts: "Every real number is an irrational number."
In logical terms, this translates to the claim that the set of real numbers is entirely equivalent to the set of irrational numbers ($\mathbb{R} = \mathbb{R} \setminus \mathbb{Q}$). For this to be true, the set of rational numbers ($\mathbb{Q}$) would have to be an empty set ($\emptyset$). However, we know that the set of rational numbers is infinitely large. Therefore, the statement contains a fundamental logical fallacy.
Step 3: Proof by Counterexample
To formally disprove a universal affirmative statement ("every $X$ is $Y$"), we only need to provide a single valid counterexample [By the rules of deductive logic and proof by contradiction].
- Let us select the number $4$.
- $4$ is a real number because it represents a continuous quantity that can be plotted on the one-dimensional real number line.
- However, $4$ can be expressed in the fractional form $\frac{4}{1}$, where both $4$ and $1$ are integers, and the denominator is not zero.
- Therefore, $4$ is a rational number, which explicitly means it is not an irrational number.
Step 4: Visualizing the Subsets of Real Numbers
The following Venn diagram illustrates the composition of the real number system, clearly demonstrating that irrational numbers only make up one portion of the entire set.
Final Solution: False. The set of real numbers is comprised of both rational and irrational numbers. Therefore, a real number can be rational, meaning it is incorrect to state that every real number is irrational. For example, $5$ is a real number, but it is a rational number, not an irrational one.
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.
- 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.
- 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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