NCERT Solutions for Class 9 Maths Exercise 1.2 Question 1
Understanding the Question 🧐
In this question, we need to check if three given statements about number systems are true or false. This involves understanding the relationship between different types of numbers: real numbers, rational numbers, and irrational numbers. Let’s provide a clear justification for each statement.
State whether the following statements are true or false. Justify your answers.
Part (i): Every irrational number is a real number. 📝
Answer: True
Justification: The definition of real numbers includes both rational and irrational numbers. Think of “Real Numbers” as a big box that contains two smaller boxes: one for “Rational Numbers” and one for “Irrational Numbers”.
- Any number that is irrational must, by definition, be inside the big box of Real Numbers.
- For example, &&\sqrt{2}&& is an irrational number, and it is also a real number. Similarly, &&\pi&& is an irrational number and also a real number.
- Therefore, the statement is correct. The collection of all real numbers is composed of all rational and all irrational numbers.
Part (ii): Every point on the number line is of the form &&\sqrt{m}&&, where &&m&& is a natural number. 📝
Answer: False
Justification: This statement claims that every single number on the number line (positive, negative, zero, fractions) can be written as the square root of a natural number. We can prove this false with a counterexample.
- Consider the negative numbers on the number line, for instance, &&-3&&.
- A natural number (&&m&&) is a positive integer (&&1, 2, 3, \dots&&).
- The square root of a positive number (&&\sqrt{m}&&) is always a positive number. It can never be negative.
- Since negative numbers like &&-3&& exist on the number line but cannot be expressed as &&\sqrt{m}&& for any natural number &&m&&, the statement is false.
Part (iii): Every real number is an irrational number. 📝
Answer: False
Justification: This statement is the reverse of the logic we used in Part (i) and is incorrect. The set of real numbers contains both rational and irrational numbers.
- To prove this is false, we just need to find one real number that is NOT irrational.
- Let’s take the number &&5&&. It is a real number because it’s on the number line.
- However, &&5&& is a rational number because it can be written as a fraction &&\frac{5}{1}&&.
- Since we found a real number (&&5&&) that is rational (and therefore not irrational), the statement that *every* real number is irrational is false.
Conclusion and Key Points ✅
To summarize our findings from these ncert solutions:
- The set of Real Numbers is the complete set of both rational and irrational numbers.
- Not all numbers on the number line can be expressed as &&\sqrt{m}&& where &&m&& is a natural number, because this form cannot produce negative numbers.
- Real numbers are not exclusively irrational; they also include all rational numbers.
- Real Numbers = Rational Numbers + Irrational Numbers.
- Every irrational number is a real number. (True)
- Every real number is NOT necessarily irrational. (False)
- The square root of a natural number is always non-negative.
FAQ
Q: What is a real number?
A: A real number is any number that can be represented on a continuous number line. It includes all rational numbers (like &&\frac{1}{2}$$, &&-5$$, &&0$$) and all irrational numbers (like &&\pi$$, &&\sqrt{2}$$).
Q: What is the key difference between rational and irrational numbers?
A: A rational number can be written as a fraction &&\frac{p}{q}$$, where &&p&& and &&q&& are integers and &&q \neq 0&&. An irrational number cannot be expressed as such a fraction; its decimal representation is non-terminating and non-repeating.
Q: Why is the statement ‘Every irrational number is a real number’ true?
A: This is true by definition. The set of real numbers is formed by combining the set of all rational numbers and the set of all irrational numbers. Therefore, any number that is irrational is automatically part of the larger set of real numbers.
Q: Why can’t every point on the number line be represented as &&\sqrt{m}&& for a natural number &&m&&?
A: The number line includes negative numbers. However, the square root of a natural number (&&m > 0&&) is always a non-negative number. For example, &&-3&& is on the number line but it cannot be expressed as &&\sqrt{m}&&.
Q: Give an example of a real number that is not irrational.
A: The number &&7&& is a real number, but it is not irrational. It is a rational number because it can be written as the fraction &&\frac{7}{1}&&.
Further Reading
For a deeper understanding of number systems, you can refer to the official NCERT textbooks, which provide the foundational concepts for these exercises.
