NCERT Solutions for Class 9 Maths Exercise 1.1 Question 1
Understanding the Question 🧐
This question asks us to determine if zero (&&0&&) qualifies as a rational number. To answer this, we must first understand the definition of a rational number. These ncert solutions will break down the concept step-by-step.
Is zero a rational number? Can you write it in the form &&\frac{p}{q}&&, where &&p&& and &&q&& are integers and &&q \neq 0&&?
Step-by-Step Solution 📝
Let’s analyze the question based on the definition of a rational number.
1. Definition of a Rational Number:
A number is called a rational number if it can be written in the form &&\frac{p}{q}&&, where:
- &&p&& and &&q&& are integers.
- The denominator, &&q&&, is not equal to zero (&&q \neq 0&&).
2. Checking if Zero Fits the Definition:
Now, let’s see if the number &&0&& meets these two conditions.
- Condition 1: Can we write &&0&& in the form &&\frac{p}{q}&&?
Yes, we can. We can set the numerator &&p&& to &&0&& and the denominator &&q&& to any non-zero integer. For example: &&\frac{0}{1}&&, &&\frac{0}{2}&&, &&\frac{0}{5}&&, &&\frac{0}{-3}&& All of these fractions are equal to &&0&&. - Condition 2: Are &&p&& and &&q&& integers, and is &&q \neq 0&&?
In our examples above:- For &&\frac{0}{1}&&: &&p=0&& (an integer) and &&q=1&& (an integer, and &&1 \neq 0&&).
- For &&\frac{0}{-3}&&: &&p=0&& (an integer) and &&q=-3&& (an integer, and &&-3 \neq 0&&).
Conclusion and Key Points ✅
Yes, zero is a rational number. It can be written in the form &&\frac{p}{q}&& where &&p=0&& and &&q&& is any non-zero integer. For example, &&0 = \frac{0}{1}&&.
Any integer can be a rational number because it can be written as a fraction with a denominator of &&1&&. Since &&0&& is an integer, it can be written as &&\frac{0}{1}&&, making it a rational number.
- A rational number must be expressible as a fraction &&\frac{p}{q}&&.
- The numerator &&p&& and denominator &&q&& must be integers.
- Crucially, the denominator &&q&& can never be zero (&&q \neq 0&&).
- Zero divided by any non-zero number is always zero.
FAQ
Q: What is the definition of a rational number?
A: A rational number is any number that can be expressed as a fraction &&\frac{p}{q}&&, where &&p&& (the numerator) and &&q&& (the denominator) are integers, and the denominator &&q&& is not equal to zero (&&q \neq 0&&).
Q: Is zero an integer?
A: Yes, zero (&&0&&) is an integer. It is a whole number that is neither positive nor negative.
Q: How can you write zero in the form p/q?
A: You can write zero as &&\frac{0}{1}&&, &&\frac{0}{2}&&, &&\frac{0}{-5}&&, or &&0&& divided by any non-zero integer. In all these cases, the numerator &&p&& is &&0&&, and the denominator &&q&& is a non-zero integer.
Q: Why can’t the denominator be zero in a rational number?
A: Division by zero is undefined in mathematics. If the denominator &&q&& in the fraction &&\frac{p}{q}&& were zero, the expression would have no meaning, which is why the definition of a rational number explicitly states that &&q \neq 0&&.
Q: What is the value of 0 divided by any non-zero integer?
A: The value of &&0&& divided by any non-zero integer is always &&0&&. For example, &&\frac{0}{7} = 0&&.
Q: Are all integers rational numbers?
A: Yes, all integers are rational numbers because any integer ‘&&n&&’ can be written in the form &&\frac{n}{1}&&. For example, the integer &&5&& can be written as &&\frac{5}{1}&&.
Further Reading
For more information on Number Systems, you can refer to the official NCERT textbook or visit the NCERT website.
