NCERT Solutions for Class 9 Maths Exercise 1.1 Question 4
Understanding the Question 🧐
This question tests our knowledge of the number system. We need to check if the given statements are true or false and provide a logical reason for our answer. Understanding the definitions of natural numbers, whole numbers, integers, and rational numbers is key. This ncert solutions guide will clarify these concepts.
State whether the following statements are true or false. Give reasons for your answers.
(i) Every natural number is a whole number.
(ii) Every integer is a whole number.
(iii) Every rational number is a whole number.
Part (i): Every natural number is a whole number 📝
Answer: True
Reason: Let’s look at the definitions:
- Natural Numbers (N): These are the counting numbers: &&\{1, 2, 3, 4, …\}&&.
- Whole Numbers (W): These are all the natural numbers plus zero: &&\{0, 1, 2, 3, 4, …\}&&.
As you can see, the collection of whole numbers contains every single natural number. Therefore, the statement is true.
Part (ii): Every integer is a whole number 📝
Answer: False
Reason: Let’s define integers:
- Integers (Z): These include all whole numbers and their negative counterparts: &&\{…, -3, -2, -1, 0, 1, 2, 3, …\}&&.
- Whole Numbers (W): These are only non-negative: &&\{0, 1, 2, 3, …\}&&.
To prove a statement false, we just need one counterexample. Consider the number &&-2&&. It is an integer, but it is not a whole number because whole numbers cannot be negative. Therefore, not every integer is a whole number.
Part (iii): Every rational number is a whole number 📝
Answer: False
Reason: Let’s define rational numbers:
- Rational Numbers (Q): Any number that can be written in the form &&\frac{p}{q}&&, where &&p&& and &&q&& are integers and &&q \neq 0&&. This includes fractions and decimals that terminate or repeat.
- Whole Numbers (W): These are only &&\{0, 1, 2, 3, …\}&& and do not include fractions.
Let’s use a counterexample. The number &&\frac{1}{2}&& is a rational number (since it’s in &&\frac{p}{q}&& form), but it is not a whole number. Therefore, not every rational number is a whole number.
Conclusion and Key Points ✅
The relationship between these number sets is hierarchical. Natural numbers are a subset of whole numbers, which are a subset of integers, which are themselves a subset of rational numbers. This can be visualized as nesting circles, with each set containing the one before it.
Think of the number systems like nesting dolls:
- The smallest doll is Natural Numbers.
- It fits inside the Whole Numbers doll (which is just the natural numbers plus a spot for 0).
- That fits inside the Integers doll (which adds all the negatives).
- That fits inside the Rational Numbers doll (which adds all the fractions).
A smaller doll is always part of the bigger one, but a bigger doll is not part of the smaller one!
- Natural (N): &&1, 2, 3, …&&
- Whole (W): &&0, 1, 2, 3, …&&
- Integer (Z): &&…, -2, -1, 0, 1, 2, …&&
- Rational (Q): Any number of the form &&\frac{p}{q}&& (e.g., &&5, -2, 0, \frac{3}{4}, 0.25&&).
FAQ
Q: What is the difference between natural numbers and whole numbers?
A: Natural numbers are the counting numbers starting from &&1&& (&&1, 2, 3, …&&). Whole numbers are the set of natural numbers plus the number &&0&& (&&0, 1, 2, 3, …&&). The only difference is the inclusion of zero.
Q: What are integers?
A: Integers include all whole numbers (&&0, 1, 2, …&&), and their negative counterparts (&&…, -3, -2, -1&&). They do not include fractions or decimals.
Q: Why isn’t a negative number like -5 considered a whole number?
A: The definition of whole numbers includes only &&0&& and positive counting numbers. They cannot be negative. Therefore, &&-5&& is an integer, but not a whole number.
Q: Why isn’t a fraction like 3/4 considered a whole number?
A: Whole numbers are complete, non-fractional numbers. A number like &&\frac{3}{4}&& represents a part of a whole, so it is a rational number but not a whole number.
Further Reading
For more information on the Number System, you can refer to the official NCERT textbook or visit the NCERT website.
