NCERT Solutions for Class 9 Maths Exercise 5.1 Question 7
Hello, future mathematicians! This guide from ncert solutions will clarify Question 7 of Exercise 5.1. It’s a conceptual question that tests our understanding of what makes an axiom a ‘universal truth’. Let’s explore why Euclid’s fifth axiom is so fundamental.
| Question | Why is Axiom 5, in the list of Euclid’s axioms, considered a ‘universal truth’? |
|---|---|
| Axiom 5 Statement | “The whole is greater than the part.” |
| Core Reason | The statement is self-evident and true in all situations and for any kind of magnitude, not just in geometry. |
Question 7: Why is Axiom 5, in the list of Euclid’s axioms, considered a ‘universal truth’? (Note that the question is not about the fifth postulate.)
How to Justify Euclid’s Axiom 5 as a Universal Truth 🤔
Follow this logical thought process to understand and explain the concept.
- Step 1: State Euclid’s Axiom 5
Clearly write down the axiom: ‘The whole is greater than the part.’ This is the statement we need to justify. - Step 2: Define ‘Universal Truth’
Understand that a universal truth is a statement that is self-evident and holds true in any context, for any objects or quantities, not just in geometry. - Step 3: Provide a Geometric Example
Consider a line segment &&AC&& as the ‘whole’. If you take a smaller section of it, like segment &&AB&& (where &&B&& is between &&A&& and &&C&&), this is a ‘part’. Visually and logically, the length of &&AC&& is greater than the length of &&AB&&. - Step 4: Provide a Non-Geometric Example
Think of a real-world object, like a pizza. The entire pizza is the ‘whole’. Any single slice taken from it is a ‘part’. The whole pizza is obviously larger than any one of its slices. - Step 5: Formulate the Conclusion
Since this principle applies universally to everything—from geometric figures to physical objects and even abstract concepts—it is accepted as a self-evident and universal truth that does not require proof.
Detailed Explanation 📝
Euclid’s Axiom 5 states: “The whole is greater than the part.”
This statement is considered a ‘universal truth’ because it is a self-evident principle that holds true in any context imaginable, not just within the field of geometry. It doesn’t require any proof because it aligns with our intuitive understanding of quantities and magnitudes.
Let’s consider some examples:
- In Geometry: Let a line segment &&AC&& be the ‘whole’. If we take a point &&B&& that lies on &&AC&&, then the segment &&AB&& is a ‘part’ of &&AC&&. It is obvious that the length of &&AC&& is greater than the length of &&AB&&. We can write this as &&AC > AB&&.
- In Arithmetic: The number 10 (the whole) is greater than the number 7 (which is a part of 10, since &&7+3=10&&).
- In the Real World: A country like India (the whole) is geographically larger than any of its states, like Madhya Pradesh (a part). A full pizza (the whole) is larger than a single slice of it (a part).
Since this principle is true for any object or concept in the universe, it is an undeniable and universal truth. It is an axiom upon which other logical statements can be built.
Common Mistakes to Avoid 🖍️
The most common error is confusing Axiom 5 with Postulate 5. The question explicitly notes this. Remember:
- Axiom 5: “The whole is greater than the part.” (A simple, universal truth).
- Postulate 5: A complex statement about parallel lines, which is an assumption specific to Euclidean geometry.
Always read the question carefully to distinguish between the two.
FAQ (Frequently Asked Questions)
Q: What is Euclid’s Axiom 5?
A: Euclid’s Axiom 5 states, “The whole is greater than the part.” It’s a fundamental principle used in geometry and logic.
Q: What is the difference between an axiom and a postulate?
A: An axiom is a self-evident truth that applies to all fields of study (e.g., ‘the whole is greater than the part’). A postulate is a specific assumption made for geometry that may not apply elsewhere (e.g., ‘a straight line can be drawn between any two points’).
Q: Is Euclid’s fifth POSTULATE the same as the fifth AXIOM?
A: No, they are very different. Axiom 5 is “The whole is greater than the part.” Postulate 5 is a much more complex statement about two straight lines being intersected by a third line, which forms the basis for the concept of parallel lines. The question specifically refers to the axiom.
Q: Can you give another real-world example of this axiom?
A: Certainly. A full year (the whole) is greater than any month within it (a part). The population of a country (the whole) is greater than the population of any single city within it (a part).
Further Reading
To deepen your understanding of Euclid’s postulates and axioms, you can refer to the official NCERT Class 9 Maths textbook available at https://ncert.nic.in/.
