NCERT Solutions for Class 9 Maths Exercise 1.3 Question 7
Understanding the Question 🧐
This question asks us to write three numbers whose decimal expansions have two important properties:
- Non-terminating: The decimal digits must go on forever without ending.
- Non-recurring (or non-repeating): There should be no single digit or block of digits that repeats itself in a cycle forever.
Numbers that fit this description are known as irrational numbers. This is what distinguishes them from rational numbers (fractions), which are always either terminating or have a repeating pattern.
Write three numbers whose decimal expansions are non-terminating non-recurring.
The Strategy: Creating a Non-Repeating Pattern 📝
The easiest way to answer this question is to invent numbers that follow a pattern, but the pattern itself changes, so it never repeats. We can do this by systematically adding more digits to a sequence.
Three Examples of Irrational Numbers ✅
Here are three examples of numbers whose decimal expansions are non-terminating and non-recurring. You can create infinitely many such numbers!
Example 1:
&&0.01001000100001…&&
Why it works: The number of zeros between the ones increases by one each time (one zero, then two, then three, and so on). Because the pattern keeps changing, no block of digits can ever repeat. The “…” indicates it is non-terminating.
Example 2:
&&0.720720072000720000…&&
Why it works: In this example, the block “72” is separated by an increasing number of zeros. This ensures the overall sequence never settles into a repeating cycle.
Example 3:
&&1.23223222322223…&&
Why it works: Here, the number of ‘2’s between each ‘3’ increases by one. This is another simple way to construct a decimal that is non-terminating and non-recurring.
Conclusion
The numbers we have written are all examples of irrational numbers. These ncert solutions show how easy it is to construct your own examples by creating a decimal that goes on forever and has a pattern that never repeats itself in a fixed cycle.
- Pi (&& \pi &&): &&\pi \approx 3.1415926535…&& The digits of Pi go on forever with no known repeating pattern.
- Square Root of 2 (&& \sqrt{2} &&): &&\sqrt{2} \approx 1.4142135623…&& This is another classic example of an irrational number.
- Rational Numbers: Decimals are terminating (e.g., &&0.5&&) or non-terminating repeating (e.g., &&0.333…&&).
- Irrational Numbers: Decimals are non-terminating and non-repeating (e.g., &&0.101001…&&).
FAQ
Q: What are the two main properties of the numbers requested in the question?
A: The numbers must have decimal expansions that are both non-terminating (they go on forever) and non-recurring (they do not have a repeating pattern of digits).
Q: What is the mathematical term for these types of numbers?
A: These numbers are called irrational numbers.
Q: Is the number &&0.121212…&& an example of a non-terminating non-recurring decimal?
A: No. While it is non-terminating, it is recurring (or repeating) because the block of digits ’12’ repeats forever. This makes it a rational number, not an irrational one.
Q: Can you give an example of a famous irrational number?
A: Pi (&&\pi&&), which is approximately &&3.14159…&&, and the square root of 2 (&&\sqrt{2}&&), approximately &&1.41421…&&, are two of the most famous irrational numbers. Their decimal expansions continue infinitely without any repeating pattern.
Q: How can I easily create my own example of an irrational number?
A: You can create a pattern that changes predictably so it never repeats. For instance, start with a block of digits and add an extra digit to the block each time you write it. An example is &&0.808008000800008…&&, where the number of zeros between the eights increases by one each time.
Further Reading
The discovery of irrational numbers was a major event in the history of mathematics. To learn more, you can refer to the official NCERT textbooks and read about the difference between rational and irrational numbers.
