NCERT Solutions for Class 9 Maths Exercise 1.3 Question 8
Understanding the Question 🧐
The goal is to find three different irrational numbers that exist between two given rational numbers, &&\frac{5}{7}&& and &&\frac{9}{11}&&.
To do this, we need a clear plan:
- First, we’ll convert the two fractions into their decimal forms. This will give us a clear start and end point for our search.
- Then, we’ll create three new numbers between these two decimal values.
- We must ensure our new numbers are irrational, meaning their decimal expansions are non-terminating and non-recurring.
Find three different irrational numbers between the rational numbers &&\frac{5}{7}&& and &&\frac{9}{11}&&.
Step 1: Finding the Decimal Expansions 📝
Let’s perform the long division for each fraction to see their decimal values.
For &&\frac{5}{7}&&:
By dividing 5 by 7, we get a repeating decimal:
&&\frac{5}{7} = 0.714285714285… = 0.\overline{714285}&&
For &&\frac{9}{11}&&:
By dividing 9 by 11, we get another repeating decimal:
&&\frac{9}{11} = 0.818181… = 0.\overline{81}&&
So, our task is to find three irrational numbers between &&0.714285…&& and &&0.818181…&&.
Step 2: Constructing Three Irrational Numbers ✅
Now we can create numbers that start with digits between 0.71 and 0.81, and then add a non-repeating tail to make them irrational. There are infinite possibilities, so the answers below are just examples.
First Irrational Number:
Let’s pick a number that starts with ‘0.73’ (which is between 0.71 and 0.81).
&&0.73073007300073…&&
This number is irrational because the number of zeros between each ’73’ increases, so the pattern never repeats.
Second Irrational Number:
Let’s pick a number that starts with ‘0.75’.
&&0.751751175111…&&
Here, the number of ‘1’s after each ’75’ increases, ensuring the decimal is non-recurring.
Third Irrational Number:
Let’s pick a number that starts with ‘0.80’.
&&0.80800800080000…&&
This familiar pattern of adding an extra zero each time is a simple way to create an irrational number that is clearly between our two boundaries.
Conclusion
The three irrational numbers we have found, for example:
- &&0.73073007300073…&&
- &&0.751751175111…&&
- &&0.80800800080000…&&
…all lie between &&\frac{5}{7}&& and &&\frac{9}{11}&&. These ncert solutions illustrate that between any two distinct rational numbers, there is an infinite number of irrational numbers.
- Convert the given rational numbers to their decimal forms.
- Identify the starting digits of the range you need to fit your numbers into.
- Create new decimal numbers that begin with digits inside that range, and then add a non-terminating, non-repeating pattern to their tails.
FAQ
Q: What is the first step to find irrational numbers between two fractions?
A: The first step is to convert both fractions into their decimal forms. This helps you understand the numerical range you need to find numbers within.
Q: What are the decimal forms of &&\frac{5}{7}&& and &&\frac{9}{11}&&?
A: The decimal form of &&\frac{5}{7}&& is &&0.\overline{714285}&&, and the decimal form of &&\frac{9}{11}&& is &&0.\overline{81}&&.
Q: What makes a number irrational?
A: A number is irrational if its decimal representation is both non-terminating (it goes on forever) and non-recurring (it has no repeating pattern of digits).
Q: Is &&0.75&& an irrational number between &&\frac{5}{7}&& and &&\frac{9}{11}&&?
A: No. Although &&0.75&& is between &&\frac{5}{7}&& and &&\frac{9}{11}&&, it is a terminating decimal, which means it is a rational number (it equals &&\frac{3}{4}&&). The question specifically asks for irrational numbers.
Q: How many irrational numbers are there between &&\frac{5}{7}&& and &&\frac{9}{11}&&?
A: There are infinitely many irrational numbers between any two different rational numbers.
Further Reading
The concept that both rational and irrational numbers are densely packed on the number line is a key idea in understanding Real Numbers. To learn more, you can refer to the official NCERT textbooks.
