NCERT Solutions for Class 9 Maths Exercise 1.3 Question 1
Understanding the Question 🧐
In this question, we need to do two things for each given fraction: first, convert it into its decimal form by performing division. Second, we need to classify the decimal as one of two types:
- Terminating: The decimal comes to an end (the remainder becomes 0).
- Non-Terminating Repeating: The decimal goes on forever, but a block of digits repeats itself endlessly (the remainder never becomes 0 and starts repeating).
Write the following in decimal form and say what kind of decimal expansion each has.
Part (i): &&\frac{36}{100}&& 📝
Dividing by 100 is straightforward; we just move the decimal point two places to the left.
&& \frac{36}{100} = 0.36 &&
Since the decimal ends after the digit 6, the division process terminates (remainder is 0).
Kind of decimal expansion: Terminating
Part (ii): &&\frac{1}{11}&& 📝
We perform long division to find the decimal form of &&\frac{1}{11}&&.
When we divide 1 by 11, we find that the remainder starts repeating. The sequence of digits ’09’ will repeat forever.
&& \frac{1}{11} = 0.090909… = 0.\overline{09} &&
Since the decimal does not end and has a repeating block of digits (’09’), it is non-terminating and repeating.
Kind of decimal expansion: Non-terminating repeating
Part (iii): &&4\frac{1}{8}&& 📝
First, convert the mixed fraction into an improper fraction.
&& 4\frac{1}{8} = \frac{(4 \times 8) + 1}{8} = \frac{33}{8} &&
Now, we perform the division of 33 by 8.
&& \frac{33}{8} = 4.125 &&
The division ends when we get a remainder of 0.
Kind of decimal expansion: Terminating
Part (iv): &&\frac{3}{13}&& 📝
We perform long division to find the decimal form of &&\frac{3}{13}&&.
On dividing 3 by 13, we get a long sequence of remainders that eventually repeats. The block of digits ‘230769’ will repeat itself.
&& \frac{3}{13} = 0.230769230769… = 0.\overline{230769} &&
The decimal does not end and has a repeating block.
Kind of decimal expansion: Non-terminating repeating
Part (v): &&\frac{2}{11}&& 📝
Similar to part (ii), we perform long division for &&\frac{2}{11}&&.
The division shows a repeating block of digits ’18’.
&& \frac{2}{11} = 0.181818… = 0.\overline{18} &&
The decimal is non-terminating and has a repeating pattern.
Kind of decimal expansion: Non-terminating repeating
Part (vi): &&\frac{329}{400}&& 📝
We perform the division of 329 by 400.
&& \frac{329}{400} = 0.8225 &&
The division process ends with a remainder of 0.
Kind of decimal expansion: Terminating
Conclusion and Key Points ✅
This exercise demonstrates the two possible types of decimal representations for rational numbers (fractions). As shown in these ncert solutions, every rational number can be written as either a terminating decimal or a non-terminating repeating decimal.
- Terminating: The remainder becomes 0 during division.
- Non-terminating repeating: The remainder never becomes 0 and a pattern of remainders emerges, leading to a repeating block of digits.
- The bar over a block of digits (e.g., &&0.\overline{09}&&) is called a vinculum and indicates the repeating part.
FAQ
Q: What is a terminating decimal expansion?
A: A terminating decimal expansion is a decimal number that has a finite number of digits after the decimal point. This occurs when the long division process ends with a remainder of 0.
Q: What is a non-terminating repeating decimal expansion?
A: A non-terminating repeating (or recurring) decimal is a decimal number that continues forever, with a specific sequence of one or more digits repeating indefinitely. The long division process for such a fraction never reaches a remainder of 0.
Q: How can you tell if a fraction will have a terminating decimal without dividing?
A: A rational number (fraction) in its simplest form will have a terminating decimal expansion if and only if the prime factorization of its denominator contains no primes other than 2 and 5.
Q: Are all fractions (rational numbers) either terminating or repeating?
A: Yes, a fundamental property of rational numbers is that their decimal representation must be either terminating or non-terminating repeating.
Q: Can a decimal be non-terminating and non-repeating?
A: Yes, decimals that are non-terminating and non-repeating are called irrational numbers. Famous examples include Pi (&&\pi&&) and the square root of 2 (&&\sqrt{2}&&).
Further Reading
For a deeper understanding of real numbers and their decimal expansions, you can refer to the official NCERT textbooks.
