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Q1(ii):
Write the following in decimal form and say what kind of decimal expansion each has :
(ii) $\frac{1}{11}$
Solution :
Initial Setup & Mathematical Objective
The objective is to express the rational number $\frac{1}{11}$ in its decimal form and classify the nature of its decimal expansion. To achieve this, we apply the standard long division algorithm, dividing the numerator ($1$) by the denominator ($11$).
Step 1: Execution of the Long Division Algorithm
We set up the division $1 \div 11$. Since the dividend ($1$) is strictly less than the divisor ($11$), we place a decimal point in the quotient and append zeros to the dividend to proceed with the fractional parts.
- Tenths place: Bring down a $0$ to make the dividend $10$. Since $10 < 11$, the quotient digit is $0$. The remainder is $10$.
- Hundredths place: Bring down another $0$ to make the dividend $100$. Divide $100$ by $11$. The maximum multiple is $11 \times 9 = 99$. The quotient digit is $9$, and the remainder is $100 - 99 = 1$.
- Thousandths place: Bring down a $0$ to the remainder $1$, making it $10$. Since $10 < 11$, the quotient digit is $0$. The remainder is $10$.
- Ten-thousandths place: Bring down another $0$ to make it $100$. Divide $100$ by $11$. The quotient digit is $9$, and the remainder is $100 - 99 = 1$.
The visual representation of this algorithmic process is detailed below:
Step 2: Analysis of the Remainder and Quotient Sequences
By tracking the division process, we can observe a strict mathematical pattern in the remainders and the resulting quotient digits.
| Iteration | Current Dividend | Quotient Digit | Remainder |
|---|---|---|---|
| 1 | 10 | 0 | 10 |
| 2 | 100 | 9 | 1 |
| 3 | 10 | 0 | 10 |
| 4 | 100 | 9 | 1 |
[Per the properties of rational numbers], because the remainder sequence $(10, 1, 10, 1, \dots)$ begins to repeat, the sequence of digits in the quotient will also repeat infinitely. The repeating block of digits in the quotient is $09$.
Step 3: Classification of the Decimal Expansion
Because the remainder never becomes zero, the decimal expansion is non-terminating. Furthermore, because a specific block of digits ($09$) repeats infinitely, the expansion is repeating (or recurring). Mathematically, this is denoted by placing a vinculum (bar) over the repeating block of digits:
$ \frac{1}{11} = 0.090909\dots = 0.\overline{09} $
Final Solution: The decimal form of $\frac{1}{11}$ is $0.\overline{09}$, and it has a non-terminating repeating (recurring) decimal expansion.
More Questions from Class 9 Mathematics Number Systems EXERCISE 1.3
- Q1(i): Write the following in decimal form and say what kind of decimal expansion each has : (i) $\frac{36}{100}$
- Q1(iii): Write the following in decimal form and say what kind of decimal expansion each has : (iii) $4\frac{1}{8}$
- Q1(iv): Write the following in decimal form and say what kind of decimal expansion each has : (iv) $\frac{3}{13}$
- Q1(v): Write the following in decimal form and say what kind of decimal expansion each has : (v) $\frac{2}{11}$
- Q1(vi): Write the following in decimal form and say what kind of decimal expansion each has : (vi) $\frac{329}{400}$
- Q2: You know that $\frac{1}{7} = 0.\overline{142857}$. Can you predict what the decimal expansions of $\frac{2}{7}, \frac{3}{7}, \frac{4}{7}, \frac{5}{7}, \frac{6}{7}$ are, without actually doing the long division? If so, how? [Hint : Study the remainders while finding the value of $\frac{1}{7}$ carefully.]
- Q3(i): Express the following in the form $\frac{p}{q}$, where $p$ and $q$ are integers and $q \neq 0$. (i) $0.\overline{6}$
- Q3(ii): Express the following in the form $\frac{p}{q}$, where $p$ and $q$ are integers and $q \neq 0$. (ii) $0.4\overline{7}$
- Q3(iii): Express the following in the form $\frac{p}{q}$, where $p$ and $q$ are integers and $q \neq 0$. (iii) $0.\overline{001}$
- Q4: Express $0.99999 ....$ in the form $\frac{p}{q}$. Are you surprised by your answer? With your teacher and classmates discuss why the answer makes sense.
- Q5: What can the maximum number of digits be in the repeating block of digits in the decimal expansion of $\frac{1}{17}$? Perform the division to check your answer.
- Q6: Look at several examples of rational numbers in the form $\frac{p}{q}$ ($q \neq 0$), where $p$ and $q$ are integers with no common factors other than 1 and having terminating decimal representations (expansions). Can you guess what property $q$ must satisfy?
- Q7: Write three numbers whose decimal expansions are non-terminating non-recurring.
- Q8: Find three different irrational numbers between the rational numbers $\frac{5}{7}$ and $\frac{9}{11}$.
- Q9(i): Classify the following numbers as rational or irrational : (i) $\sqrt{23}$
- Q9(ii): Classify the following numbers as rational or irrational : (ii) $\sqrt{225}$
- Q9(iii): Classify the following numbers as rational or irrational : (iii) $0.3796$
- Q9(iv): Classify the following numbers as rational or irrational : (iv) $7.478478...$
- Q9(v): Classify the following numbers as rational or irrational : (v) $1.101001000100001...$
CBSE Solutions for Class 9 Mathematics Number Systems
Chapters in CBSE - Class 9 Mathematics
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