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Q1(iv):
Write the following in decimal form and say what kind of decimal expansion each has :
(iv) $\frac{3}{13}$
Solution :
Initial Setup & Given Expression
We are tasked with converting the rational number $\frac{3}{13}$ into its decimal form and determining the specific nature of its decimal expansion. To achieve this, we will perform the long division algorithm, dividing the numerator ($3$) by the denominator ($13$).
Step 1: Executing the Long Division Algorithm
Since the dividend ($3$) is strictly less than the divisor ($13$), we place a decimal point in the quotient and append zeros to the dividend to proceed with the division.
Let us break down the sequential arithmetic of the division:
- $30 \div 13 = 2$ with a remainder of $4$ $\quad (13 \times 2 = 26)$
- $40 \div 13 = 3$ with a remainder of $1$ $\quad (13 \times 3 = 39)$
- $10 \div 13 = 0$ with a remainder of $10$ $\quad (13 \times 0 = 0)$
- $100 \div 13 = 7$ with a remainder of $9$ $\quad (13 \times 7 = 91)$
- $90 \div 13 = 6$ with a remainder of $12$ $\quad (13 \times 6 = 78)$
- $120 \div 13 = 9$ with a remainder of $3$ $\quad (13 \times 9 = 117)$
Step 2: Analyzing the Sequence of Remainders
The sequence of remainders generated during the division is: $3, 4, 1, 10, 9, 12$. At the 6th step, the remainder is $3$, which is the exact value of our original dividend. [Per the properties of rational numbers], whenever a remainder repeats in long division, the sequence of digits in the quotient will also begin to repeat infinitely in the exact same order.
The block of digits in the quotient corresponding to this cycle is $230769$. Therefore, the decimal expansion can be written using a vinculum (bar) over the repeating block:
$\frac{3}{13} = 0.230769230769... = 0.\overline{230769}$
Step 3: Theoretical Justification & Classification
We can verify the nature of this decimal expansion without division by analyzing the prime factorization of the denominator. [By the Fundamental Theorem of Arithmetic and the properties of rational numbers], a rational number $\frac{p}{q}$ (in simplest form) will have a terminating decimal expansion if and only if the prime factorization of $q$ is of the form $2^n \times 5^m$, where $n$ and $m$ are non-negative integers.
In our given fraction $\frac{3}{13}$:
- The denominator is $q = 13$.
- The prime factorization of $13$ is simply $13^1$.
- Because the prime factorization contains a prime number other than $2$ or $5$, the decimal expansion cannot terminate.
Since it does not terminate but represents a rational number, it must be a non-terminating repeating (or recurring) decimal.
Final Solution: The decimal form of $\frac{3}{13}$ is $0.\overline{230769}$, and it has a non-terminating repeating 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(ii): Write the following in decimal form and say what kind of decimal expansion each has : (ii) $\frac{1}{11}$
- Q1(iii): Write the following in decimal form and say what kind of decimal expansion each has : (iii) $4\frac{1}{8}$
- 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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