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Q9(iv):
Classify the following numbers as rational or irrational :
(iv) $7.478478...$
Solution :
Initial Setup & Given Number
We are tasked with classifying the given real number as either rational or irrational based on its decimal expansion.
Let the given number be $x$:
$x = 7.478478...$
Step 1: Analyzing the Decimal Expansion
By observing the sequence of digits after the decimal point, we can identify a distinct repeating pattern. The block of digits $478$ repeats infinitely. Therefore, the number can be expressed using bar notation over the repeating block:
$x = 7.\overline{478}$
[Per the fundamental theorem of real number decimal expansions, any number that exhibits a non-terminating but repeating (recurring) decimal expansion is, by definition, a rational number. Conversely, non-terminating and non-repeating decimals are irrational.]
Step 2: Algebraic Proof of Rationality
To rigorously prove that $x$ is rational, we must demonstrate that it can be expressed in the form $\frac{p}{q}$, where $p$ and $q$ are integers ($\in \mathbb{Z}$) and $q \neq 0$.
Let our initial equation be:
$x = 7.478478... \quad \text{--- (Equation 1)}$
Since the periodicity (the number of digits in the repeating block) is $3$, we multiply both sides of Equation 1 by $10^3$ (which is $1000$) to shift the decimal point past the first repeating block:
$1000x = 7478.478478... \quad \text{--- (Equation 2)}$
Next, we subtract Equation 1 from Equation 2 to eliminate the infinite repeating decimal part:
$1000x - x = 7478.478478... - 7.478478...$
$999x = 7471.000000...$
Solving for $x$, we isolate the variable:
$x = \frac{7471}{999}$
Step 3: Evaluating the Result
We have successfully expressed $x$ as a fraction $\frac{7471}{999}$.
- $p = 7471$, which is an integer ($p \in \mathbb{Z}$).
- $q = 999$, which is an integer ($q \in \mathbb{Z}$).
- $q \neq 0$ ($999 \neq 0$).
[By the formal definition of rational numbers ($\mathbb{Q}$), any number that satisfies these conditions is strictly rational.]
Visual Classification Flowchart
Final Solution: The number $7.478478...$ is a rational number.
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(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(v): Classify the following numbers as rational or irrational : (v) $1.101001000100001...$
