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Q8:
Find three different irrational numbers between the rational numbers $\frac{5}{7}$ and $\frac{9}{11}$.
Solution :
Step 1: Decimal Expansion of the Given Rational Numbers
To find irrational numbers between two rational numbers, we must first determine the decimal expansions of the given boundaries. This allows us to establish a clear numerical range.
- First Rational Number ($\frac{5}{7}$):
Performing long division of $5$ by $7$:
$5 \div 7 = 0.714285714285\dots$
Since the block of digits "$714285$" repeats infinitely, we express this as $0.\overline{714285}$. - Second Rational Number ($\frac{9}{11}$):
Performing long division of $9$ by $11$:
$9 \div 11 = 0.818181\dots$
Since the block of digits "$81$" repeats infinitely, we express this as $0.\overline{81}$.
Step 2: Establishing the Range and Criteria
[Per the fundamental properties of the Real Number System], an irrational number is defined as a number whose decimal expansion is strictly non-terminating and non-repeating. Therefore, our objective is to construct three distinct decimal numbers, say $x$, $y$, and $z$, that satisfy two conditions:
- Range Condition: $0.\overline{714285} < x, y, z < 0.\overline{81}$
- Irrationality Condition: The decimals must not terminate and must not settle into a repeating periodic block.
Step 3: Construction of the Irrational Numbers
We can systematically construct non-terminating, non-repeating decimals by choosing starting digits that fall strictly between $0.71$ and $0.81$, and then appending a pattern of digits separated by an increasing number of zeros. This guarantees the sequence never repeats a fixed block.
- First Irrational Number ($x$):
Choose a prefix greater than $0.714\dots$, such as $0.72$.
Construct the non-repeating pattern: $x = 0.720720072000720000\dots$ - Second Irrational Number ($y$):
Choose a prefix greater than $0.72$ but less than $0.81$, such as $0.75$.
Construct the non-repeating pattern: $y = 0.750750075000750000\dots$ - Third Irrational Number ($z$):
Choose a prefix greater than $0.75$ but less than $0.818\dots$, such as $0.80$.
Construct the non-repeating pattern: $z = 0.808008000800008\dots$
Step 4: Visualizing the Real Number Line
The following high-precision number line maps the exact spatial relationship between the rational boundaries and the newly constructed irrational numbers.
Note: There are infinitely many irrational numbers between any two rational numbers. The three numbers constructed above are just one valid set of examples.
Final Solution: Three different irrational numbers between $\frac{5}{7}$ and $\frac{9}{11}$ are $0.720720072000\dots$, $0.750750075000\dots$, and $0.80800800080000\dots$.
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.
- 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) $1.101001000100001...$
