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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.
Solution :
Step 1: Variable Assignment and Equation Formulation
To express the non-terminating, repeating decimal $0.99999...$ (which can also be written as $0.\overline{9}$) in the rational form $\frac{p}{q}$, we begin by assigning it to a variable $x$.
Let $x = 0.99999... \quad \text{--- (Equation 1)}$
Step 2: Algebraic Multiplication
We observe the periodicity of the repeating decimal. Since exactly one digit (the digit $9$) repeats infinitely, we multiply both sides of Equation 1 by $10^1$ (or $10$). [Per the fundamental properties of equality and base-10 positional notation, multiplying by 10 shifts the decimal point one place to the right].
$10x = 9.99999... \quad \text{--- (Equation 2)}$
Step 3: Subtraction and Isolation of the Variable
To eliminate the infinitely repeating fractional part, we subtract Equation 1 from Equation 2. Because the infinite tail of $9$s is identical in both equations, their difference is exactly zero.
$10x = 9.99999...$
$- \quad x = 0.99999...$
$9x = 9.00000...$
Now, solve for $x$ by dividing both sides by $9$:
$x = \frac{9}{9}$
$x = 1$
Step 4: Expressing in $\frac{p}{q}$ Form
The integer $1$ can be expressed as a rational number in the form $\frac{p}{q}$ where $p$ and $q$ are integers and $q \neq 0$.
$x = \frac{1}{1}$
Step 5: Analytical Discussion & Theoretical Justification
The result $0.99999... = 1$ often causes initial surprise because they appear to be distinct numbers. However, in the real number system, they are two different representations of the exact same mathematical value. This makes perfect logical sense when analyzed through multiple mathematical frameworks:
- Argument 1: Fractional Equivalency
Consider the fraction $\frac{1}{3}$. In decimal form, $\frac{1}{3} = 0.33333...$
If we multiply both sides of this equation by $3$, we get:
$3 \times \frac{1}{3} = 3 \times 0.33333...$
$1 = 0.99999...$ - Argument 2: Infinite Geometric Series (Real Analysis)
The decimal $0.999...$ can be expanded as a sum of fractions:
$0.9 + 0.09 + 0.009 + \dots = \frac{9}{10} + \frac{9}{100} + \frac{9}{1000} + \dots$
This is an infinite geometric series where the first term $a = 0.9$ and the common ratio $r = 0.1$.
[Per the formula for the sum of an infinite geometric series, $S = \frac{a}{1 - r}$ for $|r| < 1$]:
$S = \frac{0.9}{1 - 0.1} = \frac{0.9}{0.9} = 1$. - Argument 3: The Archimedean Property
In the real number system, if two numbers are strictly different, there must exist a non-zero distance between them. The difference between $1$ and $0.999...$ is $0.000...$ with an infinite string of zeros. Since there is no "final" position to place a $1$, the difference is exactly $0$. Therefore, the numbers are identical.
Final Solution: The repeating decimal $0.99999...$ expressed in the rational form $\frac{p}{q}$ is $\frac{1}{1}$. The value is exactly equal to $1$, which is mathematically sound and justified by the properties of infinite geometric series, fractional equivalency, and the definition of limits in the real number system.
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) $0.\overline{001}$
- 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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