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Q1(i):
Write the following in decimal form and say what kind of decimal expansion each has :
(i) $\frac{36}{100}$
Solution :
Initial Setup & Given Variables
We are tasked with evaluating the rational number provided in fractional form:
$x = \frac{36}{100}$
The objective is to convert this fraction into its equivalent decimal representation and mathematically classify the nature of its decimal expansion.
Step 1: Conversion to Decimal Form
To convert a fraction to a decimal, we perform the division of the numerator by the denominator. When the denominator is a perfect power of $10$ (i.e., $10^k$), the division can be executed by shifting the decimal point of the numerator $k$ places to the left.
- Numerator: $36$ (which can be written as $36.0$)
- Denominator: $100 = 10^2$
Since the exponent $k = 2$, we shift the decimal point two places to the left:
$x = \frac{36}{100} = 0.36$
Step 2: Analyzing the Nature of the Decimal Expansion
A decimal expansion is classified based on its termination behavior:
- Terminating Decimal: The division process yields a remainder of zero after a finite number of steps, resulting in a finite number of digits after the decimal point.
- Non-Terminating Repeating Decimal: The division process never yields a zero remainder, but a digit or a block of digits repeats infinitely.
Because the value $0.36$ ends exactly after the hundredths place (two decimal digits), it is classified as a terminating decimal expansion.
Step 3: Theoretical Justification via Prime Factorization
[Per the Fundamental Theorem of Arithmetic and the properties of rational numbers], a rational number expressed in its simplest form $\frac{p}{q}$ will have a terminating decimal expansion if and only if the prime factorization of its denominator $q$ is strictly of the form:
$q = 2^n \times 5^m$
where $n$ and $m$ are non-negative integers.
Let us analyze the denominator $q = 100$:
$100 = 10 \times 10 = (2 \times 5) \times (2 \times 5) = 2^2 \times 5^2$
Here, $n = 2$ and $m = 2$. Because the prime factors of the denominator consist exclusively of $2$ and $5$, the mathematical theorem guarantees that the fraction $\frac{36}{100}$ must resolve into a terminating decimal.
Step 4: Visual Representation of the Decimal
To conceptualize $0.36$ geometrically, we can map the fraction onto a $10 \times 10$ Cartesian grid representing $100$ equal units. Shading $36$ of these units provides a precise visual truth of the decimal's magnitude.
Final Solution: The decimal form of $\frac{36}{100}$ is $0.36$, and it has a terminating decimal expansion.
More Questions from Class 9 Mathematics Number Systems EXERCISE 1.3
- 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(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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