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Q9(ii):
Classify the following numbers as rational or irrational :
(ii) $\sqrt{225}$
Solution :
Initial Setup & Mathematical Definitions
We are tasked with classifying the real number $\sqrt{225}$ as either a rational or an irrational number. To do this rigorously, we must evaluate the radical expression and apply the formal definitions of the real number system.
- Rational Number: A number that can be expressed in the form $\frac{p}{q}$, where $p$ and $q$ are integers ($p, q \in \mathbb{Z}$) and $q \neq 0$.
- Irrational Number: A real number that cannot be expressed as a simple fraction of two integers. Its decimal expansion is non-terminating and non-repeating.
Step 1: Prime Factorization of the Radicand
The number inside the square root symbol is called the radicand. Here, the radicand is $225$. To simplify the square root, we first determine the prime factorization of $225$ [Per the Fundamental Theorem of Arithmetic].
We systematically divide $225$ by the smallest possible prime numbers:
- $225$ ends in $5$, but let us check divisibility by $3$. The sum of the digits is $2 + 2 + 5 = 9$, which is divisible by $3$.
- $225 \div 3 = 75$
- $75 \div 3 = 25$
- $25 \div 5 = 5$
- $5 \div 5 = 1$
Thus, the prime factorization of $225$ is:
$225 = 3 \times 3 \times 5 \times 5 = 3^2 \times 5^2$
Step 2: Simplification of the Radical Expression
Substitute the prime factorization back into the radical expression. Using the property of radicals $\sqrt{a \times b} = \sqrt{a} \times \sqrt{b}$ (where $a, b \ge 0$):
$\sqrt{225} = \sqrt{3^2 \times 5^2}$
$\sqrt{225} = \sqrt{3^2} \times \sqrt{5^2}$
Since the square root of a squared positive real number is the number itself ($\sqrt{x^2} = x$ for $x \ge 0$):
$\sqrt{225} = 3 \times 5 = 15$
Step 3: Geometric Interpretation of a Perfect Square
Geometrically, finding the square root of $225$ is equivalent to finding the side length of a square whose total area is $225$ square units. Because the side length evaluates to exactly $15$ (a whole number), $225$ is classified as a perfect square.
Step 4: Classification Based on Number Theory
We have established that $\sqrt{225} = 15$. We must now determine if $15$ fits the definition of a rational number.
Any integer $n$ can be written as a fraction by placing it over $1$ ($n = \frac{n}{1}$). Therefore:
$15 = \frac{15}{1}$
In this fraction:
- $p = 15$, which is an integer ($15 \in \mathbb{Z}$).
- $q = 1$, which is an integer ($1 \in \mathbb{Z}$).
- The denominator $q \neq 0$ ($1 \neq 0$).
Because $\sqrt{225}$ can be expressed exactly as the ratio of two integers, it perfectly satisfies the criteria for being a rational number.
Final Solution: $\sqrt{225}$ simplifies to $15$, which can be written as $\frac{15}{1}$. Therefore, $\sqrt{225}$ 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(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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