RBSE Solutions Class 9 Maths Chapter 2 Number System Exercise 2.1

Rajasthan Board RBSE Class 9 Maths Solutions Chapter 2 Number System Ex 2.1

Question 1. Classify the following numbers as rational or irrational:
(i) \( \sqrt{23} \)
(ii) \( \sqrt{225} \)
(iii) 0.3796
(iv) 7.478478...
(v) 1.101001000100001...
Answer:
(i) \( \sqrt{23} \)
The number 23 is not a perfect square. This means that \( \sqrt{23} \) cannot be written as a simple fraction, so it is an irrational number. When you find its decimal, it goes on forever without a pattern.
(ii) \( \sqrt{225} \)
The number 225 is a perfect square, as \( 15 \times 15 = 225 \). So, \( \sqrt{225} = 15 \). Since 15 can be written as \( \frac{15}{1} \), it is a rational number. Rational numbers are those that can be expressed as a fraction \( \frac{p}{q} \), where \( q \neq 0 \).
(iii) 0.3796
This decimal number stops after a few digits. Any decimal that ends can be easily written as a fraction. For example, \( 0.3796 = \frac{3796}{10000} \). Therefore, it is a rational number.
(iv) 7.478478...
In this decimal, the digits '478' keep repeating. A decimal that goes on forever but has a repeating pattern can always be written as a fraction. Thus, it is a rational number.
(v) 1.101001000100001...
This decimal number goes on forever, and there is no repeating pattern in its digits. Numbers like these, which never end and never repeat, cannot be written as a fraction. So, it is an irrational number.
In simple words: Rational numbers either end in decimal form or have a repeating pattern, and can be written as a fraction. Irrational numbers go on forever without any repeating pattern and cannot be written as a simple fraction.

🎯 Exam Tip: Remember that square roots of non-perfect squares are always irrational. Also, any number that has a non-terminating, non-repeating decimal expansion is irrational.

Question 2. Write three numbers whose decimal expansions are non-terminating non-recurring.
Answer: Three numbers whose decimal expansions are non-terminating and non-recurring (meaning they are irrational numbers) are:
1. 0.01001000100001... (Here, the number of zeros between the ones increases each time, preventing a simple repeat.)
2. 0.02002000200002... (Similar to the first example, the pattern of zeros grows, making it non-repeating.)
3. 0.03003000300003... (This also creates an unpredictable sequence, confirming its irrational nature.)
There are infinitely many such numbers, and you can create them by ensuring there is no fixed block of digits that repeats.
In simple words: We need to list three numbers that never end and never repeat any part of their decimal. These are called irrational numbers.

🎯 Exam Tip: To create a non-terminating, non-recurring decimal, make sure to change the number of zeros (or any other digit) in a sequence, preventing a consistent block from repeating.

Question 3. Write the decimal form of each of the following fractions.
(iii) \( 4\frac{1}{8} \)
(iv) \( \frac{3}{13} \)
(v) \( \frac{2}{11} \)
(vi) \( \frac{329}{400} \)
Answer:
(iii) \( 4\frac{1}{8} \)
First, convert the mixed fraction to an improper fraction: \( 4\frac{1}{8} = \frac{(4 \times 8) + 1}{8} = \frac{32 + 1}{8} = \frac{33}{8} \).
Now, perform the division: \( 33 \div 8 = 4.125 \).
This is a terminating decimal expansion because the division ends after a few steps.
(iv) \( \frac{3}{13} \)
When we divide 3 by 13 using long division, we get:
\( \frac{3}{13} = 0.230769230769... \)
Here, the block of digits '230769' repeats. So, we can write it as \( 0.\overline{230769} \).
This is a non-terminating repeating decimal expansion because the decimal goes on forever with a repeating pattern.
(v) \( \frac{2}{11} \)
When we divide 2 by 11 using long division, we get:
\( \frac{2}{11} = 0.181818... \)
Here, the block of digits '18' repeats. So, we can write it as \( 0.\overline{18} \).
This is a non-terminating repeating decimal expansion.
(vi) \( \frac{329}{400} \)
When we divide 329 by 400 using long division, we get:
\( \frac{329}{400} = 0.8225 \).
This is a terminating decimal expansion because the division ends. Fractions whose denominators have only prime factors of 2 or 5 will always result in terminating decimals.
In simple words: To find the decimal form, divide the top number by the bottom number. If the division stops, it's a terminating decimal. If digits keep repeating, it's a non-terminating repeating decimal, and you put a bar over the repeating part.

🎯 Exam Tip: For fractions, if the prime factors of the denominator are only 2s and 5s, the decimal expansion will terminate. Otherwise, it will be non-terminating and repeating.

Question 4. Express the following in the form \( \frac{p}{q} \), where p and q are integers and \( q \neq 0 \).
(i) \( 0.\overline{3} \)
(ii) \( 0.\overline{47} \)
(iii) \( 1.\overline{27} \)
(iv) \( 1.2\overline{35} \)
Answer:
(i) To express \( 0.\overline{3} \) in \( \frac{p}{q} \) form:
Let \( x = 0.\overline{3} \)
\( \implies x = 0.3333... \) (Equation i)
Since one digit is repeating, multiply Equation (i) by 10:
\( \implies 10x = 3.3333... \) (Equation ii)
Now, subtract Equation (i) from Equation (ii):
\( 10x - x = 3.3333... - 0.3333... \)
\( \implies 9x = 3 \)
\( \implies x = \frac{3}{9} \)
\( \implies x = \frac{1}{3} \)
(ii) To express \( 0.\overline{47} \) in \( \frac{p}{q} \) form:
Let \( x = 0.\overline{47} \)
\( \implies x = 0.474747... \) (Equation i)
Since two digits are repeating, multiply Equation (i) by 100:
\( \implies 100x = 47.474747... \) (Equation ii)
Now, subtract Equation (i) from Equation (ii):
\( 100x - x = 47.474747... - 0.474747... \)
\( \implies 99x = 47 \)
\( \implies x = \frac{47}{99} \)
(iii) To express \( 1.\overline{27} \) in \( \frac{p}{q} \) form:
Let \( x = 1.\overline{27} \)
\( \implies x = 1.272727... \) (Equation i)
Since two digits are repeating, multiply Equation (i) by 100:
\( \implies 100x = 127.272727... \) (Equation ii)
Now, subtract Equation (i) from Equation (ii):
\( 100x - x = 127.272727... - 1.272727... \)
\( \implies 99x = 126 \)
\( \implies x = \frac{126}{99} \)
\( \implies x = \frac{14}{11} \)
(iv) To express \( 1.2\overline{35} \) in \( \frac{p}{q} \) form:
Let \( x = 1.2\overline{35} \)
\( \implies x = 1.2353535... \) (Equation i)
First, multiply by 10 to move the non-repeating digit (2) past the decimal point:
\( \implies 10x = 12.353535... \) (Equation ii)
Now, since two digits (35) are repeating, multiply Equation (ii) by 100:
\( \implies 100 \times (10x) = 100 \times (12.353535...) \)
\( \implies 1000x = 1235.353535... \) (Equation iii)
Now, subtract Equation (ii) from Equation (iii):
\( 1000x - 10x = 1235.353535... - 12.353535... \)
\( \implies 990x = 1223 \)
\( \implies x = \frac{1223}{990} \)
In simple words: To change a repeating decimal into a fraction, first set the decimal equal to 'x'. Then, multiply 'x' by powers of 10 to line up the repeating parts. Subtract the original 'x' equation from the multiplied one to remove the repeating part, which lets you solve for 'x' as a simple fraction.

🎯 Exam Tip: Pay close attention to how many digits are repeating and if there are any non-repeating digits after the decimal point. This determines whether you multiply by 10, 100, or 1000 and which equations you subtract.

Question 5. Find three different irrational numbers between the rational numbers \( \frac{5}{7} \) and \( \frac{9}{11} \).
Answer:
First, convert the given rational numbers to their decimal forms:
\( \frac{5}{7} = 0.714285714285... = 0.\overline{714285} \)
\( \frac{9}{11} = 0.818181... = 0.\overline{81} \)
We need to find three irrational numbers between 0.714285... and 0.818181.... Irrational numbers are non-terminating and non-repeating decimals.
Here are three examples of irrational numbers between them:
1. 0.720720072000... (The number of zeros between the 72 blocks increases, making it non-repeating.)
2. 0.750750075000... (Similar to the first, the pattern changes each time.)
3. 0.801001000100001... (This number is greater than 0.714285... and less than 0.818181... and has no repeating block.)
There are infinitely many irrational numbers between any two rational numbers.
In simple words: First, turn the fractions into decimals to see their range. Then, pick three numbers that fall between these decimals but make sure they never end and never repeat in any regular way, making them irrational.

🎯 Exam Tip: To construct an irrational number, use a pattern that changes each time, like increasing the number of zeros between a set of digits (e.g., 0.1010010001...). Ensure it falls within the required range.