GSEB Class 9 Maths Solutions Chapter 1 Number Systems Exercise 1.1

Question 1. Is zero a rational number? Can you write it in the form of \( \frac { p }{ q } \), where p and q are integers and \( q \ne 0 \)?
Answer: Yes, zero is a rational number, as 0 can be expressed in the form of \( \frac { p }{ q } \), where p and q represent integers and \( q \ne 0 \). We are able to write \( \frac { 0 }{ 1 } = \frac { 0 }{ 2 } = \frac { 0 }{ 3 } \), and so on.
In simple words: Zero is a rational number because you can write it as a fraction, like 0 over 1, where both numbers are whole numbers and the bottom number is not zero.

Exam Tip: A key point to remember is that any number expressible as \( \frac{p}{q} \), with p and q as integers and \( q \neq 0 \), is rational.

Question 2. Find six rational numbers between 3 and 4.
Answer: Many rational numbers can exist between 3 and 4.
Rational numbers between 3 and 4:
We use the formula \( \frac { 1 }{ 2 }(a + b) \) to find rational numbers between a and b.
First rational number:
\( = \frac { 1 }{ 2 }(3 + 4) \) (Here \( a = 3 \) and \( b = 4 \))
\( = \frac { 7 }{ 2 } \)
Second rational number between 3 and \( \frac { 7 }{ 2 } \):
\( = \frac { 1 }{ 2 }(3 + \frac { 7 }{ 2 }) \) (Here \( a = 3 \) and \( b = \frac { 7 }{ 2 } \))
\( = \frac { 1 }{ 2 }(\frac { 6+7 }{ 2 }) \)
\( = \frac { 13 }{ 4 } \)
Third rational number between 3 and \( \frac { 13 }{ 4 } \):
\( = \frac { 1 }{ 2 }(3 + \frac { 13 }{ 4 }) \) (Here \( a = 3 \) and \( b = \frac { 13 }{ 4 } \))
\( = \frac { 1 }{ 2 }(\frac { 12+13 }{ 4 }) \)
\( = \frac { 25 }{ 8 } \)
Fourth rational number between 3 and \( \frac { 25 }{ 8 } \):
\( = \frac { 1 }{ 2 }(3 + \frac { 25 }{ 8 }) \) (Here \( a = 3 \) and \( b = \frac { 25 }{ 8 } \))
\( = \frac { 1 }{ 2 }(\frac { 24+25 }{ 8 }) \)
\( = \frac { 1 }{ 2 } \times \frac { 49 }{ 8 } \)
\( = \frac { 49 }{ 16 } \)
Fifth rational number between 3 and \( \frac { 49 }{ 16 } \):
\( = \frac { 1 }{ 2 }(3 + \frac { 49 }{ 16 }) \) (Here \( a = 3 \) and \( b = \frac { 49 }{ 16 } \))
\( = \frac { 1 }{ 2 }(\frac { 48+49 }{ 16 }) \)
\( = \frac { 97 }{ 32 } \)
Sixth rational number between 3 and \( \frac { 97 }{ 32 } \):
\( = \frac { 1 }{ 2 }(3 + \frac { 97 }{ 32 }) \) (Here \( a = 3 \) and \( b = \frac { 97 }{ 32 } \))
\( = \frac { 1 }{ 2 }(\frac { 96+97 }{ 32 }) \)
\( = \frac { 1 }{ 2 }(\frac { 193 }{ 32 }) \)
\( = \frac { 193 }{ 64 } \)
So, six rational numbers between 3 and 4 are:
\( \frac { 193 }{ 64 }, \frac { 97 }{ 32 }, \frac { 49 }{ 16 }, \frac { 25 }{ 8 }, \frac { 13 }{ 4 }, \frac { 7 }{ 2 } \)
Alternative method:
We need to find \( n = 6 \) rational numbers.
So, \( n + 1 = 6 + 1 = 7 \)
Therefore, we can write 3 as \( \frac { 3 }{ 1 } = \frac { 3 \times 7 }{ 1 \times 7 } = \frac { 21 }{ 7 } \)
And 4 as \( \frac { 4 }{ 1 } = \frac { 4 \times 7 }{ 1 \times 7 } = \frac { 28 }{ 7 } \)
Six rational numbers between 3 and 4 are:
\( \frac { 22 }{ 7 }, \frac { 23 }{ 7 }, \frac { 24 }{ 7 }, \frac { 25 }{ 7 }, \frac { 26 }{ 7 }, \frac { 27 }{ 7 } \)
In simple words: To find rational numbers between two given numbers, you can either take their average repeatedly or multiply both numbers (and their denominators) by a number slightly larger than how many numbers you need to find.

Exam Tip: There are multiple methods to find rational numbers between two given numbers; choose the one you find most efficient and make sure to show all steps clearly.

Question 3. Find five rational numbers between \( \frac { 3 }{ 5 } \) and \( \frac { 4 }{ 5 } \)
Answer: We need to find 5 rational numbers between \( \frac { 3 }{ 5 } \) and \( \frac { 4 }{ 5 } \).
Here, \( n = 5 \). So, \( n + 1 = 5 + 1 = 6 \).
Multiply both the numerator and denominator by 6.
For \( \frac { 3 }{ 5 } \): \( \frac { 3 }{ 5 } \times \frac { 6 }{ 6 } = \frac { 18 }{ 30 } \)
For \( \frac { 4 }{ 5 } \): \( \frac { 4 }{ 5 } \times \frac { 6 }{ 6 } = \frac { 24 }{ 30 } \)
Hence, rational numbers between \( \frac { 18 }{ 30 } \) and \( \frac { 24 }{ 30 } \) are:
\( \frac { 19 }{ 30 }, \frac { 20 }{ 30 }, \frac { 21 }{ 30 }, \frac { 22 }{ 30 }, \frac { 23 }{ 30 } \)
In simple words: To find rational numbers between two fractions, you need to make their denominators larger by multiplying the top and bottom by a chosen number, then pick numbers that fit in between.

Exam Tip: When finding rational numbers between fractions, it's often easiest to make the denominators common and larger by multiplying by \( (n+1) \), where \( n \) is the number of rational numbers required.

Question 4. State whether the following statements are true or false. Give reasons for your answers.
(i) Every natural number is a whole number
(ii) Every integer is a whole number.
(iii) Every rational number is a whole number.
Answer:
(i) True, as the set of whole numbers includes all natural numbers.
(ii) False, because -1 is an integer, however, it is not a whole number.
(iii) False, because \( \frac { 2 }{ 3 } \) is a rational number, however, it is not a whole number.
In simple words: Natural numbers are counting numbers (1, 2, 3...), whole numbers include zero (0, 1, 2, 3...), integers include negative numbers, and rational numbers are fractions. Each set is larger and includes the previous one, but not the other way around.

Exam Tip: Understand the definitions and relationships between natural numbers, whole numbers, integers, and rational numbers to correctly identify true or false statements. A counter-example is often the easiest way to disprove a statement.