Cbse Class 9 Maths Chapter 1 Exercise 1.1 Solutions

Yes, zero is a rational number and it can be represented in the form of \( \frac{P}{q} \) as \( \frac{0}{1}, \frac{0}{2} \) or \( \frac{0}{\mathrm{z}} \) where z is an integer and \( \mathrm{z} \neq 0 \).

There are infinite rational numbers between 3 and 4.
So, we can represent 3 and 4 in terms of fractions as an example:
(Multiplying \(\&\) dividing by 8)
\(3=\frac{3 \times 8}{8}=\frac{24}{8}\)
\(4=\frac{4 \times 8}{8}=\frac{32}{8}\)
NOTE: We can multiply \(\&\) divide by other numbers as well.
We are multiplying and dividing at the same time because it is easier to write other fractions in between. The decimal numbers between 3 \(\&\) 4 can also be written for e.g. 3.1, 3.2, 3.3, etc.
Now, we can write the six rational numbers between 3 and 4 as:
\(\frac{25}{8}, \frac{26}{8}, \frac{27}{8}, \frac{28}{8}, \frac{29}{8}, \frac{30}{8}\)
Now let us try dividing with some other number.(Multiplying and dividing by 10 )
\(3=\frac{3 \times 10}{10}=\frac{30}{10}\)
\(4=\frac{4 \times 10}{10}=\frac{40}{10}\)
And now we can write the rational numbers between 3 and 4 as \( \frac{31}{10}, \frac{32}{10}, \frac{33}{10}, \frac{34}{10} \ldots \). and so on
So, you can take any number.

Between \( \frac{3}{5} \) and \( \frac{4}{5} \) there exist infinite rational numbers \(\&\) we know that rational numbers are fractions, whole numbers and decimals.
Trick: To find "n" rational numbers between any two numbers, multiply \( \& \) divide both by "\( \mathrm{n}+1 \)".
Between \( \frac{3}{5} \) and \( \frac{4}{5} \) there exist infinite rational numbers \(\&\) we know that rational numbers are fractions, whole numbers and decimals.
Trick: To find "n" rational numbers between any two numbers, multiply \(\&\) divide both by "\( \mathrm{n}+1 \)".
Example: If we need to find 5 rational numbers between \( \frac{3}{5} \) and \( \frac{4}{5} \), we multiply and divide both the numbers by 6 to obtain the set of rational numbers.
\(\frac{3}{5}=\frac{3 \times 6}{5 \times 6}=\frac{18}{30}\)
\(\frac{4}{5}=\frac{4 \times 6}{5 \times 6}=\frac{24}{30}\)
Hence, five rational numbers between \( \frac{3}{5} \) and \( \frac{4}{5} \) are:
\(\frac{19}{30}, \frac{20}{30}, \frac{21}{30}, \frac{22}{30}, \frac{23}{30}\)
Now its not necessary to only multiply and divide by \( \mathrm{n}+1 \), you can do by any number
\(\frac{3}{5}=\frac{3 \times 10}{5 \times 10}=\frac{30}{50}\)
greater than n. For example if you multiply and divide by 10
\(\frac{4}{5}=\frac{4 \times 10}{5 \times 10}=\frac{40}{50}\)
So the rational numbers between 3 and 4 now will be
\(\frac{31}{50}, \frac{32}{50}, \frac{33}{50}, \frac{34}{50}, \frac{35}{50}, \frac{36}{50}, \frac{37}{50}\)
Hence, five rational numbers.

(i) True, since the collection of whole numbers contains all natural numbers.
Natural numbers are numbers starting from \(1,\) i.e \(1, 2, 3, 4, 5, 6, \ldots\ldots\) And whole numbers are numbers starting from \(0\). i.e, \( 0,1,2,3,4,5 \ldots\ldots\) And you can see that all natural numbers are within whole numbers.

(ii) False, as integers may be negative or positive but whole numbers are always positive.
Now integers are numbers that are both negative and positive and include zero also, i.e., \( \ldots\ldots -3,-2,-1,0,1,2,3,4,\ldots\ldots \)
And whole numbers are numbers starting from \(0\). i.e., \( 0,1,2,3,4,5\ldots\ldots\) And clearly, negative numbers are missing from whole numbers.

(iii) False, as rational numbers may be fractional but whole numbers are not fractional. Now, Rational numbers are numbers that can be expressed in \( \frac{\mathrm{p}} {\mathrm{q}} \) form where \(\mathrm{q}\) not equal to zero. So they include fractions on the other hand whole numbers are numbers starting from \(0\). i.e, \( 0,1,2,3,4,5 \ldots\ldots \)
Therefore, Whole numbers does not contain all rational numbers.