# NCERT Class 9 Maths Number Systems

The chapter 1 starts with an introduction to the number system using some examples (using number lines), followed by exercise problems. Next, the chapter deals with a detailed explanation of irrational numbers followed by coverage of real numbers and their decimal expansion. Later, the chapter explains in detail about the representation of numbers on a number line, followed by operations on real numbers. The solutions to the exercise problems are provided by us and are downloadable in the PDF format.

## Chapter 1 Ex.1.1 Question 1

Is zero a rational number? Can you write it in the form \begin{align}\frac{p}{q}\end{align} where \begin{align}p \end{align} and \begin{align}q \end{align} are integers and \begin{align}q \ne 0\end{align}?

### Solution

Steps:

Yes, zero is a rational number.

Zero can be written as:

\begin{align}\frac{0}{{{\rm{Any}}\,{\rm{non - zero}}\,{\rm{integer}}}}\end{align}

\begin{align}\text{Example}:\frac{0}{1} = \frac{0}{{ - 2}}\,\,\end{align}

Which is in the form of \begin{align}\frac{p}{q}\end{align}, where \begin{align}p \end{align} and \begin{align}q \end{align} are integers and \begin{align}q \ne 0\end{align}.

## Chapter 1 Ex.1.1 Question 2

Find six rational numbers between $$3$$ and $$4$$.

### Solution

Steps:

We can find any number of rational numbers between two rational numbers. First of all we make the denominators same by multiplying or dividing the given rational numbers by a suitable number. If denominator is already same then depending on number of rational no. we need to find in question, we add one and multiply the result by numerator and denominator.

\begin{align} 3&=\frac{3\times 7}{7}\,\,\,\text{and}\,\,\,\,\,4=\frac{4\times 7}{7} \\ 3&=\frac{21}{7}\,\,\,\,\,\,\,\,\,\text{and }\,\,\,\,4=\frac{28}{7} \\ \end{align}

We can choose $$6$$ rational numbers as:

\begin{align}\frac{22}{7},\frac{23}{7},\frac{24}{7},\frac{25}{7},\frac{26}{7}\,\,\text{and}\,\,\frac{27}{7}\end{align}

## Chapter 1 Ex.1.1 Question 3

Find five rational numbers between \begin{align}\frac{3}{4} \text{ and } \frac{4}{5}\end{align}

### Solution

Steps:

Since we make the denominator same first, then

\begin{align} \frac{3}{4} &=\frac{3 \times 5}{4 \times 5} \\ &=\frac{15}{20} \\ \frac{4}{5} &=\frac{4 \times 4}{5 \times 4} \\ &=\frac{16}{20} \end{align}

Now, we have to find $$5$$ rational numbers.

\begin{align}\therefore\,\frac{15}{20}& =\frac{15\times 6}{20\times 6} \\ &=\frac{90}{120} \\ \frac{16}{20} &=\frac{16\times 6}{20\times 6} \\ & =\frac{96}{120} \\ \end{align}

$$\therefore$$ Five rational numbers between  $$\frac{3}{4}$$ and $$\frac{4}{5}$$ are

\begin{align}\frac{91}{120}, \frac{92}{120}, \frac{93}{120}, \frac{94}{120} \text { and } \frac{95}{120} \end{align}

## Chapter 1 Ex.1.1 Question 4

State whether the following statements are true or false. Give reasons for your answers.

### Solution

(i) Every natural number is a whole number.

Steps:

True, because the set of natural numbers is represented as\begin{align}\text{N= {1, 2, 3…….}}\end{align}and the set of whole numbers is \begin{align}\text{W = {0, 1, 2, 3 ………}.}\end{align}We see that every  natural number is present in the set of whole numbers. Also, we can see that the as compared to the set of natural numbers, the set of whole numbers contains just one extra number and that number is $$0$$.

(ii) Every integer is a whole number.

Steps:

False. Negative integers are not present in the set of whole numbers.

(iii) Every rational number is a whole number.

Steps:

False. For example \begin{align}\frac{1}{2}\end{align} is a rational number, which is not a whole number.