In this chapter we will learn the concept of negative rational number with examples and solved practice problems.

To understand the chapter, we will first review the basics of rational numbers.

## What are rational numbers ?

The numbers which can be represented in the form of P / Q are called rational numbers.

Numbers like \mathtt{\frac{3}{7} ,\ \frac{-5}{6} \ ,\ \frac{1}{-9}} are examples of rational numbers.

If you want to learn **rational number** in detail, click the red link.

## Negative rational numbers

A **rational number is negative** when the **numerator and denominator have different sign**.

i.e. the negative rational number has;

(i) positive numerator and negative denominator

(ii) negative numerator and positive denominator

Hence, numbers like \mathtt{\frac{-10}{13} \ \&\ \frac{11}{-9}} are examples of **negative rational number**.

### Are negative integers part of negative rational numbers ?

Let us understand negative integers first.

Numbers less than 0 that are neither decimal or fraction are negative integers.

Number like -1, -2, -3, -4, -5, . . . are examples of negative integers.

These **numbers are negative rational number because**;

(a) they can be **represented in the form of P / Q **by inserting 1 in denominator.

(b) **numerator is negative and denominator is positive**.

**For example;**

Is negative 3 a negative rational number ?

Yes !!!

First represent -3 in form of P/Q by inserting 1 in the denominator.

\mathtt{-3\ \Longrightarrow \ \frac{-3}{1}}

Here the **numerator is negative and denominator is positive**, hence negative three is a **negative rational number**.

#### Is negative five a negative rational number ?

**Yes !!**

First convert the number in form of P / Q by inserting 1 in denominator.

\mathtt{-5\ \Longrightarrow \ \frac{-5}{1}}

You can see that **numerator and denominator have different signs**.

Hence, – 5 is a **negative rational number**.

**Conclusion**

Negative integer are part of negative rational number.

### Are natural numbers part of negative rational number ?

**NO !!**

When we represent natural number in the form of P/Q, we get **numerator and denominator of same sign** which is characteristic of positive rational number.**For example;**

Consider the natural number 7.

Convert the number in form of P/Q by inserting 1 as denominator.

\mathtt{7\ \Longrightarrow \ \frac{7}{1}}

Note that both the numerator & denominator are positive.

Hence, the given number is a positive rational number.

## Negative Rational Number – Solved Example

Identify the negative rational number among the given numbers.

(i) \mathtt{\frac{5}{-17}}

Here numerator is positive and denominator is negative.

Since both numerator and denominator have different sign, the given number is **negative rational number**.

(ii) \mathtt{\frac{-11}{-13}}

Here both numerator & denominator are negative.

Both the negative sign cancel out each other to generate positive rational number.

Since, both the numerator and denominator contain same sign. the given number is **positive rational number**.

(iii) \mathtt{\frac{-19}{1}}

Here numerator is negative and denominator is positive.

Since the numerator and denominator is of different sign, the given number is **negative rational number**.

(iv) \mathtt{\frac{101}{203}}

Here both numerator and denominator have same sign. So the number is **non negative rational number**.