# Negative rational number

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.