Positive rational numbers

In this chapter we will learn the concept of positive rational numbers with examples.

To understand the chapter, we will first start with the revision of the basics of rational number.

What are rational numbers ?

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

Numbers like $\mathtt{\frac{2}{3} ,\ \frac{7}{5} \ ,\ \frac{11}{13}}$ are some examples of rational number.

If you want to learn about rational numbers in detail, click the red link.

A rational number is positive when both the sign of numerator and denominator is same.

That is, the positive rational number has;

(i) either both positive numerator and denominator, or;

(ii) both negative numerator and denominator.

Hence, numbers like $\mathtt{\frac{5}{3} ,\ \frac{-11}{-13} \ ,\ \frac{2}{7}}$ are examples of positive rational number.

Why rational number with negative numerator and denominator is positive ?

The rational number with negative sign in numerator and denominator cancel out each other to produce positive rational number.

Are all natural numbers example of positive integers ?

Let us first classify the natural numbers.

All positive integers like 1, 2, 3, 4, 5 . . . . are the natural numbers.

These natural numbers are also positive rational numbers because;

(i) By inserting denominator 1, we can represent in the form of P / Q.

(ii) Both numerator and denominator are positive.

For example;
Consider the natural number 22.

Insert 1 in the denominator so that it get represented in the form of P / Q.

$\mathtt{22\ \Longrightarrow \ \frac{22}{1}}$

You can see that $\mathtt{\ \frac{22}{1}}$ contains numerator and denominator of same sign.

Hence, the number 22 is a positive rational number.

Conclusion
All the natural numbers are part of positive rational numbers.

Are all negative integers part of positive rational number ?

Let us first understand the negative integers.

The numbers less than 0 which are neither fraction or decimal are negative integers.

Numbers like -1, -2, -3, -4, -5 . . . etc. are example of negative integers.

These negative integers are also negative rational numbers since;

(i) They can be represented in the form of P / Q by inserting denominator 1.

(ii) Here numerator is negative number & denominator is positive, so the rational number is non positive.

For example;
Consider the negative integer -50.

Insert 1 in denominator to represent in the form of P / Q.

$\mathtt{-50\ \Longrightarrow \ \frac{-50}{1}}$

You can see that the rational number is negative since the numerator and denominators are of different signs.

I hope you understood the above concept. Let us now practice some questions.

Positive Rational Number – Solved examples

Among the given numbers, identify the positive rational numbers.

( i ) $\mathtt{\frac{7}{11}}$

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

(ii) $\mathtt{\frac{-2}{15}}$

Here the numerator and denominator have different signs, hence the rational number is not positive.

(iii) $\mathtt{\ \frac{-33}{-19}}$

Here both the numerator an denominator have same sign. Hence the given rational number is positive.

(iv) $\mathtt{\ \frac{50}{3}}$

Both numerator & denominator have same sign. Hence, the given number is positive rational number.

(v) $\mathtt{\frac{70}{-13}}$

Here numerator is positive number and denominator is negative number.

Since both numerator and denominator have different sign, the given rational number is not positive.