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Properties of Rational Numbers

We will learn some useful properties of rational numbers.

Property 1:

If a/b is a rational number and m is a nonzero integer, then

$\frac{a}{b}$ = $\frac{a × m}{b × m}$

In other words, a rational number remains unchanged, if we multiply its numerator and denominator by the same non-zero integer.

For examples:

$\frac{-2}{5}$ = $\frac{(-2) × 2}{5 × 2}$ = $\frac{-4}{10}$ , $\frac{(-2) × 3}{5 × 3}$ = $\frac{-6}{15}$ , $\frac{(-2) × 4}{5 × 4}$ = $\frac{-8}{20}$ and so on ……

Therefore, $\frac{-2}{5}$ = $\frac{(-2) × 2}{5 × 2}$ = $\frac{(-2) × 3}{5 × 3}$ = $\frac{(-2) × 4}{5 × 4}$ and so on ……

Property 2:

If $\frac{a}{b}$ is a rational number and m is a common divisor of a and b, then

$\frac{a}{b}$ = $\frac{a ÷ m}{a ÷ m}$

In other words, if we divide the numerator and denominator of a rational number by a common divisor of both, the rational number remains unchanged.

For examples:

$\frac{-32}{40}$ = $\frac{-32 ÷ 8}{40 ÷ 8}$ = $\frac{-4}{5}$

Property 3:

Let $\frac{a}{b}$ and $\frac{c}{d}$ be two rational numbers.

Then $\frac{a}{b}$ = $\frac{c}{d}$ ⇔ $\frac{a × d}{b × c}$ .

$Properties of Rational Numbers$

a × d = b × c

For examples:

If $\frac{2}{3}$ and $\frac{4}{6}$ are the two rational numbers then, $\frac{2}{3}$ = $\frac{4}{6}$ ⇔ (2 × 6) = (3 × 4).

Note:

Except zero every rational number is either positive or negative.

Every pair of rational numbers can be compared.

Property 4:

For each rational number m, exactly one of the following is true:

(i) m > 0 (ii) m = 0 (iii) m < 0

For examples:

The rational number $\frac{2}{3}$ is greater than 0.

The rational number $\frac{0}{3}$ is equal to 0.

The rational number $\frac{-2}{3}$ is less than 0.

Property 5:

For any two rational numbers a and b, exactly one of the following is true:

(i) a > b (ii) a = b (iii) a < b

For examples:

If $\frac{1}{3}$ and $\frac{1}{5}$ are the two rational numbers then, $\frac{1}{3}$ is greater than $\frac{1}{5}$ .

If $\frac{2}{3}$ and $\frac{6}{9}$ are the two rational numbers then, $\frac{2}{3}$ is equal to $\frac{6}{9}$ .

If $\frac{-2}{7}$ and $\frac{3}{8}$ are the two rational numbers then, $\frac{-2}{7}$ is less than $\frac{3}{8}$ .

Property 6:

If a, b and c be rational numbers such that a > b and b > c, then a > c.

For examples:

If $\frac{3}{5}$ , $\frac{17}{30}$ and $\frac{-8}{15}$ are the three rational numbers where $\frac{3}{5}$ is greater than $\frac{17}{30}$ and $\frac{17}{30}$ is greater than $\frac{-8}{15}$ , then $\frac{3}{5}$ is also greater than $\frac{-8}{15}$ .