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

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 $$\frac{-8}{15}$$.

So, the above explanations with examples help us to understand the useful properties of rational numbers.

Rational Numbers

What is Rational Numbers?

Is Every Rational Number a Natural Number?

Is Zero a Rational Number?

Is Every Rational Number an Integer?

Is Every Rational Number a Fraction?

Positive Rational Number

Negative Rational Number

Equivalent Rational Numbers

Equivalent form of Rational Numbers

Rational Number in Different Forms

Properties of Rational Numbers

Lowest form of a Rational Number

Standard form of a Rational Number

Equality of Rational Numbers using Standard Form

Equality of Rational Numbers with Common Denominator

Equality of Rational Numbers using Cross Multiplication

Comparison of Rational Numbers

Rational Numbers in Ascending Order

Representation of Rational Numbers on the Number Line

Addition of Rational Number with Same Denominator

Addition of Rational Number with Different Denominator

Properties of Addition of Rational Numbers

Subtraction of Rational Number with Different Denominator

Subtraction of Rational Numbers

Properties of Subtraction of Rational Numbers

Simplify Rational Expressions Involving the Sum or Difference

Multiplication of Rational Numbers

Properties of Multiplication of Rational Numbers

Rational Expressions Involving Addition, Subtraction and Multiplication

Division of Rational Numbers

Rational Expressions Involving Division

Properties of Division of Rational Numbers

To Find Rational Numbers