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### Definition of Rational Numbers:

A rational number is any number that can be expressed as the quotient or fraction \(\frac{p}{q}\) of two integers, with the denominator q not equal to zero.

In short, rational number represents a ratio of two integers.

**Example:**

3.5 is a rational number because 3.5 = \(\frac{7}{2}\) (it can be written as a fraction)

\(\frac{7}{0}\) is not a rational number as the denominator can’t be 0.

Since q may be equal to 1, every integer is a rational number.

Every Natural Number n can be written as \(\frac{n}{1}\), which is a rational number, but a rational number such as \(\frac{1}{n}\) need not be a natural number.

**Notation:**

The set of all rational numbers is usually denoted by **Q**.

**Relationship between different types of numbers:**

Natural numbers includes set of elements like **N = {1, 2, 3, 4,….}** and Whole numbers are **W = {0, 1, 2, 3, 4,….}**.

Combining —1, —2, —3, —4, —5, … etc with whole numbers we get a set of numbers called integers **Z = {…, -2, -1, 0, 1, 2,….}**

Fractional numbers include numbers in the form of **\(\frac{p}{q}\)** where q is not equal to 0.

All these numbers can be related as follows:

Rational numbers includes natural numbers, whole numbers, integers and fractions. They are denoted by Q.

Every Natural number is a rational number but a rational number need not be a natural number.

Zero is a rational number.

Every fraction is a rational number but a rational number need not be fraction.

Every Integer is a rational number but a rational number need not be integer.

In general, there are two types of rational numbers:

1) Positive Rationals

2) Negative Rationals

**Positive Rationals:**

A rational number is said to be positive if its numerator and denominator are either both positive (+ve) or both negative (-ve).

**Example:** \(\frac{11}{3}\) and \(\frac{-17}{-6}\) are both positive rationals.

**Negative Rationals:**

A rational number is said to be negative if its numerator and denominator are of opposite signs.

**Example:** \(\frac{-7}{3}\) and \(\frac{5}{-2}\) are both negative rationals.

#### Properties of Rational Numbers:

**Property 1:**

If \(\frac{a}{b}\) is a rational number and m is a nonzero integer then \(\frac{a}{b}\) =\(\frac{(a X m)}{(b X m)}\)

**Example:** \(\frac{-19}{4}\) is a rational number and \(\frac{-19}{4}\) can also be written as,

\(\frac{(-19) x 2}{4 x 2}\) = \(\frac{(-19) x 3}{4 x 3}\) = \(\frac{(-19) x 4}{4 x 4}\) = ……

That implies\(\frac{-19}{4}\) = \(\frac{-38}{8}\) = \(\frac{-57}{12}\) = \(\frac{-76}{16}\) = …..

Such rational numbers are called * Equivalent rational numbers*.

**Property 2:**

If \(\frac{a}{b}\) is a rational number and m is a common divisor of a and b then \(\frac{a}{b}\) = a/m/b/m

**Example:**\(\frac{-21}{14}\) is a rational number and \(\frac{-21}{14}\) can also be written as,

(-21)/7/14/7 = \(\frac{-3}{2}\)

A rational number \(\frac{a}{b}\) is said to be in standard form if a and b are integers having no common divisor other than 1 and b is positive (or) if rational number is in its lowest terms.

To express a given rational number in standard form:

Step 1) Make the denominator of the given rational number positive.

Step 2) Divide both the numerator and denominator by their HCF.

**Example:**

Express \(\frac{72}{-40}\) in standard form.

**Solution:** \(\frac{72}{-40}\) = \(\frac{72 x (-1)}{(-40) x (-1)}\) = \(\frac{-72}{40}\)

The greatest common divisor of 72 and 40 is 8.

Therefore, \(\frac{-72}{40}\) = -72/8/40/8 = \(\frac{-9}{5}\)

Hence, \(\frac{-9}{5}\) is the standard form as both -9 and 5 have no other common divisors other than 1.

**Property 3:**

Let \(\frac{a}{b}\) and \(\frac{c}{d}\) be two rational numbers. Then, \(\frac{a}{b}\) = \(\frac{c}{d}\) <=> (a x d) = (b x c)

**Example:** \(\frac{-3}{2}\) = \(\frac{-6}{4}\)

(-3 x 4) = -12 and (2 x -6) = -12

<=> (-3 x 4) = (2 x -6)

Every positive rational number is greater than 0.

Every negative rational number is less than 0.

#### Comparison of Rational Numbers:

There are two methods for comparing rational numbers. They are:

**Method 1:**

Step 1) Express each rational number with a positive denominator.

Step 2) Find the LCM of the positive denominators.

Step 3) Express each of the given rational numbers with this LCM as common denominator.

Step 4) Then number having greater numerator is greater.

**Example:** Compare \(\frac{-7}{5}\) and \(\frac{5}{-2}\).

Solution: Step 1: By expressing the rationals in standard form we have,

\(\frac{-7}{5}\) = \(\frac{-7}{5}\) and \(\frac{5}{-2}\) = \(\frac{-5}{2}\)

Step 2: LCM of 5 and 2 is 5 x 2 = 10. (Since, both 2 and 5 are primes so their LCM is their product)

Step 3: By expressing each rational number with LCM as denominator we have,

\(\frac{-7}{5}\) = \(\frac{-7 x 2}{5 x 2}\) = \(\frac{-14}{10}\)

\(\frac{-5}{2}\) = \(\frac{-5 x 5}{2 x 5}\) = \(\frac{-25}{10}\)

Step 4: Comparing numerators of both rationals we have,

-14 > -25

Therefore, \(\frac{-7}{5}\) > \(\frac{5}{-2}\).

**Method 2:**

If a and b are integers and c and d are positive integers, then

\(\frac{a}{c}\) > \(\frac{b}{d}\), if and only if ad > bc.

\(\frac{a}{c}\) < \(\frac{b}{d}\), if and only if ad < bc.

\(\frac{a}{c}\) = \(\frac{b}{d}\), if and only if ad = bc.

**Example:** Compare \(\frac{17}{-7}\) and \(\frac{-11}{13}\).

Solution: By expressing the rationals in standard form we have,

\(\frac{17}{-7}\) = \(\frac{-17}{7}\) and \(\frac{-11}{13}\) = \(\frac{-11}{13}\)

On cross multiplication -17 x 13 = -221 and 7 x -11 = -77

-221 < -77

Therefore, \(\frac{17}{-7}\) < \(\frac{-11}{13}\).

**Arranging rational numbers:**

Rational numbers can be arranged in **ascending or descending order** by comparing the rational numbers by applying any one of the method explained above.

For **example**, arrange \(\frac{-3}{2}\), \(\frac{5}{-9}\), \(\frac{-15}{7}\) in ascending order.

Writing each rational number with a positive denominator, we have \(\frac{-3}{2}\), \(\frac{-5}{9}\), \(\frac{-15}{7}\).

LCM of 2, 9 and 7 is 126

By expressing each rational number with common denominator we have, \(\frac{-3}{2}\) = \(\frac{-3 x 63}{2 x 63}\) = \(\frac{-189}{126}\)

\(\frac{-5}{9}\) = \(\frac{-5 x 14}{9 x 14}\) = \(\frac{-70}{126}\)

\(\frac{-15}{7}\) = \(\frac{-15 x 18}{7 x 18}\) = \(\frac{-270}{126}\)

Since, -270 < -189 < -70 therefore, \(\frac{-15}{7}\) < \(\frac{-3}{2}\) < \(\frac{-5}{9}\)

Hence, the rational numbers in ascending order are \(\frac{-15}{7}\), \(\frac{-3}{2}\), \(\frac{-5}{9}\)

**Example2:** Arrange \(\frac{-7}{5}\), \(\frac{11}{-9}\), \(\frac{-13}{5}\) in descending order by cross multiplication method.

On cross multiplication of \(\frac{-7}{5}\), \(\frac{11}{-9}\) we have,

-7 x -9 = 63 and 5 x 11 = 55.

Since, 63 > 55 (ad > bc) therefore, \(\frac{-7}{5}\) > \(\frac{11}{-9}\).

Similarly, on cross multiplication of \(\frac{-11}{9}\), \(\frac{-13}{5}\) we have,

-11 x 5 = -55 and 9 x -13 = -117.

Since, -55 > -117 (ad > bc) therefore, \(\frac{-11}{9}\) > \(\frac{-13}{5}\).

By comparing \(\frac{-7}{5}\), \(\frac{-13}{5}\) we have,

\(\frac{-7}{5}\) > \(\frac{-13}{5}\)

From the above results we have,

\(\frac{-7}{5}\) > \(\frac{-11}{9}\) > \(\frac{-13}{5}\)

Hence, the rational numbers in descending order are \(\frac{-7}{5}\), \(\frac{-11}{9}\), \(\frac{-13}{5}\)