Image Credit: Pixabay.com |

So far, we have learned about the following set of numbers

*Natural Number, *N = {1, 2, 3, …}

*Whole number*, W = {0, 1, 2, 3, …}

*Integers, *Z = {..,-2,-1,0,1,2, …}

We have also learned about fractions and their applications, so it is important to include them in set of numbers.

*Rational number is the set of numbers that includes both integers and fractions.*

*Q = { p/q : p∈ Z, q∈ Z , q ≠ 0 }*

➤The word ‘**rational**’ arises from the term ‘**ratio**’ & ‘**Q’** comes from the word ‘**quotient**‘.

➤The **decimal numbers** can be written as an ordinary fraction and, hence, are **rational numbers**.

➤The **integers** can be written as a fraction with denominator 1 and, hence, are **rational numbers***.*

*“A rational number is defined as a number that can be expressed in the form p/q , where p and q are integers and q ≠ 0.”*

## Equivalent Rational Numbers

By multiplying/dividing the numerator and denominator of a rational number by the same non zero integer, we obtain another rational number equivalent to the given rational number. This is exactly like obtaining equivalent fractions.

e.g.,

−3/5 = −3×**2**/5×**2** = −6/10

−3/5 = −3×(**−2**)/5×(**−2**) = 6/−10

−3/5 = −3×**3**/5×**3** = −9/15

∴ −3/5 = −6/10 = 6/−10 = −9/15 are equivalent rational numbers.

## Positive & Negative Rational Numbers

**Positive rational numbers** – Rational numbers whose both numerator and denominator are positive integers or negative integers (division of two negative integers result in positive integer) are called positive rational numbers.

e.g., 2/7, 3/5

**Negative rational numbers** – Rational numbers whose either numerator or denominator is negative integer are called negative rational numbers.

e.g., −3/5, 4/−7

## Rational Number in a Standard form

➤ A rational number is said to be in the standard form if its denominator is a positive integer and the numerator and denominator have no common factor other than 1. e.g., 3/2, 5/7

➤ To reduce the rational number to its standard form, we divide its numerator and denominator by their HCF ignoring the negative sign, if any.

e.g.,

→

→

## Rational Numbers on Number Line

➤ If a rational number is **proper fraction **and

→ **positive** then, it lie between 0 & 1, (e.g., 3/5)

→ **negative** then, it lie between 0 & -1, (e.g., −3/4)

➤ If a rational number is **improper fraction** – we should convert it to mixed fraction (integer+ fraction) first then, the rational number lie between the integer and the successive integer. e.g., lie between 1 & 2, −7/3(−21/3) lie between −2 & −3.

## Comparing Rational Numbers

➤ Using number line – Rational number on the right side of other rational number would be greater .

➤ Find equivalent rational numbers having same denominator (LCM) then, we can compare them as like fractions .

➤ To compare two negative rational numbers, we can compare them by ignoring their negative signs and then reversing the order.

e.g., to compare −7/5 and −5/3 , we first compare 7/5 and 5/3 . We get 7/5 < 5/3 and conclude that −7/5 > −5/3 .

➤ Between a negative rational number and a positive rational number, it is obvious that positive rational number is greater.

## Rational Numbers between two Rational Numbers

➤ Number of integers between two integers are limited (finite). e.g., between 2 and 7, there are 4 integers.

➤ We can find unlimited number of rational numbers (infinite) between any two rational numbers.

→ Convert the given rational numbers into like fractions, then we can find rational numbers between them.

e.g., find rational numbers between 1/3 & 3/5

We can write the given numbers as,

Then, rational numbers between are

Also, we can write the given numbers as,

10/30 < 11/30 < … < 17/30 < 18/30

Now, we can say that, rational numbers between 1/3 & 3/5 are —

So, we can insert as many rational numbers as we want between two rational numbers.

## Addition of Rational Nombers

➧ If rational numbers are like fractions, simply add their numerator keeping the denominator same. e.g., 5/12 + 7/12 = (5+7)/12 = 12/12 = 1

➧ If rational numbers are unlike fractions, find the LCM of their denominators and convert the numbers into like fractions and then, add as above. e.g.,

→ 5/6 + 6/5 = 25/30 + 36/30 = (25+36)/30 = 61/30

→ −5/6 + 6/5 = −25/30 + 36/30 = (−25+36)/30 = 11/30

### Additive Inverse

Opposite (or Negative) of a given rational number is called its additive inverse because when we add them together, the sum we got is zero.

e.g., 4/7 + (−4/7) = 0

∴ Additive inverse of 4/7 is −4/7 and vice versa.

## Subtraction of Rational Numbers

➧ If rational numbers are like fractions, simply subtract their numerator keeping the denominator same.

e.g., 5/12 − 7/12 = (5−7)/12 = −2/12 = −1/6

➧ If rational numbers are unlike fractions, find the LCM of their denominators and convert the numbers into like fractions and then, add as above. e.g.,

## Multiplication of Rational Numbers

➧ While multiplying a rational number by a positive integer, we multiply the numerator by that integer, keeping the denominator unchanged.

e.g., 2/3 × (−4) = −8/3 ,

3/5 × 4 = 12/5

➧ We multiply two rational numbers in the following way:

→ Multiply the numerators of the two rational numbers.

→ Multiply the denominators of the two rational numbers.

→ Write the product as

e.g., 2/3 × −4/5 = (2×−4)/(3×5) = −8/15,

4/3 × 2/−5 = (4×2)/(3×−5) = 8/−15 = −8/15 (in standard form)

### Multiplicative Inverse

Inverse (or Reciprocal) of a given rational number is called its multiplicative inverse because when we multiply them together, the product we got is one.

e.g.,

∴ Multiplicative inverse of is and vice versa.

## Division of Rational Numbers

To divide a rational number by other rational number, we multiplying the rational number with the reciprocal of the other .

e.g.,