Rational Numbers between Two Rational Numbers
Natural Numbers: Natural Numbers are counting numbers. We can represent Natural Numbers indefinitely to the right of 1 on the number line. Whole Numbers: Whole Numbers are Natural Numbers including zero. We can represent Whole Numbers indefinitely to the right of Zero on the number line. Integers: Integers are a collection of numbers consisting of all Natural Numbers, their negatives, and zero. We can represent Integers in definitely on both sides of Zero on the number line. Rational Numbers: Rational number is a number that is expressed in the form $\frac{\text{p}}{\text{q}}$ , where p and q are Integers and q ≠ 0. In case of a Rational number, the denominator tells us the number of equal parts into which the first unit has been divided, while the numerator tells us ‘how many’ of these parts have been considered. Lowest form of a Rational number A rational number  $\frac{\text{p}}{\text{q}}$ is said to be lowest form or simplest form if p and q have no common factor other than 1. Equality of Rational numbers If two rational numbers $\frac{\text{p}}{\text{q}}$ and $\frac{\text{r}}{\text{s}}$ are said to be equal then p x s = q x r (or) $\frac{\text{p}}{\text{q}}$ = $\frac{\text{r}}{\text{s}}$. We can also represent Rational Numbers in definitely on both sides of Zero on the number line. Addition of Rational Numbers Numbers with same denominators Sum of two rational numbers have the same denominator, follow the following steps. 1) Obtain the numerators of two given rational numbers and their common denominator. 2) Add the numerator of two given rational numbers obtained in step 1. 3) Write a rational number whose numerator is the sum of two given rational numbers obtained in step 2 and whose denominator is the common denominator of the given rational numbers. Numbers with distinct denominators Sum of two rational numbers which do not have the same denominator, follow the following steps. 1) Obtain the rational nmbers and see whether their denominators are positive or not. If the denominator of one or both of the numbers is negative rewrite the denominators becomes positive. 2) Obtain the denominators of the rational numbers in step 1. 3) Find the LCM of denominators in step 2. 4) Express each one of the rational numbers in step 1 so that the LCM obtained in step 3 becomes their common denominator. 5) Write a rational number whose numerator is equal to the sum of the numerators of rational numbers obtained in step 4 and denominator as the LCM obtained in step 3. 6) The rational number obtained in step 5 is the required sum. Subtraction of Rational Numbers If $\frac{\text{a}}{\text{b}}$and $\frac{\text{c}}{\text{d}}$ are two rational numbers, then subtracting $\frac{\text{c}}{\text{d}}$ from $\frac{\text{a}}{\text{b}}$ means adding additive inverse (negative) of  to $\frac{\text{a}}{\text{b}}$. The subtracting of $\frac{\text{c}}{\text{d}}$ from $\frac{\text{a}}{\text{b}}$ is written as  $\frac{\text{a}}{\text{b}}$ - $\frac{\text{c}}{\text{d}}$ Thus, we have $\frac{\text{a}}{\text{b}}$- $\frac{\text{c}}{\text{d}}$ = $\frac{\text{a}}{\text{b}}$ + (- $\frac{\text{c}}{\text{d}}$), [∴ Additive inverse of $\frac{\text{c}}{\text{d}}$ is - $\frac{\text{c}}{\text{d}}$] Multiplication of Rational Numbers The product of two given fractions is a fraction whose numerator is the product of the numerators of the given fractions and whose denominator is the product of the denominators of the given fractions. Product of two given fractions = $\frac{\text{Product of their numerators}}{\text{Product of their denominators}}$ Division of Rational Numbers Division of fractions is the inverse of multiplication. If m and n two rational numbers such that n ≠ 0, then the result of dividing m by n is the rational number obtained on multiplying m by the reciprocal of n. When m is divided by n, we write m ÷ n. Thus m ÷ n = m × $\frac{\text{1}}{\text{n}}$. Rational Numbers between two Rational Numbers If m and n be two rational numbers such that m < n then $\frac{\text{1}}{\text{2}}$ (m + n) is a rational number between m and n. There are a finite number of Natural Numbers between any two Natural Numbers. Similarly there are a finite number of Whole numbers between any two Whole Numbers. But there are infinitely many Rational Numbers between any two Rational Numbers. The idea of mean helps us to find Rational Numbers between two Rational Numbers.

Summary

Natural Numbers: Natural Numbers are counting numbers. We can represent Natural Numbers indefinitely to the right of 1 on the number line. Whole Numbers: Whole Numbers are Natural Numbers including zero. We can represent Whole Numbers indefinitely to the right of Zero on the number line. Integers: Integers are a collection of numbers consisting of all Natural Numbers, their negatives, and zero. We can represent Integers in definitely on both sides of Zero on the number line. Rational Numbers: Rational number is a number that is expressed in the form $\frac{\text{p}}{\text{q}}$ , where p and q are Integers and q ≠ 0. In case of a Rational number, the denominator tells us the number of equal parts into which the first unit has been divided, while the numerator tells us ‘how many’ of these parts have been considered. Lowest form of a Rational number A rational number  $\frac{\text{p}}{\text{q}}$ is said to be lowest form or simplest form if p and q have no common factor other than 1. Equality of Rational numbers If two rational numbers $\frac{\text{p}}{\text{q}}$ and $\frac{\text{r}}{\text{s}}$ are said to be equal then p x s = q x r (or) $\frac{\text{p}}{\text{q}}$ = $\frac{\text{r}}{\text{s}}$. We can also represent Rational Numbers in definitely on both sides of Zero on the number line. Addition of Rational Numbers Numbers with same denominators Sum of two rational numbers have the same denominator, follow the following steps. 1) Obtain the numerators of two given rational numbers and their common denominator. 2) Add the numerator of two given rational numbers obtained in step 1. 3) Write a rational number whose numerator is the sum of two given rational numbers obtained in step 2 and whose denominator is the common denominator of the given rational numbers. Numbers with distinct denominators Sum of two rational numbers which do not have the same denominator, follow the following steps. 1) Obtain the rational nmbers and see whether their denominators are positive or not. If the denominator of one or both of the numbers is negative rewrite the denominators becomes positive. 2) Obtain the denominators of the rational numbers in step 1. 3) Find the LCM of denominators in step 2. 4) Express each one of the rational numbers in step 1 so that the LCM obtained in step 3 becomes their common denominator. 5) Write a rational number whose numerator is equal to the sum of the numerators of rational numbers obtained in step 4 and denominator as the LCM obtained in step 3. 6) The rational number obtained in step 5 is the required sum. Subtraction of Rational Numbers If $\frac{\text{a}}{\text{b}}$and $\frac{\text{c}}{\text{d}}$ are two rational numbers, then subtracting $\frac{\text{c}}{\text{d}}$ from $\frac{\text{a}}{\text{b}}$ means adding additive inverse (negative) of  to $\frac{\text{a}}{\text{b}}$. The subtracting of $\frac{\text{c}}{\text{d}}$ from $\frac{\text{a}}{\text{b}}$ is written as  $\frac{\text{a}}{\text{b}}$ - $\frac{\text{c}}{\text{d}}$ Thus, we have $\frac{\text{a}}{\text{b}}$- $\frac{\text{c}}{\text{d}}$ = $\frac{\text{a}}{\text{b}}$ + (- $\frac{\text{c}}{\text{d}}$), [∴ Additive inverse of $\frac{\text{c}}{\text{d}}$ is - $\frac{\text{c}}{\text{d}}$] Multiplication of Rational Numbers The product of two given fractions is a fraction whose numerator is the product of the numerators of the given fractions and whose denominator is the product of the denominators of the given fractions. Product of two given fractions = $\frac{\text{Product of their numerators}}{\text{Product of their denominators}}$ Division of Rational Numbers Division of fractions is the inverse of multiplication. If m and n two rational numbers such that n ≠ 0, then the result of dividing m by n is the rational number obtained on multiplying m by the reciprocal of n. When m is divided by n, we write m ÷ n. Thus m ÷ n = m × $\frac{\text{1}}{\text{n}}$. Rational Numbers between two Rational Numbers If m and n be two rational numbers such that m < n then $\frac{\text{1}}{\text{2}}$ (m + n) is a rational number between m and n. There are a finite number of Natural Numbers between any two Natural Numbers. Similarly there are a finite number of Whole numbers between any two Whole Numbers. But there are infinitely many Rational Numbers between any two Rational Numbers. The idea of mean helps us to find Rational Numbers between two Rational Numbers.