What Have We Discussed?

1. A number that can be expressed in the form pq, where p and q are integers and q ≠ 0, is called a rational number. The numbers −27, 38 ,3 etc. are numbers.

2. All integers and fractions are rational numbers.

3. If the numerator and denominator of a rational number are multiplied or divided by a integer, we get a rational number which is said to be equivalent to the given rational number. For example −37=−3×27×2=−614.So, we say −614 is the equivalent form of −37. Also note that −614=−6÷214÷2=−37

4. Rational numbers are classified as and Negative rational numbers. When the numerator and denominator, both, are positive integers, it is a positive rational number. When either the numerator or the denominator is a negative integer, it is a negative rational number. For example,38 is a positivenegative rational number whereas −89 is a negativepositive rational number

5. The number 0 is is not neither a positive nor a negative rational number.

6. 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. The numbers −13, 27 etc. are in form.

7. There are unlimitedlimited number of rational numbers between two rational numbers.

8. Two rational numbers with the same denominator can be added by adding their numerators, keeping the denominator same. Two rational numbers with different denominators are added by first taking the LCM of the two denominators and then converting both the rational numbers to their equivalent forms having the LCM as the denominator. For example, −23+38=−1624+924=−16+924=−724. Here, LCM of 3 and 8 is .

9. While subtracting two rational numbers, we add the additive inverse of the rational number to be subtracted to the other rational number.

Thus, 78−23=78 + additive inverse of 23=78+−23=21+−1624=524

10. To multiply two rational numbers, we multiply their numerators and denominators separately, and write the product as productofnumeratorsproductofdenominators

11. To divide one rational number by the other non-zero rational number, we multiply the rational number by the reciprocal of the other. Thus,

−72÷43=−72× (reciprocal of 43) = −72×34=−.