Natural numbers = The counting numbers are called natural numbers. E.g. 1,2,3,4,5……………… Whole numbers = All natural numbers together with 0. E.g. 0,1,2,3,4,5,6,…………… Integers = All natural numbers, 0 and negative of counting numbers are called integers. E.g. -1,-2,-3,-4,-5,0,1,2,3,4,5………………… Fractions = The numbers in the form of where a and b are not whole numbers and b is not equal to 0. E.g. 2/3, 3/8,11/5………………… a b Let us recall something-

Define Rational Numbers:

Define Rational Numbers The number of the form , , where p and q are integers and q is not equal to 0, are called rational numbers. P q

Rational Numbers A rational number is a real number that can be written as a ratio of two integers. A rational number written in decimal form is terminating or repeating.

Addition of rational numbers:

Addition of rational numbers With same denominators In order to add the rational numbers with same denominators we follow the following steps Step 1 Obtain the numerators of two given rational numbers and their common denominator. Step 2 Add the numerators obtained in step 1 Step 3 Write the rational no. whose numerator is the sum obtained in step 2 and whose denominator is the common denominator of given rational numbers. p r p+r q q q

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Add and We have See an Example 5 9 -13 9 5 9 + -13 9 = 5 + (-13) 9 -8 9 = [5+(-13)=(8)]

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With different denominators Step 1 See whether the denominators are positive or not. If the denominator of one or two are negative rewrite it so that it becomes positive. Step 2 Obtain the denominators of rational of the rational numbers in then Step 1 Step 3 Find the LCM of denominators obtained in Step 2. Step 4 Express each one of rational number in step 1 so that LCM obtained in the step 3 become their common denominator. Step 5 Write a rational number whose numerator s equal to the sum of numerators of rational numbers obtained n step 4 and denominators as the LCM obtained in the step 3. Step 6 The rational number obtained in the step 5 is the required sum. Addition of rational numbers

Subtraction of rational numbers If and are two rational numbers, then subtracting from means adding additive inverse (negative) of to . The subtraction of from is written as - .thus, we have a b c d a b c d c d a b a b c d c d a b a b c d - = [ ] - +

See an Example:

See an Example Subtract from The additive inverse of is 3 4 5 6 3 4 3 4 5 3 5 (-3) 6 4 6 4 - = - 5 3 5 2 (-3) 3 10 (-9) 10+(-9) 1 6 4 6 2 4 3 12 12 12 12 - = = = = + +

Multiplication of Rational Numbers:

Multiplication of Rational Numbers The product of two rational numbers is defined below Product of two rational numbers = Thus we have, Product of their numerators Product of their denominators a c (a c) b d (b d) =

See an example :

See an example 5 7 (2 5) 10 (3 7) 21 =

Reciprocal of a number :

Reciprocal of a number If the product of two rational numbers is 1 then each one is called the reciprocal of the other. Thus, the reciprocal of is and we write = . a b a b b a b a [ ] - 1

See some example :

See some example Find the reciprocal of 13 7 7 13 = 1. 2. -8 9 9 -8 = 3. -6 1 -6 = 4. 5 12 12 5 =

Division of rational numbers:

Division of rational numbers If and are two rational numbers such that is not equal to 0, then we define: a b c d c d a c a d b d b c =

See an example:

See an example 2 15 3 3 15 2 = = = 3 15 2 5 1 7 10

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