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WELCOME CHAPTER - 2 RATIONAL NUMBERS REAL NUMBERS • Numbers that can be found on the number line. Two Kinds of Real Numbers • Rational Numbers • Irrational Numbers RATIONAL NUMBERS • A rational number is a real number that can be written as a ratio of two integers. • Each rational number can be written in the form x/y ,where x and y are integers. • A rational number written in decimal form is terminating or repeating. EXAMPLES OF RATIONAL NUMBERS • • • • • ½ -3.5 -7/6 0.345… 3.14 • If we multiplying the numerator and denominator of a rational number by the same integer, we can get another form of the same rational number. In the language of algebra , a/b = ax/bx ADDITION AND MULTIPLICATION • 2+2=4 & 2*2= 4 • 3 + 3/2 = 9/2 & 3 * 3/2 = 9/2 Can you find other pairs of numbers such that the sum and product are equal? ADDITION AND SUBTRACTION • • • • Let us find the sum of 2/3 and 1/5 2/3 = 2*5/3*5 = 10/15 1/5 = 1*3/5*3 = 3/15 2/3 + 1/5 = 10/15 + 3/15 =10 + 3 /15 =(2*5 + 1* 3)/3*5 There fore, a/b + p/q = aq + bp / bq When the sum is one • There are many pairs of fractions whose sum is 1 such as • ½ +½ =1 • ¼+¾ =1 • NOTE : If we take the reciprocals of the pairs of numbers having sum and product equal, there sum is one. • Example : 3 + 3/2 = 9/2 and 3 * 3/2 = 9/2 Reciprocal of 3 = 1/3 Reciprocal of 3/2 = 2/3 Sum of the Reciprocals = 1/3 + 2/3 =1 Note :If we take a pair of numbers whose sum is one, then the product and sum of their reciprocals are equal. Multiplication and Division • In algebraic method multiplication and division of functions can be said as a/b * p/q = ap / bq In the case of division a/b / p/q = aq / bp Equal Fractions • To check equality of two fractions , it is enough to check whether their simplified forms are equal. • Example : consider the fractions 36/48 and 42/56 36/48 = 3/4 42/56 = 3/4 Then 36/48 = 42/56 • If a/b = p/q ,then aq = bp • If a/b = p/q ,then a/p =b/q COMPARISON • If the fractions a/b and p/q are equal, then aq = pb • If two fractions have same denominator, greater fraction has greater numerator. • If aq > pb , then a/b > p/q • If aq < bp , then a/b < p/q Decimal Forms • In the decimal form of any fraction, the digits after the decimal point will be terminating or non terminating. REFERENCES • Gilbert, Jimmie; Linda, Gilbert (2005). Elements of Modern Algebra (6th ed.). Belmont, CA: Thomson Brooks/Cole. pp. 243–244. ISBN 0534-40264-X. • SCERT Text book, Class 5,(2015).