3GENERAL OBJECTIVES At the end of the lesson students will enable to: 1.Understand terminating decimal expansions.2. Understand non-terminating repeating expansions.
4SPECIFIC OBEJECTIVESstudents can differentiate terminating and non terminating repeating decimal expansions.
5Previous knowledge testing Teacher:-Dear students ,what is a rational number? Students: 2/5 a rational number ? T:-Change it into decimal expansion. S: - 2/5=0.4 T:- Change 1/8 into decimal expansion. S:- 1/8=0.125 T: -Change 1/3 into decimal expansion. S:- 1/3=0.3333……………
6INTRODUCTION OF THE TOPIC Dear students , as you know that 2/5=0.4 and 1/8=0.125These are the terminating decimal expansions.To understand this topic Let us consider the following examples0.4=4/10=4/1010.125=125/1000=125/1030.375=375/1000=375/1030.1025=1025/10000=1025/104As we know that 2 and 5 are the only prime factors of 10Any positive integral power of 10, say10n can be expressed as2nx 5ncontd
7Therefore if the numerators and denominators are co-prime and denominator are in the form of 2mx5n then the rational numbers having terminating decimal expansions.For examples:-(1) 1.512=1512/103=23x33x7/23x53=33x7/53=189/20x53(2) =277624/104=23x32x7x531/24x54=32x7x531/2x54=33453/21x54From the above discussion that the denominators of the rational numbers having terminating decimal expansions are expressible in the form 2mx5nConclusion:-If x be a rational number whose decimal expansion terminates. Then x can expressed in form p/q, where p and q are co-prime , and the prime factorisation q is in the form 2mx5n, where m, n are non negative integers.
8Let us now consider the following rational numbers:- (1) 1/3= …………5/3=1.6666……….1/7= …….The above rational numbers are non terminating and repeating.These numbers numbers are non terminating and repeating because the denominators of these rational numbers are not in form of 2mx 5nConclusion:- If x =p/q be a rational number, such that the prime factorisation of q is not of the form 2mx 5n where m,n are non negative integers. Then x has a decimal expansion which is non-terminating repeating.
9STUDENT ACTIVITYWithout long division , state whether the following rational numbers having terminating decimal expansion or a non-terminating repeating decimal expansion:Group1 : 23/8125/44115/16003/320Group2 : 13/312577/21029/3432/15Group3: 17/6189/12549/50015/525
10Home AssignmentEvery students should write any ten rational numbers and determine the nature of the decimal expansions.