Information about Terminating non terminating class 10 groupA

Published on September 29, 2015

Author: ashu7047

Source: slideshare.net

2. eXamPleS of noneXamPleS of non terminating DecimalSterminating DecimalS (a) 2.666... is a non-terminating repeating(a) 2.666... is a non-terminating repeating decimaldecimal (b) 0.141414 ... is a non-terminating repeating(b) 0.141414 ... is a non-terminating repeating decimaldecimal 1/3 in decimal form is 0.333…... The repeating1/3 in decimal form is 0.333…... The repeating of numbers indicates that an infinite number ofof numbers indicates that an infinite number of repeating 3's are required to represent thisrepeating 3's are required to represent this fraction. fraction.

3. EvEry rational numbEr isEvEry rational numbEr is EithEr a tErminating orEithEr a tErminating or rEpEating dEcimalrEpEating dEcimal For any given divisor, only finitely many differentFor any given divisor, only finitely many different remainders can occur. In the example above, the 74 possibleremainders can occur. In the example above, the 74 possible remainders are 0, 1, 2, …, 73. If at any point in the divisionremainders are 0, 1, 2, …, 73. If at any point in the division the remainder is 0, the expansion terminates at that point.the remainder is 0, the expansion terminates at that point. If 0 never occurs as a remainder, then the division processIf 0 never occurs as a remainder, then the division process continues forever, and eventually a remainder must occurcontinues forever, and eventually a remainder must occur that has occurred before. The next step in the division willthat has occurred before. The next step in the division will yield the same new digit in the quotient, and the same newyield the same new digit in the quotient, and the same new remainder, as the previous time the remainder was the same.remainder, as the previous time the remainder was the same. Therefore the following division will repeat the same results.Therefore the following division will repeat the same results.

4. EvEry rEpEating orEvEry rEpEating or tErminating dEcimal is atErminating dEcimal is a rational numbErrational numbEr Each repeating decimal number satisfies a Each repeating decimal number satisfies a linear equationlinear equation with integer coefficients, and its with integer coefficients, and its unique solution is a rational number.unique solution is a rational number. To illustrate the latter point, the number α =To illustrate the latter point, the number α = 5.8144144144… above satisfies the equation 10000α5.8144144144… above satisfies the equation 10000α − 10α = 58144.144144… − 58.144144… =− 10α = 58144.144144… − 58.144144… = 58086, whose solution is α = 58086/9990 =58086, whose solution is α = 58086/9990 = 3227/5553227/555..

5. convErting rEpEatingconvErting rEpEating dEcimals to fractionsdEcimals to fractions Given a repeating decimal, it is possible toGiven a repeating decimal, it is possible to calculate the fraction that produced it.calculate the fraction that produced it. For example:For example: X=0.333333…….X=0.333333……. 10x=3.3333…….(multiply 10 on both10x=3.3333…….(multiply 10 on both side)side) 9x=3 (subtracting 1line from the 29x=3 (subtracting 1line from the 2

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