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Maharashtra Class 9 Answered

Root 2 is irrational no ?
Asked by omprintshop | 18 Jun, 2019, 09:05: PM
answered-by-expert Expert Answer
begin mathsize 16px style Ye open parentheses open parentheses blank close parentheses close parentheses straight s comma space square root of 2 space is space an space irrational space number. Proof colon Let space us space assume space that space square root of 2 space is space straight a space rational space number. rightwards double arrow square root of 2 space space equals space straight p over straight q space space space space space space space space space space open parentheses space where space straight p space and space straight q space are space co space minus space prime comma space straight p comma space straight q element of straight Z space and space straight q not equal to 0 space close parentheses Squaring space on space both space the space sides comma space we space get rightwards double arrow straight p squared over straight q squared space equals space 2 rightwards double arrow straight p squared space equals space 2 straight q squared space space space... space left parenthesis straight i right parenthesis rightwards double arrow straight p squared space is space divisible space by space 2. rightwards double arrow straight p space is space also space divisible space by space 2. space space left parenthesis ii right parenthesis Let space straight p space equals space 2 straight m rightwards double arrow straight p squared space equals space 4 straight m squared rightwards double arrow 2 straight q squared space equals space 4 straight m squared space space space space space space... space from space left parenthesis straight i right parenthesis rightwards double arrow straight q squared space equals space 2 straight m squared rightwards double arrow straight q squared space is space divisible space by space 2. rightwards double arrow straight q space is space also space divisible space by space 2. space space space... space left parenthesis iii right parenthesis From space left parenthesis ii right parenthesis space and space left parenthesis iii right parenthesis comma space we space get space straight p space and space straight q space are space divisible space by space 2. rightwards double arrow straight p space and space straight q space have space straight a space common space factor space 2. This space contradicts space our space assumption space that space straight p over straight q space is space rational space straight i. straight e space straight p space and space straight q space do space not space have space any space common space factor space other space than space 1. left parenthesis co space minus space prime right parenthesis. rightwards double arrow straight p over straight q space is space not space rational. rightwards double arrow square root of 2 space space is space an space irrational space number. end style
Answered by Yasmeen Khan | 19 Jun, 2019, 09:48: AM

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