Prove that root 2 - root 5 is irrational
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1ST √2
LET √2 BE RATIONAL NUMBER
I.E √2 = P/Q WHERE , P AND Q ARE COPRIMES AND Q ≠ 0
√2 = P/Q
Q√2 = P
SQUARING ON BOTH THE SIDES WE GET 2Q² = P²
THEREFORE 2 DIVIDES P² AND P ALSO SATISFY THE THEORUM IF P DIVIDES a² THEN P DIVIDES a ALSO {THEORUM 1.3 }
SO, WE CAN WRITE P = 2C
SUBSTITUTING FOR P WE GET 2Q² = 4C² THAT IS , Q² = 2C²
BUT P AND Q ARE COPRIMES AND HAVING 2 AS COMMON FACTOR
THIS CONTRADICTS THAT OUR ASSUMPTION IS WRONG √2 IS IRRATIONAL .
√5 IS HAVING THE SAME PROCESS AS OF √2 .
LET √2 BE RATIONAL NUMBER
I.E √2 = P/Q WHERE , P AND Q ARE COPRIMES AND Q ≠ 0
√2 = P/Q
Q√2 = P
SQUARING ON BOTH THE SIDES WE GET 2Q² = P²
THEREFORE 2 DIVIDES P² AND P ALSO SATISFY THE THEORUM IF P DIVIDES a² THEN P DIVIDES a ALSO {THEORUM 1.3 }
SO, WE CAN WRITE P = 2C
SUBSTITUTING FOR P WE GET 2Q² = 4C² THAT IS , Q² = 2C²
BUT P AND Q ARE COPRIMES AND HAVING 2 AS COMMON FACTOR
THIS CONTRADICTS THAT OUR ASSUMPTION IS WRONG √2 IS IRRATIONAL .
√5 IS HAVING THE SAME PROCESS AS OF √2 .
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