If numbers α and β are irrational, and α+β rational, prove that the numbers α-β and α+2β are irrational?

1 Answer
MattyMatty
Mar 3, 2018

#"See explanation"#

Explanation:

#"The sum of a rational and irrational number is irrational."#
#"So if "gamma = alpha+beta" is rational, then we have that"#
#gamma + beta = alpha + 2 beta" is irrational, and also "gamma - 2 beta = alpha-beta#
#"is irrational, because "beta and -2 beta" are irrational ."#

#"To see that the sum of a rational and irrational number is"#
#"irrational, we put"#
#R = a/b " (rational number R can be written as the quotient of"#
#"two integers a and b)"#
#R + I = a/b + I " (with I an irrational number)"#
#"Now suppose "R+I" is rational, then we can write it as"#
#R + I = c/d#
#=> I = c/d - a/b = (cb-ad)/(bd)#
#"So "I" would be rational which is in contradiction with our"#
#"assumptions, so "R + I" is irrational."#

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