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Prove R is rational and S is irrational then R plus S must be irrational?
Wiki User
∙ 13y ago
Let R + S = T, and suppose that T is a rational number.
The set of rational number is a group.
This implies that since R is rational, -R is rational [invertibility].
Then, since T and -R are rational, T - R must be rational [closure].
But T - R = S which implies that S is rational.
That contradicts the fact that y is an irrational number. The contradiction implies that the assumption [that T is rational] is incorrect.
Thus, the sum of a rational number R and an irrational number S cannot be rational.
Wiki User
∙ 9y ago
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