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If r is rational (r = 0) and x is rational, prove that r+x and rx are irrational.
Proof:
We want to prove that r + x R but r + x Q i.e we want r + x R/Q.
What we know:
r Q and also r R r R as Q R.
x Q but x R x R.
Hence x + r R. But since x Q, x + r Q also. This implies x + r R/Q
as desired.
1
Proof:
We want to prove that r + x R but r + x Q i.e we want r + x R/Q.
What we know:
r Q and also r R r R as Q R.
x Q but x R x R.
Hence x + r R. But since x Q, x + r Q also. This implies x + r R/Q
as desired.
1
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