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Irrational numbers/Structural properties/Exercise
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Let
u
∈
R
{\displaystyle {}u\in \mathbb {R} }
and
v
∈
Q
{\displaystyle {}v\in \mathbb {Q} }
.
Show that
u
{\displaystyle {}u}
is an irrational number if and only if
u
+
v
{\displaystyle {}u+v}
is irrational.
Suppose now also that
v
≠
0
{\displaystyle {}v\neq 0}
. Show that
u
{\displaystyle {}u}
is irrational if and only if
u
⋅
v
{\displaystyle {}u\cdot v}
is irrational.
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