Can anyone help me with this?

"if x is an irrational number, is 2x rational or irrational? explain"

(Headbang)

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- Oct 30th 2009, 07:45 AMokamiirrational numbers
Can anyone help me with this?

"if x is an irrational number, is 2x rational or irrational? explain"

(Headbang) - Oct 30th 2009, 07:52 AMCaptainBlack
If $\displaystyle 2x$ were a rational number then there exist integers $\displaystyle a$ and $\displaystyle b$ such that:

$\displaystyle 2x=\frac{a}{b}$

but then:

$\displaystyle x=\frac{a}{2b}$

but that means $\displaystyle x$ is also rational.

So if $\displaystyle x$ is irrational $\displaystyle 2x$ must also be irrational as otherwise we have a contradiction (we would have $\displaystyle x$ both rational and irrational and that is not possible).

CB