Question Number 9510 by FilupSmith last updated on 11/Dec/16
$$\mathrm{Prove}\:\mathrm{if}\:\mathrm{a}\:\mathrm{rational}\:\mathrm{times}\:\mathrm{an}\:\mathrm{irrational} \\ $$$$\mathrm{can}\:\mathrm{result}\:\mathrm{in}\:\mathrm{a}\:\mathrm{rational}. \\ $$
Answered by geovane10math last updated on 11/Dec/16
$${A}\:{rational}\:{times}\:{an}\:{irrational}\:\boldsymbol{{alyaws}}\: \\ $$$${is}\:{a}\:{irrational}.\: \\ $$$${q}_{{m}} =\:{rational} \\ $$$${i}_{{m}} \:=\:{irrational} \\ $$$${Suppose}\:{that},\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\boldsymbol{{q}}_{\boldsymbol{{m}}} \centerdot\boldsymbol{{i}}_{\boldsymbol{{m}}} \:=\:\boldsymbol{{q}}_{\boldsymbol{{n}}} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\boldsymbol{{i}}_{\boldsymbol{{m}}} \:=\:\frac{\boldsymbol{{q}}_{\boldsymbol{{n}}} }{\boldsymbol{{q}}_{\boldsymbol{{m}}} }\:\:\:\:\:\left(\mathrm{1}\right) \\ $$$$\frac{{q}_{{n}} }{{q}_{{m}} }\:{is}\:{rational},\:{so}\:\left(\mathrm{1}\right)\:{is}\:{false} \\ $$$$ \\ $$$$ \\ $$