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Question Number 9510 by FilupSmith last updated on 11/Dec/16

Prove if a rational times an irrational  can result in a rational.

$$\mathrm{Prove}\:\mathrm{if}\:\mathrm{a}\:\mathrm{rational}\:\mathrm{times}\:\mathrm{an}\:\mathrm{irrational} \\ $$$$\mathrm{can}\:\mathrm{result}\:\mathrm{in}\:\mathrm{a}\:\mathrm{rational}. \\ $$

Answered by geovane10math last updated on 11/Dec/16

A rational times an irrational alyaws   is a irrational.   q_m = rational  i_m  = irrational  Suppose that,                               q_m ∙i_m  = q_n                            i_m  = (q_n /q_m )     (1)  (q_n /q_m ) is rational, so (1) is false

$${A}\:{rational}\:{times}\:{an}\:{irrational}\:\boldsymbol{{alyaws}}\: \\ $$$${is}\:{a}\:{irrational}.\: \\ $$$${q}_{{m}} =\:{rational} \\ $$$${i}_{{m}} \:=\:{irrational} \\ $$$${Suppose}\:{that},\: \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\boldsymbol{{q}}_{\boldsymbol{{m}}} \centerdot\boldsymbol{{i}}_{\boldsymbol{{m}}} \:=\:\boldsymbol{{q}}_{\boldsymbol{{n}}} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\boldsymbol{{i}}_{\boldsymbol{{m}}} \:=\:\frac{\boldsymbol{{q}}_{\boldsymbol{{n}}} }{\boldsymbol{{q}}_{\boldsymbol{{m}}} }\:\:\:\:\:\left(\mathrm{1}\right) \\ $$$$\frac{{q}_{{n}} }{{q}_{{m}} }\:{is}\:{rational},\:{so}\:\left(\mathrm{1}\right)\:{is}\:{false} \\ $$$$ \\ $$$$ \\ $$

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