Answer:
A rational times an irrational is irrational.
Step-by-step explanation:
Let x be an irrational number and
[tex]p=\frac{m}{n}[/tex] be a rational number
We want to show that [tex]xp[/tex] is irrational through contradiction:
Assume [tex]xp = q =\frac{a}{b}[/tex] is rational. Rearranging, we get ;
[tex]\frac{xm}{n} = \frac{a}{b} \\\\Cross \:multiply\\bmx = an\\\\Divide \:both\:sides\:of\:the\:equation\:by\:bm\\\\x = \frac{an}{bm}[/tex]
which is a contradiction.
Therefore, a rational times an irrational is irrational.
