$\begingroup$I checked in many sources and I saw "Multiplication is closed under Rational Numbers Q". But consider $$ a = \frac{1}{7} ; \;\;\; b = \frac{22}{1} ;$$
both a, b are individually rational (either repeating or terminating decimal vlaues) $$ a = 0.\overline{142857} ; \;\;\;b = 22.0 ; $$ but their product $$ \frac{22}{7}=3.14159265359 ...$$ which is clearly irrational .
Then how is multiplication closed on rational numbers??
$\endgroup$63 Answers
$\begingroup$You are wrong: $\displaystyle\frac{22}7=3.142857142857142857\ldots$ and this periodic. Not to mention that by definition $\dfrac{22}7$ is rational.
$\endgroup$1$\begingroup$$\frac {22} 7 = 3.\overline{142857}$, which is a rational approximation of $\pi$ but not exactly $\pi$.
$\endgroup$$\begingroup$Rational numbers by definition, are numbers that can be expressed as the quotient of two integers. Since $22$ and $7$ are integers, $\frac{22}{7}$ is rational. The fact that this ratio approximates $\pi$ is just an interesting coincidence.
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