Can the product of two rational numbers be an irrational number? (Kindly see the example in description)

Question

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??

Answer 1

$\frac {22} 7 = 3.\overline{142857}$, which is a rational approximation of $\pi$ but not exactly $\pi$.

Answer 2

You are wrong: $\displaystyle\frac{22}7=3.142857142857142857\ldots$ and this decimal expansion is periodic. Not to mention that by definition $\dfrac{22}7$ is rational.

Answer 3

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.

Answer 4

Isn't is just absurd to think otherwise - The very fact that you can write a number in $\frac{p}{q}$ makes it rational number, and if you have two such numbers there multiplication or sum can also be written in the same form, so hence its rational.

Irrational comes only when you cant write in fractional form, but they are already in fractional form. What did you expect.