The function of this blog is to present mathematical proofs as early as possible. When I to speak mathematical proof, that does not need to be rigorous; sometimes basic reasoning by words alone would suffice specifically in the early grades.

You are watching: The product of two rational numbers is rational

In this post, i am walk to display you a straightforward proof that have the right to be probably offered as a difficulty for center school students. That is, once the students already understand the an interpretation (and meaning) of reasonable numbers.

A reasonable number is a number which have the right to be expressed together a fraction whose numerator and denominator are both integers and also the denominator is not equal come 0. A student will probably argue the the fractions of integral numerators and denominators, however it is no the case. The portion $latex \frac\pi2$ (not the photo above) for circumstances is a fraction, however not a rational number. From that definition, us can show that the product of 2 rational number is rational. I have actually written the proof as layman together possible.

Theorem: The product of 2 rational numbers is rational.

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Proof

If we have two rational numbers, climate both the them can be expressed as fractions whose denominator is not equal come zero. Permit the 2 fractions be $latex \fracab$ and also $latex \fraccd$. If we multiply them, your product will certainly be

$latex \displaystyle \fracabcd$

Now, since $latex a$ and also $latex b$ room integers, their product will additionally be an essence (Closure property). Also, because $latex c$ and $latex d$ space integers, and also both that them room not equal to $latex 0$, therefore, the product is an integer not equal come $latex 0$. Thus, the product is a fraction whose numerator and denominator room integers and also the denominator is integer no equal to 0. This is the an interpretation of rational number which completes the proof. $latex \blacksquare$

For Teachers

It is as much as the teacher to build questions and elicit great reasoning to arrive to the evidence above. The inquiries will count on the level the students and their prerequisite knowledge. If the students already know the closure property, climate they can give this as a reason that the product of 2 integers is always an integer. If not, then maybe, it have the right to be assumed as a truth for the time being.

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