Proof that the Product of Two Rational Numbers is Rational

The purpose of this blog is to introduce mathematical proofs as early as possible. When I say mathematical proof, it does not have to be rigorous; sometimes simple reasoning by words alone would suffice especially in the early grades.

In this post, I am going to show you a simple proof that can be probably given as a problem for middle school students. That is, when the students already understand the definition (and meaning) of rational numbers.

A rational number is a number which can be expressed as a fraction whose numerator and denominator are both integers and the denominator is not equal to 0. A student will probably argue that the fractions of integral numerators and denominators, but it is not the case. The fraction \frac{\pi}{2} (not the image above) for instance is a fraction, but not a rational number. From that definition, we can show that the product of two rational numbers is rational. I have written the proof as layman as possible.

Theorem: The product of two rational numbers is rational.

Proof

If we have two rational numbers, then both of them can be expressed as fractions whose denominator is not equal to zero. Let the two fractions be \frac{a}{b} and \frac{c}{d}. If we multiply them, their product will be

\displaystyle \frac{ab}{cd}

Now, since a and b are integers, their product will also be an integer (Closure property). Also, since c and d are integers, and both of them are not equal to 0, therefore, the product is an integer not equal to 0. Thus, the product is a fraction whose numerator and denominator are integers and the denominator is integer not equal to 0. This is the definition of rational numbers which completes the proof. \blacksquare

For Teachers

It is up to the teacher to formulate questions and elicit good reasoning to arrive to the proof above. The questions will depend on the level of students and their prerequisite knowledge. If the students already know the closure property, then they can give this as a reason that the product of two integers is always an integer. If not, then maybe, it can be assumed as a fact for the time being. 

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