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Proof

(related to Proposition: Multiplying Negative and Positive Rational Numbers)

According to the definition of negative and positive rational numbers, we can represent a rational number $x$ by an ordered pair of 0 and a positive integer $b > 0$ or two positive integers $a > 0,~b > 0$ in three ways: $$x=\begin{cases}\frac ab,~b\neq 0&\Longleftrightarrow \text{ if }x\text{ is a positive rational number}\\ \frac 0b,~b\neq 0&\Longleftrightarrow \text{ if }x\text{ equals 0}\\ \frac {-a}b,~b\neq 0&\Longleftrightarrow \text{ if }x\text{ is a negative rational number}\\ \end{cases}$$

Due to the definition of multiplying rational numbers, we have for two rational numbers $x=\frac ab$ and $y=\frac cd$, $b\neq 0$, $d\neq 0$:

$$\begin{array}{ccl} x\cdot y:=\frac {ac}{bd}.\quad\quad ( * ) \end{array}$$

Because the multiplication of rational numbers is commutative, it is sufficient to prove the following four cases, applying the rules of multiplying negative and positive integers:

$(1)$ A positive rational number times a positive rational number gives a positive rational number.

Let $x=\frac ab$ and $y=\frac cd$ with $a > 0,~b > 0,~c > 0,~d > 0$. It follows in $( * )$ that $ac > 0$ and $bd > 0$, thus the product is a positive rational number.

$(2)$ A negative rational number times a positive rational number gives a negative rational number.

Let $x=\frac ab$ and $y=\frac cd$ with $a < 0,~b > 0,~c > 0,~d > 0$. It follows in $( * )$ that $ac < 0$ and $bd > 0$, thus the product is a negative rational number.

$(3)$ A negative rational number times a negative rational number gives a positive rational number.

Let $x=\frac ab$ and $y=\frac cd$ with $a < 0,~b > 0,~c < 0,~d > 0$. It follows in $( * )$ that $ac > 0$ and $bd > 0$, thus the product is a positive rational number.

$(4)$ Zero times any rational number gives zero.

Let $x=\frac ab$ and $y=\frac cd$ with $a = 0,~b > 0,~d > 0$ and an arbitrary $c\in\mathbb Z$. It follows in $( * )$ that $ac = 0$ and $bd > 0$, thus the product is zero.

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References

Bibliography

1. Piotrowski, Andreas: Own Research, 2014