# Proof

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.

Github: ### References

#### Bibliography

1. Piotrowski, Andreas: Own Research, 2014