Given the structure $(\mathbb{R};+,\cdot)$ (the language is assumed to have equality and the parameters $\forall$, $+$, and $\cdot$), I want to define any natural, integer, and rational number. So far I have managed the following:
I have defined $0$ using the formula: $$\exists v_2 \forall v_1 (v_2 + v_1 = v_1)$$
I have defined a given natural number $n$ using the formula: $$\exists v_2 ((v_1=v_2 + \cdots + v_2) \wedge \forall v_3 (v_2 \cdot v_3 = v_3))$$ (where $v_2$ is added $n$ times).
I have defined a given rational number $\frac{m}{n}$ using the formula: $$\exists v_2 ((v_1 + \cdots +v_1 = v_2+ \cdots + v_2) \wedge \forall v_3 (v_2 \cdot v_3 = v_3))$$ (where $v_1$ is added $n$ times and $v_2$ is added $m$ times).
I am stuck, however, in trying to define negative integers and negative rationals. Any help in solving that would be much appreciated!
Thanks in advance!