Instead of having two signs $\{+,-\}$ consider mathematics with three $\{r,g,b\}$. It is $$(r)x + (g)x + (b)x = 0, \ x\in(0,\infty).\tag{1}$$ where $$0= (r)0 = (g)0 = (b)0,$$ is the additive zero-element. It holds $$(r)x + (g)y + (b)z = \text{sign}(\max(x,y,z))[\max(x,y,z) - \text{mid}(x,y,z)].\tag{2}$$ Multiplication is given by the table below.
Smaller-than relation still denoted by $<$ is clock-wise $$(r)x < (g)x < (b)x < (r)x < ...$$ giving the following number line.
What are the differences to usual mathematics/analysis? Is that approach consistent? What about applicability? I am prob. not the first having this idea...
Edit: In contrast to usual analysis,
addition is not associative
$<$ is not transitive
there is no $\max$ or $\min$ among three different signs
every number has one unique $n$-th root
$x+y=0$, $x$ fixed, has two solutions
every integer has two successors and two predecessors
a successor is also a predecessor of the same number
It looks (1) does not work. Better use $$(r)x + (g)x = (g)x + (b)x = (r)x + (b)x= 0.$$
An example for addition:
A nice way to define the addition and multiplication is the following $$ G=\{(\alpha, x)|\alpha\in \mathbb{N}_3, x\in\mathbb{R}^+\}\\ (\alpha)x:=(\alpha,x)\\ (\alpha)x+(\beta)y=\begin{cases} (\alpha)x+y,&\alpha=\beta\\ (\alpha)x-y,&\alpha\neq\beta, x>y\\ (\beta)y-x,& \alpha\neq\beta, y>x \end{cases}\\ (\alpha)x\cdot(\beta)y=(\alpha+\beta)x\cdot y$$