Can there be numbers with more "signs" than plus or minus?

Question

I will first begin with the question.

What if there are more than just plus and minus basics signs.

What if complex number i is just number with other sign?

If we represent multiplying - to - in turns, and think of minus as turn in opposite direction, we'll get this

Then if we draw complex plane, and assume that i is sign s and -i is sign ns, and assume that this sign is half turn, we'll get our usual complex number

Does it have any sense?

Answer 1

What if there are more than just plus and minus basics signs?

Various authors have attempted to construct new number systems with multiple "signs". For example, in my prior answer you will find some links to so-called PolySign numbers that were discussed frequently on sci.math. A quick introduction to PolySign numbers is given in Eitzen's paper linked there.

It turns out that PolySign numbers (and most similar constructions) are isomorphic to certain ring direct sums of $\Bbb R$ and $\Bbb C,\,$ as follows immediately from old results of Dedekind and Weierstrass classifying finite dimensional algebras over $\Bbb R$ without nilpotents. There you will also find links to other related classification results on algebras that play a key role on this and related matters.

Answer 2

Well, multiplying by $-1$ is already commonly interpreted as a half turn so multiplying by $i$ would be better interpreted as a quarter (half half) turn.

You could invent a new notation for complex numbers and denote $i$ by $1$ with a new sign. However, I don't think that it would achieve much.

In some contexts, e.g. electrical engineering, $j$ is sometimes used in place of $i$ since $i$ is reserved for current. I was introduced to $j$ first in school; it was two years later in university that I first encountered $i$ and for a while it seemed unfamiliar.

Such a move would probably not be popular as the current notation is pretty well agreed on. We already live with multiple notations for differentiation, having multiple notations for complex numbers would probably be seen as a move in the wrong direction.

Answer 3

You have to ask "How do we abstract the idea of positive and negative numbers?" Notice that, if we let $\mathbf P$ be the set of positive integers and $\mathbf N$ be the set of negative integers, then we can partition the set of integers into the union $\mathbb Z = \mathbf P \cup \{0\} \cup \mathbf N$ of disjoint sets.

To continue with this idea, we need to introduce units. In an abelian ring $\mathbf R$, a unit is a member $x \in \mathbf R$ that has a multiplicative inverse.

In the ring of integers, $\mathbb Z$, the units are $1$ and $-1$. Note that $\mathbf P = 1\cdot \mathbf P$ and $\mathbf N = -1 \cdot \mathbf P$.

In the ring $\mathbb Z[i] = \{a+bi : a,b \in \mathbb Z \}$, of Gaussian integers the units are $1,-1, i$ and $-i$. We can partition the Gaussian integers into the union

$$\mathbb Z[i] = \{0\} \cup \mathbf Q_1 \cup \mathbf Q_2 \cup \mathbf Q_3 \cup \mathbf Q_4$$

of disjoint sets, where

$\mathbf Q_1 = \{a+bi : a > 0, b \ge 0 \}$

$\mathbf Q_2 = i \mathbf Q_1$

$\mathbf Q_3 = i^2 \mathbf Q_1$

$\mathbf Q_4 = i^3 \mathbf Q_1$