Suppose you only know if some numbers are positive (+), negative (-) or zero (0). Can you figure out how to add and multiply them using just this information?

If you add two positive numbers you know the answer is positive, so let's write

+ ⊕ + = +

Here ⊕ stands for addition: if I used an ordinary plus sign I'd say

+ + + = +

and my brain would explode. If you add a negative number and zero you know the answer is negative, so

- ⊕ 0 = -

And so on. The only problem is this: what's a positive number plus a negative one? It can be positive, negative or zero! So you're stuck unless you introduce a new thing: "indeterminate", for when your number could be either +, - or 0. Let's call this "i", so we get

+ ⊕ - = i

Then you can fill out the whole addition table:

⊕ + 0 – i
+ + + i i
0 + 0 – i
– i – – i
i i i i i

Multiplication is actually easier: it's never indeterminate unless you *start* with something indeterminate. Let's call it ⊗. We have things like

+ ⊗ - = -

and the one that confuses kids:

- ⊗ - = +

It's also easy to see what happens with "indeterminate" numbers, like

+ ⊗ i = i

but

0 ⊗ i = 0

So we can write down the whole multiplication table:

⊗ + 0 – i
+ + 0 – i
0 0 0 0 0
– – 0 + i
i i 0 i i

And it turns out ⊕ and ⊗ are commutative and associative, and ⊗ distributes over ⊕, and so on... so we get a mathematical gadget called a 'rig', which is 'ring without negatives'. Yes, even though we have a thing called -, there's nothing you can add to + to get 0.

This weird math is actually useful for "qualitative" reasoning, when you don't know the exact numbers! That's why I care about it. Read on here:

https://johncarlosbaez.wordpress.com/2024/11/12/polarities-part-4/