John Carlos Baez on Nostr: Suppose I only care if numbers are positive (+), negative (-) or zero (0). Can I ...
Suppose I only care if numbers are positive (+), negative (-) or zero (0). Can I figure out how to add and multiply 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! Read on here:
https://johncarlosbaez.wordpress.com/2024/11/12/polarities-part-4/
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! Read on here:
https://johncarlosbaez.wordpress.com/2024/11/12/polarities-part-4/