No, it’s correct. You define the operation by it’s properties. It’s not saying that “a plus 0 = a” but “the result of applying the binary operation ‘+’ to any number with 0 should give the original number.”

• is just a symbol. You could instead write it as +(a,0)=a and +(a,S(b))=S(+(a,b)).

You have to have previously defined 1=S(0), 2=S(1), 3=S(2), and so on.

The system works perfectly, it just looks wonky in base 10. In base 3 0.333… looks like 0.1, exactly 0.1

Do you know what an irrational number is?

Sure, let’s do it in base 3. 3 in base 3 is 10, and 3^(-1) is 10^(-1), so:

1/3 in base 10 = 1/10 in base 3
0.3… in base 10 = 0.1 in base 3

Multiply by 3 on both sides:

3 × 0.3… in base 10 = 10 × 0.1 in base 3
0.9… in base 10 = 1 in base 3.

But 1 in base 3 is also 1 in base 10, so:

0.9… in base 10 = 1 in base 10

Carbohydrates are the ones with (H20)n

But snakes are legless lizards. Link to youtube

Edit: Wrong link, I’ll leave the previous one anyways because it’s also fun https://youtu.be/_5jNZyoSszE