Just out of curiosity I tried making a new operation to see its properties. It is like a variation of addition that works like this:
a ¨ b = a + b - 1
For example, I found that it has the commutative property and that
a ¨ b ¨ c ¨ d ... = a + b + c + d + ... - [number of operations done]
Then I tried to make the equivalent of the multiplication in terms of this operation. The same way as a*b is a+a+a+a...(b times), a " b = a ¨ a ¨ a ... (b times)
From here I find that
a " b = ab - b - 1
What I can't figure out is whether this formula can also be applied for negative numbers or zero.
By the way, I found that it doesn't have the commutative property.
Since it is a negative number I don't know what it would mean to repeat the first operation a negative number of times. I know that for example 5*(-3) equals -15 because it is the same as doing - 3 - 3 - 3 ... (five times), but since my operation does not have a commutative property I don't know how to look at this problem.
I realise this question might be silly but I hope someone can help me. Thank you in advance.