# What am I doing wrong in calculating point addition? Is something redundant in the formula I am using?

First of all, just to note that I am having trouble with this MOD math, so for some this question might be basic, but in my case it caused a lot of confusion.

Just for the context. In my previous question Pieter Wuille and I had a discussion in the comments (we've deleted them so you won't find them anymore) about points in a finite field and he said that it's just a convention to represent a point as a "direct" value of some field. For example, in a curve where `p = 11`, the direct values are `[0, 1, 2, ..., 10]` (I randomly denoted them as a "direct" values). He said that we can represent, for example, value of `3` also as `14`, `-8` and `-19` etc, since using `mod 11` on them they all denote the same value and it is just a convention to represent them as "direct" values. I totally agree with that.

He also said that we can do EC arithmetic (addition) with, for example, `(3, 9)` as well as with `(3, -2)` or `(3, 20)` (since `-2 mod 11`, `9 mod 11` and `20 mod 11` are all the same values) and we will end up with the same point/result after arithmetic operation. What he specifically drew my attention to is the following (I'm paraphrasing): "as long as you realize that the output points whose coordinates differ by multiple of 11 are the SAME POINTS). This seems totally logical to me and I agree with that, but...

What confuses me is that I always end up with the exactly same point regardless of the input point values. For example, in case of `(3, 2) + (9, 5) = (8, 2)`, according to what he says, if I use `14`, `-8` or `-19` in place of `3` it need to end up with some X coordinate that differ by a multiple of `11` to `8`. However, I always end up with the `(8, 2)` regardless of what are the initial coordinates.

I know that the formula for adding points in EC is as follows (taken from here). I have given it as a picture. Since we work with the EC over finite field, we must take MOD operation into account , so the modificated formulas are as the following (taken from the same site): Using these formulas I always end up with the exact same point (in my example `(8, 2)`), regardless of inputs. So there are no points on output whose coordinates differ by a multiple of 11.

I assume that there is something redundant in this formula, which consistently returns values in the set, as I previously referred, of "direct" values. I assume some MOD is redundant, or do I even need any of these MOD (although when I do not use MOD I and up with totally wrong points).

What am I doing wrong? What is redundant in the formula?

Thanks to all!