the simple way to put it is the point at infinity is 0
Did you know that the points on an elliptic curve form an Abelian group?
Now I’m glad we suffered through 2 podcasts to explain what groups are and how we don’t know why but it’s important to know something is a group!
We (on this podcast) still don’t know why this fact matters for elliptic curves yet - but it seems plausible in the context of what we discussed that we like:
1) closure under point addition (adding any two points gives us a point that is definitely on the curve)
2) infinity point is the identity (always wondered what that point meant in the coding books like nostr:npub10vlhsqm4qar0g42p8g3plqyktmktd8hnprew45w638xzezgja95qapsp42 Programming Bitcoin)
3) order of addition doesn’t matter (P1+P2) = (P2+P1)
So the points being an abelian group makes all of the arithmetic around adding points nice. VERY NICE!
Discussion
Thank you. I’m going through proof (of its existence) involving projective geometry and it’s removing all ability to discuss this with other humans.
Thank you kindly for that beautiful economy of words that eludes me.