# The Phantom of the XY-Plane

August 12, 2023 7:11 AM Subscribe

The Phantom of the XY-Plane

Every year mathematics YouTuber 3Blue1Brown runs a contest called the Summer of Math Exposition which encourages people, who might not otherwise do so, to publish explanations on math related topics. It's the third year of the contest but the first year I actually had an idea.

A lot of people get confused by complex solutions to quadratic equations when they first learn them. I think part of the reason is that most of us never really get much in the way of visual/geometric intuition for what or where those complex roots actually are. Finding solutions to quadratic equations is generally treated entirely as an algebraic exercise. So I've tried to (massively over-)explain what complex roots "mean" geometrically and give a bit of intuition for "where" they live in relation to the XY-plane.

Lots of people, probably most people, make videos but that's not a requirement. I don't know how to make videos and had neither the time nor inclination to learn. So I wrote this instead. I still can't decide if it's really obvious, everybody knows this and I'm just kind of oblivious for somehow managing to miss it for 30 years; or if it's just that solving quadratics is something that we learn so early that we never really think too much about it once we've mastered the basics.

But hey I'm not the only oblivious person on the planet so hopefully it'll be new and/or interesting to somebody.

Every year mathematics YouTuber 3Blue1Brown runs a contest called the Summer of Math Exposition which encourages people, who might not otherwise do so, to publish explanations on math related topics. It's the third year of the contest but the first year I actually had an idea.

A lot of people get confused by complex solutions to quadratic equations when they first learn them. I think part of the reason is that most of us never really get much in the way of visual/geometric intuition for what or where those complex roots actually are. Finding solutions to quadratic equations is generally treated entirely as an algebraic exercise. So I've tried to (massively over-)explain what complex roots "mean" geometrically and give a bit of intuition for "where" they live in relation to the XY-plane.

Lots of people, probably most people, make videos but that's not a requirement. I don't know how to make videos and had neither the time nor inclination to learn. So I wrote this instead. I still can't decide if it's really obvious, everybody knows this and I'm just kind of oblivious for somehow managing to miss it for 30 years; or if it's just that solving quadratics is something that we learn so early that we never really think too much about it once we've mastered the basics.

But hey I'm not the only oblivious person on the planet so hopefully it'll be new and/or interesting to somebody.

Role:

**writer programmer**Thanks!

posted by Mister_Sleight_of_Hand at 2:42 AM on August 18

posted by Mister_Sleight_of_Hand at 2:42 AM on August 18

« Older Star Wars: Galactic Starcruiser extended essay/rev... | A song (and music video) about... Newer »

posted by mpark at 10:30 AM on August 16