# 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, 2023

posted by Mister_Sleight_of_Hand at 2:42 AM on August 18, 2023

This is amazing! I started tutoring at a community college in February, and have been re-learning math to do it. I'm 58, and so I'm revisiting algebra I took in the early 80s. I also tutor social sciences and humanities, but I am discovering a real love of math and of teaching it. I feel like I'm understanding it conceptually in a way I never did as a young person. We've just been doing rational expressions with the College Algebra students, after a run at quadratics and polynomial functions. Much factoring! Much finding of roots!

I admit this lost me when complex inputs and vectors showed up—I vaguely remember vectors but the math I'm tutoring hasn't required me to revisit them—but my brain is absolutely lit up by the phantom parabola. I can't wait to show it to the trio of Advanced Math Nerds who have been shepherding me through my re-learning. Just last week I derived the quadratic formula with one of them in support! It was so cool.

We have lots of calculus tutors so I'm not needed there, but I'm loving the math I'm doing so much I may take some higher math when I have a chance. I get tuition remission as an employee.

posted by Well I never at 9:16 AM on October 28, 2023

I admit this lost me when complex inputs and vectors showed up—I vaguely remember vectors but the math I'm tutoring hasn't required me to revisit them—but my brain is absolutely lit up by the phantom parabola. I can't wait to show it to the trio of Advanced Math Nerds who have been shepherding me through my re-learning. Just last week I derived the quadratic formula with one of them in support! It was so cool.

We have lots of calculus tutors so I'm not needed there, but I'm loving the math I'm doing so much I may take some higher math when I have a chance. I get tuition remission as an employee.

posted by Well I never at 9:16 AM on October 28, 2023

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posted by mpark at 10:30 AM on August 16, 2023