Do Complex Numbers Exist?

Do complex number exist or are they just a convenient, mathematical tool that we use in science? With the exception of quantum mechanics, it is easy to get rid of complex numbers. But can you do quantum mechanics without complex numbers? A recent paper says no, you can't.

#science #physics #mathematics

0:00 Intro
01:15 The Math of Complex Numbers
03:44 The Physics of Complex Numbers
05:45 Complex Numbers in Quantum Mechanics
06:58 The New Paper
09:28 Why is it controversial?
10:26 Sponsor Message

[sirnukesalot]

A while ago I was surprised to learn that Gauss didn't like the term "imaginary" at all. Instead he wanted to call them "lateral" numbers because they described a lateral process. Seems like he knew that future nerds would get caught up arguing about this.

[clieding]

The label “imaginary” for these numbers is arbitrary, unnecessarily evocative [Apparently also provocative!], and merely a relic of the difficult history of their acceptance. Euler disapproved of the label as it came packaged with false philosophical-psychological implications. He said it would have been much better if they had been pragmatically called “orthogonal” numbers as they indicate a direction orthogonal to the “real” major axis of the two-dimensional complex plane. They can be seen as somewhat analogous to the “negative” numbers which themselves indicate a direction opposite the positive real numbers on the one-dimensional “real-number line”. The acceptance of negative numbers was also met with conflict and controversy. I would argue that the term “real” is itself also arbitrary, redundant and unnecessarily suggestive. Numbers of all kinds serve a deeper, more fundamental purpose as abstract unique identifiers which also embody the concept of sequential order; numbers being products of sequential logical operations and algorithms. It is these two major properties of numbers, each being unique and sequential, that make them useful and necessary in measuring, comparing and recording all sorts of physical quantities, properties, intensities and directions as well as representing purely abstract concepts and relationships.

[MarioBoley]

This video is an impressive didactic achievement: it manages to explain complex numbers, Euler’s identity, and the relevant aspects of quantum mechanics in ten minutes, just requiring basic mathematical prerequisites. It’s beautiful.

[Markv]

I recall in an electrical engineering class, the professor showed us all on the chalkboard the lengthy, tedious, laborious (and boring) process of only using real numbers to calculate electrical impedance in an AC circuit. After about 20 minutes, he got the answer. Then, did the same calculation, but instead used complex numbers, and it took him all of around a couple of minutes.

Basically all numbers, real and imaginary, are just tools to help us answer certain questions. Both real and imaginary have their benefits. For example, one cup of coffee has a meaning whereas sqr(-1) cups of coffee is meaningless. But for solving the equations like x^2 = -1, real numbers are worthless, ; they can only be done with imaginary numbers. And in the real world, when doing certain calculations, in the middle of those calculations you run into things like x^2 = a negative number, but eventually produce a real number answer.

[swish]

Missed opportunity: „Are complex numbers real?“ :D

[jehl]

I'm waiting for Google to add context to this video since the subject is controversial and at least one side is based on imaginary arguments.

[roobear]

The IRS is not very fond of imaginary numbers, but the Fed loves them😂

[Iohannis]

Being able to tell my teachers and professors that complex numbers don't exist would have saved me a lot of homework.

[JM-usfr]

Just a clarification: Complex number ARE a real vector space (specifically isomorphic to R^2), they just ALSO have the additional structure of a field. They also have some nice properties, like being algebraically closed (which you mentioned) and being analytically complete (so you can do calculus with them). You can also do some cool calculus you wouldn’t normally be able to do in real numbers.

[seanspartan]

At one point in history, negative numbers were considered highly controversial...

[contessa.adella]

I love the concept. I saw a suggestion a while back that the complex series are an integral part of the universe of numeracy. Like algebra, irrationals, integers, powers etc. It is just that we don’t commonly use them, but they are no less valid.

[lostinmuzak]

Really enjoy your channel. You make it so easy to understand complex (no punt intended). Thanks

[GameTimeWhy]

"Super niche nerd fights"
Fantastic.

[thwh]

Actually, I wanted to make my opinion on this subject very clear.
Instead, my advisors Dunning and Kruger advised me to limit myself to commenting on the algorithm.

[douglasstrother]

"Applied Complex Variables" by John W. Dettman is a great read: the first part covers the geometry/topology of the complex space from a Mathematicans' perspective, and the second part covers application of complex analysis to differential equations and integral transformations, etc. from a Physicists' perspective.

[philipalexander]

This woman is incredible.. she embodies a literal channel.. simplifying complex issues into understandable equations.. isn't that what life is?

[johannesrauch]

I never understood the question "Do Complex Numbers Exist?". It is a question which does not make any sense. I mean you could also ask "Does a natural number exist?". Both do exist as a mental/spiritual concept in our heads but both do not have a physical manifestation. The term "imaginary number" is simply misleading.

[MervynPartin]

When I studied electrical engineering (a long time ago), we made considerable use of complex numbers for alternating current calculations, not worrying if they existed or not, but extremely useful as a mathematical tool for describing system behaviour.
Whereas mathematicians and physicists would use the term "i" to be the square root of -1, we used "j" as the square root of -1, and it was commonly referred to as "j notation".

[procerpat]

Fascinating video. Great work, Sabine!

[ConnyNordgren]

Thank you very much for this video. It was really an inspiring and interesting one. The best explanation of complex numbers I ever heard or seen of!!