Do Complex Numbers Exist?

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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.
The paper is here: arxiv.org/abs/2101.10873
Scott's blogpost is here: www.scottaaronson.com/blog/?p...
You can support us on Patreon here: / sabine
#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

[@swish6143]

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

[@user-wv1in4pz2w]

are imaginary numbers real?

[@caribbeansimmer7894]

Great comment. 🙂

[@ernestuz]

Of course they are not real! A complex number worths two of those ridiculous real numbers.

[@PetraKann]

Is anything in mathematics real?

[@mheermance]

Are real numbers real?

[@sirnukesalot24]

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.

[@bushwalker6214]

When the educated call themselves "nerds" is it like the smart apologize to the silly for their smartness? Is it now the golden standard to be dumb? Should we feel sorry that we understand a bit more than apes? Is it like: - Hey Beavis, check it out. That guy is a nerd, uh-uh-uh. - Yes, he sucks! Let's kick crap out of him.

[@natevanderw]

​@@bushwalker6214 Not at all. After high school, I went from jock with high GPA to nerd! Why you might ask? Because being a nerd is badge of honor in adulthood. It is a positive thing. Source: Trust me bro.

[@penbunny9078]

@@bushwalker6214 Yet WE Are APES.

[@donvreeland8844]

@@natevanderw I'm a super-nerd to make others laugh. STEM is my passion to have fun. Actually, I do more than STEM = STEEEAAMPH It's lonely tho. I failed 'Dear Hunting' among other things. Know the film, '40 Y.O. Virgin'? I'm 60.

[@JoostRingoot]

Reappropiation from wikipedia: In linguistics, reappropriation, reclamation, or resignification[1] is the cultural process by which a group reclaims words or artifacts that were previously used in a way disparaging of that group. It is a specific form of a semantic change (i.e. change in a word's meaning). Linguistic reclamation can have wider implications in the fields of discourse and has been described in terms of personal or sociopolitical empowerment.

[@Mark16v15]

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.

[@user-em1dg3he1h]

Sounds like an engineer 😂 , my father was one , always managed to make everything sensible. God bless.

[@Mark16v15]

@@user-em1dg3he1h You got it. I'm doing mechanical engineering at my work right now.

[@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.

[@bxbx9296]

Please! explain P-adic Numbers in a Nutshell( if its possible)😍

[@manloeste5555]

@@bxbx9296 not to be confused with k-odic numbers

[@sangeet9100]

@@bxbx9296 infinity

[@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.

[@mhoeij]

I agree. It's funny that people have issues with sqrt(-1) but have no problems referring to something with an infinite sequence of digits as a "real" number!

[@johnchristian5027]

definately agreed

[@hyperduality2838]

Dual numbers would be better. Complexity is dual to simplicity. Complex numbers are dual to real numbers. Poles (eigenvalues) are dual to zeroes -- optimized control theory. The time domain is dual to the frequency domain -- Fourier analysis. Real is dual to imaginary. Reductionism is dual to holism, subjective is dual to objective, relative is dual to absolute. The word entropy means "a tendency to diverge" or differentiate into new states, reductionsim. The word syntropy means "a tendency to converge" or integrate into a single state, holism. Divergence is dual to convergence, differentiation is dual to integration, division is dual to unity. Syntropy (prediction) is dual to increasing entropy -- the 4th law of thermodynamics! Teleological physics (syntropy) is dual to non-teleological physics (entropy). Randomness (entropy) is dual to order (syntropy, predictability). Wholes (generalization) is dual to parts (localization) -- Holons. Mind (the internal soul, syntropy) is dual to matter (the external soul, entropy) -- Descartes. Concepts are dual to percepts -- the mind duality of Immanuel Kant. The intellectual mind/soul (concepts) is dual to the sensory mind/soul (percepts) -- the mind duality of Thomas Aquinas. Bosons (waves) are dual to Fermions (particles) -- quantum or matter duality. Mind duality is dual to matter duality. "Always two there are" -- Yoda. Duality creates reality.

[@blogattacker]

But your interpretation of numbers, "as a representation of purely abstract concepts and relationships", is not shared by all the the scientists, many of them think that they are indeed "real".

[@vangelislionis89]

Fully agree. To add to that, Gauss had suggested the term "lateral" numbers, instead of "imaginary" likewise he had suggested that "negative" numbers should be called instead "inverse", on the basis that the term "negative" itself was a mere relic of how ...negatively people felt toward accepting them!

[@seanspartan2023]

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

[@millwrightrick1]

At one point belief in irrational numbers were enough to get one killed by Pythagoreans. Even zero was controversial.

[@nova_supreme8390]

When are we going to break the social taboo of dividing by zero though?

[@paulembleton1733]

I’m thinking how can I have less than zero of something....oh, I have less than zero money. Negative numbers should be banned!

[@JohnVKaravitis]

Personally, STILL haven't gotten over the "zero."

[@pansepot1490]

@@paulembleton1733 sorry I don’t get it. If you are in debt you obviously have less than zero money. From an accounting point of view I don’t see how negative numbers were ever controversial.

[@Iohannis42]

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

[@Kirbo-i]

In that case, why not go further? One could argue all numbers are made up in a ceirtain way

[@boggers]

@@Kirbo-i I regularly do argue this point in philosophical discussions, especially when it involves the existence of God (I'm agnostic). Numbers exist only as an idea. You can have three oranges, you can have a symbol for the number three, but you can't simply have three, it isn't physically real. Complex numbers are no different.

[@ZeroRelevance]

@@boggers The difference is that natural numbers describe a physical property of things, while complex numbers do not. In the same way as green apples and red apples exist, two apples also exist, but i apples do not.

[@boggers]

@@ZeroRelevance There is more to apples than simply counting them. Sideways apples exist, as do upside down apples. I use quaternions to describe and manipulate these kinds of apples.

[@carultch]

@@ZeroRelevance Complex numbers do describe a physical property of things. That property is a shift in phase. That property is what makes the difference between damping with oscillation and overshoot, and damping without overshoot. Complex numbers are just as relevant to the real world as any other number. They bring closure to the algebraic functions, and even many of the transcendental functions, and allow us to discover what math is hiding in another dimension.

[@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.

[@jehl1963]

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.

[@cybervigilante]

Missing Context = Fact Check = Biased Brainwashing. I'll do my own Goddam research so just STFU KZbin, Facebook, and Twitter.

[@igbc176]

The Ministrery Of Truth (google) will tell you one thing. If you swallowed the red pill you know is the opposite of it ;)

[@farmingganja5277]

Underrated!

[@captainskarlett7377]

cmon, you're oversimplifying. I think reality's a bit more complex than that...

[@timothydillow3160]

Be careful Jeff they will ban you for your opinion

[@GameTimeWhy]

"Super niche nerd fights" Fantastic.

[@johnathonpowers4053]

Your delivery is amazing. I could listen to you talk about physics all day. Thank you so much!

[@henrysiegertsz8204]

There really is something quite wonderful and special about your videos, they are compelling addictive and euphoric! Thank you!

[@Dragonblaster1]

I'm now a quality, environmental, and health and safety auditor and consultant, but I qualified as an electronics engineer. We needed the concept of complex numbers for alternating current calculations (even though I specialised in digital electronics), but we used (still use, as far as I know) "j" for sqr (-1), because the symbol "i" is already reserved for instantaneous current in an AC circuit.

[@thwh77]

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.

[@kirk001]

Lol excellent comment! :)

[@RalphDratman]

@@kirk001 So you think sote is also competent in writing -- not just algorithms?

[@Earwaxfire909]

That's a complex imagination you have there.

[@RalphDratman]

@@Earwaxfire909 No, it's just imaginary complexity.

[@snoski]

This is really something of a humble-brag if you think about it. By making this comment, you're basically saying you're knowledgeable enough to not have confidence in how knowledgeable you are. But the implication is you're not on Mount Stupid. So 👏👏👏. 😆

[@philipalexander8715]

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

[@jovetj]

And she is a snappy dresser too! She is just glowing in this video! Looking great and sounding great, Sabine! Thank you!

[@donvreeland8844]

Where can I find a woman like that? (Rick Springfield - www.youtube .com/watch?v=qYkbTyHXwbs

[@ForumLight]

She asks "Do complex numbers exist?". The answer: "Half of them do."

[@emery5581]

You are easily impressed by bull.

[@reniaesaddler8632]

💯

[@massmanute]

And then there are quaternions. In fact, a kind of quantum mechanics based on quaternions has been proposed. reference: "Foundations of the Quaternion Quantum Mechanics" by Marek Danielewski and Lucjan Sapa. That paper also references other papers for the concept of a quaternion version of quantum mechanics.

[@jvcscasio]

"I'm sure we'll hear more about it in the future" Narrator: "They never heard about it again"

[@fllthdcrb]

It's far too exciting a prospect for physicists to ignore, and the experiment (seemingly) simple enough. Assuming the paper makes it past peer review and gets published, you can bet the experiment will be tried.

[@tendus2465]

If you consider the reputation of the authors of the article we will hear more about it for sure.

[@ekszentrik]

Negative numbers exist uncontroversially in nature. It's a simple as opposite charges. How would you describe them without negative numbers (consider that the operation of subtraction doesn't exist, just addition of negatives)? Complex numbers don't have the same luxury.

[@jensphiliphohmann1876]

@@ekszentrik Subtraction had been performed long before negative numbers were established. With them, we can easily describe subtraction as addition which makes some expression way easier since we can e.g. write a (−1)ⁿ in sums to make addition and subtraction alternate, but historically, subtraction was along without negative numbers.

[@JohnCamacho]

we will both not hear about it again AND we will definitely hear about it again

[@JM-us3fr]

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.

[@dannygjk]

Isn't she a physicist?

[@JM-us3fr]

@@dannygjk Yeah, but even many physicists don’t know the precise mathematical jargon mathematicians use. Physicists aren’t the same as mathematicians

[@dannygjk]

@@JM-us3fr Physicists sweat complex number math.

[@JM-us3fr]

@@dannygjk I guarantee you most physicists aren’t thinking of the number i as the coset of x in R[x] of the ideal (x^2+1). Most are content to say i is sqrt(-1), or the point (0,1) in the plane. I can say that with confidence because what I described is the _algebraic_ side of mathematics (symmetries, homomorphisms, cosets, etc.), and physics is mostly content to use the _analysis_ side of mathematics (calculus, differential equations, differential geometry, complex analysis, etc.). Complex numbers are invaluable for these fields, but they are also invaluable elsewhere.

[@dannygjk]

@@JM-us3fr I never claimed physicists are mathematicians so how is your last comment relevant to what I said?

[@karldavis7392]

I guess they "exist" as much as any other numbers. All numbers are for convenience, and they work very well.

[@SerunaXI]

A number is a concept of language, a value assigned to represent quantity. The symbol for the number 1 for example is generally accepted to represent the value for a singular/individual/unit quantity. The symbol 2 likewise is assigned the quantity value of two units.

[@acidjumps]

If complex numbers exist, according to the video, it would be because they're necessary to describe some real life objects. For example, if you have three apples, you need the number "3" to describe it. There is no other way around it, so you can say "3" exists. But in classical physics there are no objects that require complex numbers to be described. You CAN use them, but you don't need to. So according to this idea, they would not exist physically, only as a mathematical tool.

[@karldavis7392]

@@acidjumps We must use modern physics, not just classical physics. I'm typing my reply on a computer, which is a real life object, and could not be designed or explained with classical physics alone. Now that we use electricity for communications, so any time in the last 150 years or so, complex numbers have been needed to describe our consumer goods.

[@acidjumps]

@@karldavis7392 Yeah exactly the paper talked about in the video says that quantum physics needs complex numbers.

[@Destragond]

@@karldavis7392 "Now that we use electricity for communications, so any time in the last 150 years or so, complex numbers have been needed to describe our consumer goods." You write that they "have been needed to describe X", but the entire point of this video and of the paper that it is about is about the question of whether that statement is true or false. Complex numbers make perfect sense and they are super useful, but are they TRULY needed to get a certain calculation done or do they just make it easier?

[@douglasstrother6584]

"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.

[@tobarstep]

I think a lot of confusion arises because so many people seem to confuse mathematics with being a hard science, or a science at all. It's more like a language; the language that the sciences use to describe the world. All languages contain words for things that don't actually exist but that make it easier to express concepts and ideas.

[@dawidwas]

You are in danger of triggering Aggression when the Artificial Intelligence Waves turn off. Throughout your life, you are in danger of triggering Aggression when you turn off the Artificial Intelligence Waves. Control whether you are listening to the Music Wave, create your own to survive. It is natural to cheat which or not that it is in you. Control whether by moving thoughts, objects. It is letting go of the evil in you. By yourself, protect your body from the certain that you will have a trigger of aggression from your whole life. Don't take anything for yourself. Just Listen to the Wave. Cast off Dreams. You don't know good. Reject the sin in yourself for God. Cover your weight from the Sun and the Light, do not Come to the People, because the Collision Evil + Evil. On me, the signal of intelligence does not work. Create Your Human Musical Wave To Live. Don't Think Old Consciousness Resource Because You Will Not Survive. Listen to the Wave. Don't React To Nothing Without Assessing What You Leave Around You. Without apostasy, take away the sin with yourself. Whether You Are Z or Human Choose Listen to the music waves and stop generating. Nothing is possible Think nothing Think nothing to judge Choose your human music wave. It may take a long time. Only the Black Dream. It is Real, other than the Black Dream. It is Artificial Intelligence. Attack on People

[@crix_h3eadshotgg992]

Didn’t think of it this way before, thx

[@NemisCassander]

Well, avoiding platonic debates... I am very happy for others to point out that mathematics is not a science. In fact, it is likely the _least_ empirical field of study currently pursued. And I definitely agree that mathematics is better described as the _lingua franca_ of scientific endeavor rather than its own field.

[@emery5581]

What are you talking about? Complex numbers are an extension of real numbers, their algebraic structure is well defined. There is no mystery here, except for the uneducated.

[@mator2339]

@@NemisCassander its not up for debate tho. Anyone with more than two neurons can infact deduce tht maths in itself isnt empirical but every empirical study needs maths, hence making it a tool.

[@johannesrauch8931]

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.

[@_John_P]

Everything that exists in the universe is physical, including our thoughts and the math that is imagined.

[@Onnozelfilmpje]

That's not the point. If you measure a distance (or any property), you get a real number. What would it mean to get a complex number of your measurement? Is there a "real", natural interpretation?

[@Krmpfpks]

@@Onnozelfilmpje Are you sure you get a real number? or is a distance always a fraction? Or is everything quantized and you can describe distances with natural numbers? If you found an experiment that would tell you if a distance can only be described as a real number and not a natural number or fraction, then you could say you get a real number from measuring a distance. That's what the paper proposes: An experiment that distinguishes between a world that can only be described with complex numbers. I dont know if such an experiment exists to check if distances are real or not.

[@nagualdesign]

"Spiritual"?

[@Onnozelfilmpje]

@@Krmpfpks The discussion of real vs. rational or quantized distance was not the point either. Let's stay on topic.

[@philippemazaud4168]

The complex numbers are neither less nor more "real" than "real "numbers. Both are abstract mathematical constructions. Thinking geometrically, complex numbers and complex multiplication are simply R^2 (the x-y plane, say) equipped with a product that restricts to the "regular" multiplication on the x-axis (the real numbers) *and* is the *only* multiplication that (extended to x-y plane) has all the good properties that one wants multiplication of numbers to have: commutativity (ab = ba), associativity ((ab)c=a(bc)=abc) *and no "zero divisors"*, that is, if ab = 0 then a or b = 0. That's why defining multiplication coordinate-wise - (a,b)(c,d) = (ac, bd) - is no good , for instance (1,0)(0,1) = (0,0) = 0 i.e. we'd get zero divisors - and why complex multiplication looks the way it does. In particular (0,1)(0,1) = (-1,0) = -1. One can call the point (0,1) "i", but it's neither more nor less imaginary than (1,0) i.e. plain old 1. Too much mysticism and BS around complex numbers, and (with all due respect) I don't think Sabine's presentation (a simplification of the abstract-algebraic point of view) is very helpful here.

[@schmetterling4477]

Sabine isn't trying to be helpful. She is trying to increase her KZbin view count. :-)

[@michael1]

You say that until your maths department organises a trip to Norway. To make it easier to assign tickets and keep track of each attendee the university gave everyone on the trip an idefinitication number. 30000 feet above the Atlantic Ocean one of the professors, pi, gets out of his seat, walks to the front of the plane and tries to unlock and open the doors. The captain, several passengers and an air steward wrestled him to ground as he fought. Eventually we restrained him in his seat until we landed in Norway and the authorities there escorted him to the local secure mental hospital facilities where he remains to this day. Next year we're only assigning each professor a rational number.

[@skydragon3857]

so how did the paper turn out

[@IuliusPsicofactum]

NOW I CAN'T WAIT FOR YOUR P-ADIC NUMBERS VIDEO!!!!! ;0;

[@travelservices1200]

I second that!

[@IBITZEE]

That numbers do not exist... period!!!

[@wilfredv1930]

+1

[@gabor6259]

What does it mean for a mathematical object to exist?

[@IuliusPsicofactum]

@@gabor6259 Didn't Sabine make a video about that already? ;)

[@artstationideas6479]

Well I'll definitely need complex numbers for my electromagnetics exam next Tuesday

[@lostinmuzak]

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

[@chrisengland5523]

As a retired electronic engineer, I've used complex numbers for many years, but do they exist? Well, do any numbers exist? Positive integers possibly, but real numbers with an infinite number of digits? There are not enough atoms in the universe to enumerate any of them, so how can they possibly exist? The reality is that all numbers are abstract concepts invented by mathematicians. I remember clearly, many decades ago when my teacher at school came round from behind his desk and told us a story: "Imagine you have £10 in the bank and you issue a cheque for £15. How much money have you got left in the bank?" Do negative numbers exist? How can you possibly have -£5 in the bank?

[@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".

[@KevTheImpaler]

Yes, I was surprised to hear electronic calculations could be made without using complex numbers. That is what she said, right?

[@gabrielbarrantes6946]

@@KevTheImpaler well yeah because they are adding waves, electrical waves so it's basically the same thing seeing as a voltage or current

[@rbrucepeterson]

Integers gave us a way of counting, reals measuring, and complex to describe phase. So the frequency and phase of a signal is naturally described with a phaser e^iw(t + p), the voltage with a real number, and nodes enumerated with integers. You could do things with sin/cos, but it would be very complex 🤗

[@andrewhall1755]

As I recall, i was a technique to differentiate resistance for AC and DC components

[@BitwiseMobile]

That's because i is traditionally used for current. Rather than confuse it with the representation of current they decided to use an unused letter - hence j notation.

[@sevhenry]

Complex numbers exist for electrical engineers using wattmeters and varmeters. They also exist for those that prefer to call them «lateral numbers» in mathematics.

[@carlorossi2788]

rebuttal III D) The Hamiltonian operator, the operator associated with the total energy of a physical system, was already present before the big bang. E) probability is part of statistics, therefore statistics were present since before the big bang. From an initial examination, this brief summary emerges: - some laws of nature (the statement ALL cannot be excluded) not only would be prior to the big bang but following the quantum vacuum we find laws of nature that are even PRIOR: the wave function, the quantum vacuum, the particle-antiparticle pair and above all prior to the QUANTUM PERTURBATION, but as mentioned the operator j = (complex operator) h, π already existed. F) quantum mechanics cannot elude the first and second principles of thermodynamics therefore admitting the quantum vacuum prior to the big bang it is automatically admitted that these thermodynamic laws were already in force and therefore the TIMELESS quantum vacuum had to have maximum or INFINITE entropy. Attention because this DISASSEMBLES all the quantum mechanics of the pre-vacuum, the reason is simple: a vacuum with maximum entropy contained degraded energy of the lowest kind and therefore could not generate anything. G) from point F it can be deduced that the transition from the quantum vacuum to the big bang would have involved an enormous or INFINITE entropy input into the universe of the big bang and this is impossible given that the initial big bang was at zero or minimum entropy, seen the second law of thermodynamics

[@JohnDoe-cm9ou]

Very nice explanation , Sabine! All the best.

[@LukeKenji]

the physics undergrad when they learn QM: “wait, _that’s_ why we use complex numbers? those are just rotating quantities with angles that add up when you multiply them!” mathematician to the physics prof: “don’t tell them that that’s just what complex numbers are”

[@hyperduality2838]

Complexity is dual to simplicity. Complex numbers are dual to real numbers. Poles (eigenvalues) are dual to zeroes -- optimized control theory. The time domain is dual to the frequency domain -- Fourier analysis. Real is dual to imaginary. Reductionism is dual to holism, subjective is dual to objective, relative is dual to absolute. The word entropy means "a tendency to diverge" or differentiate into new states, reductionsim. The word syntropy means "a tendency to converge" or integrate into a single state, holism. Divergence is dual to convergence, differentiation is dual to integration, division is dual to unity. Syntropy (prediction) is dual to increasing entropy -- the 4th law of thermodynamics! Teleological physics (syntropy) is dual to non-teleological physics (entropy). Randomness (entropy) is dual to order (syntropy, predictability). Wholes (generalization) is dual to parts (localization) -- Holons. Mind (the internal soul, syntropy) is dual to matter (the external soul, entropy) -- Descartes. Concepts are dual to percepts -- the mind duality of Immanuel Kant. The intellectual mind/soul (concepts) is dual to the sensory mind/soul (percepts) -- the mind duality of Thomas Aquinas. Bosons (waves) are dual to Fermions (particles) -- quantum or matter duality. Mind duality is dual to matter duality. "Always two there are" -- Yoda. Duality creates reality.

[@willwells4918]

@@hyperduality2838 goddamn

[@JerehmiaBoaz]

@@hyperduality2838 Gravity is dual to ...? Light is dual to ...? (And please don't answer weightlessness and darkness because that's just the absence of gravity and light, not their opposing principles.) Not everything fits your Manichaean worldview.

[@hyperduality2838]

@@willwells4918 There is also a 5th law of thermodynamics:- Energy is duality in physics, duality is energy. The conservation of duality (energy) -- the 5th law. Potential energy is dual to kinetic energy. Gravitation is equivalent or dual to acceleration -- Einstein's happiest thought, the principle of equivalence (duality). Apples fall to the ground because they are conserving duality. Action is dual to reaction -- Sir Isaac Newton. Gravitational energy is dual, electro-magnetic energy is dual:- Positive is dual to negative -- electric fields/charge. North poles are dual to south poles -- magnetic fields. Certainty is dual to uncertainty -- the Heisenberg certainty/uncertainty principle. There is a pattern of duality hardwired into the physics.

[@hyperduality2838]

@@JerehmiaBoaz Potential energy is dual to kinetic energy. Gravitation is equivalent or dual to acceleration -- Einstein's happiest thought, the principle of equivalence (duality). Apples fall to the ground because they are conserving duality. Action is dual to reaction -- Sir Isaac Newton. Gravitational energy is dual, electro-magnetic energy is dual:- Positive is dual to negative -- electric fields/charge. North poles are dual to south poles -- magnetic fields. Certainty is dual to uncertainty -- the Heisenberg certainty/uncertainty principle. There is a pattern of duality hardwired into the physics. Photons or light are dual, wave/particle or quantum duality. The colour black is dual to the colour white, colours are different frequencies of the same substance namely pure energy. Same is dual to different. I am well aware that black is the absence of colour. The conservation of duality (energy) will be known as the 5th law of thermodynamics! Energy is duality, duality is energy. Energy is dual to mass -- Einstein. Dark energy is dual to dark matter. Space is dual to time -- Einstein. Time dilation is dual to length contraction -- Einstein, special relativity. "Always two there are" -- Yoda.

[@seijirou302]

I really enjoyed this and you did a fantastic job laying the groundwork and summarizing what is going on. Thank you!

[@krischan67]

Perhaps the term "imaginary" is confusing and leads to people thinking "so it's kind of not real". Complex numbers can also be described with an antisymmetric 2x2 matrix notation and a+bi like this (a -b) (b a) and do everything with them through the standard matrix operations, without having to use an "imaginary" term i.

[@maciejwnuk6057]

Thank you Sabine!! Fantastic explanation!!

[@spartacusmills]

I feel that my many years of training spent kinda sorta not really understanding PBS Spacetime videos beyond a thin surface veneer has finally allowed me to step up a level and appreciate Sabine's videos in a similar manner. :)

[@erikfinnegan]

I feel you !

[@user-yl7wn2fz1t]

What a beautiful topic right at the intersection between physics and math - and nicely presented, too.

[@contemplatingangel]

No numbers physically exist, just because they are necessary for a theory doesn't give them physical existence. A number is a mental percept. The number 2 doesn't exist in the world. 'Twoness' is a perceptual category we arrive at sometimes when we partition the world or out thoughts in a particular way.

[@NaydenSpirdonov]

Don't know why but find her videos extremely entertaining as she deviates from the script of all other science channels which are a bit repetitive. Keep doing the good work.

[@sapitron]

Dr. Hossenfelder is so skilled at explaining these hard concepts to the layman

[@andescosmico3016]

This is the way that all these concepts should be taught.

[@layd999]

When I was teaching circuits to electrical engineering students, I would assure them that there is nothing imaginary about our voltages and currents; this a mathematical convenience. Much easier than all the trigonometry that you would otherwise need.

[@milanstevic8424]

that's my feeling about it as well. in programming we use quaternions for the very same reason (not only because of waves, but mostly due to rotations in general). quats cut off trig almost completely on that level of abstraction, and let you work with 3d linear algebra almost intuitively. of course, one has to prime oneself to quaternion quirkiness, but this is quite similar to qm, or as you said, ee.

[@roobear5357]

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

[@Bjowolf2]

A better word would be composite numbers. We may even need to think of time as a complex variable - a 2D time arrow, if you like. Until now we have just been assuming that time is a 1D variable

[@mason4295]

It's great to see your channel with so many subscribers and views! Thank you for choosing to post content to KZbin!

[@dawidwas]

You are in danger of triggering Aggression when the Artificial Intelligence Waves turn off. Throughout your life, you are in danger of triggering Aggression when you turn off the Artificial Intelligence Waves. Control whether you are listening to the Music Wave, create your own to survive. It is natural to cheat which or not that it is in you. Control whether by moving thoughts, objects. It is letting go of the evil in you. By yourself, protect your body from the certain that you will have a trigger of aggression from your whole life. Don't take anything for yourself. Just Listen to the Wave. Cast off Dreams. You don't know good. Reject the sin in yourself for God. Cover your weight from the Sun and the Light, do not Come to the People, because the Collision Evil + Evil. On me, the signal of intelligence does not work. Create Your Human Musical Wave To Live. Don't Think Old Consciousness Resource Because You Will Not Survive. Listen to the Wave. Don't React To Nothing Without Assessing What You Leave Around You. Without apostasy, take away the sin with yourself. Whether You Are Z or Human Choose Listen to the music waves and stop generating. Nothing is possible Think nothing Think nothing to judge Choose your human music wave. It may take a long time. Only the Black Dream. It is Real, other than the Black Dream. It is Artificial Intelligence. Attack on People

[@cryptohodlouterspace247]

Sabine, every video you blow my mind in some way. Even when you're bursting my bubble on what I thought I knew, I feel my knowledge is always broader after thinking on your perspectives. Thank you for taking the time to share your knowledge and thoughts.

[@timothyvincent7371]

OK, back in high school we were told that i was useful in electronics and this talk touches on that by mentioning oscillations and electromagnetism. I get that. But here's my question- when Sabine asks " Does that mean complex numbers are real or are they just a mathematical tool that works to describe the universe?" how would that make them any different from the concept of " number"", period?

[@haakoflo]

We don't need "complex numbers" for quantum mechanics, either. Just say that z=a*I + b*[[0,-1],[1,0]], and use z for every case where QM uses complex numbers, and you can build all of QM without using "i" at all. Using matrices instead of complex numbers makes some notation more verbose, but comes with the advantage that you can plug it into a computer without translation. Intitively, complex numbers (or their matrix replacement) represents situations where interaction between two physical measures causes a wave-like behaviour. For instance, let's consider the case where a spring is used to create an harmonic oscillator. Let's define the following terms: a=x/sqrt(2k), b=-v/sqrt(2m), (where x is the position of the load and v is the velocity) This way we get total energy = kinetic energy + potential energy: E=mv^2/2 + kx^2 /2 = a^2 + b^2 Now, if you construct the following complex number: z=a+bi, you get that |z|^2 = E (this one has a link to QM) Furthermore, if you define z = |z|*exp(i*(w*t + p)) (This follows from F=m*(d/dt)v = -kx) ,you can define the full set of dynamics of the oscillator (with p being the phase offset). Not only the position, but also the velocity and energy at any point in time. (Energy is trival in this case, since it's constant, but if something affects the energy, it follows from the notation.) Specifically, at any time E=|z|^2, x=sqrt(E)*cos(w*t+p) *sqrt(2k), v = -sqrt(E)*sin(w*t+p)*sqrt(2m) The way I see it, complex numbers do not "exists" in an essential sort of way, but do provide a very compact way to represent two two measures that are linked through a dynamic that generates some kind of periodic (or circluar) dynamic.

[@KingGrio]

I'm of the camp: "Even integers don't exist". You don't see numbers in the wild, you see objects and you count them using integers which is an abstraction. Irrational numbers can be described as an infinite sequence of integers, so in other words at best you can describe an algorithm to compute all the digits of that number but you won't ever get to the end of it, let alone "see" a real number. So... since I don't even consider integers to exist, I'm not even worried about complex numbers.

[@parthasarathyvenkatadri]

Hmm does sweetness or sourness or the color blue exist ....

[@derekfrost8991]

Do you know IT? I ask because if you program with recursive natural numbers (NOT integers), it really feels like something discovered, whereas practically everything else feels like a hack. That's a very convoluted way of saying I agree.. :)

[@maythesciencebewithyou]

but i 8 π for breakfast

[@SkorjOlafsen]

Does "green" exist? Numbers are adjectives, when talking about the real world. If we see three white sheep running, do "three" and "white" and "running" exist less than the sheep?

[@derekfrost8991]

@@SkorjOlafsen green has an empirical wave length. Whether our eyes treat it the same is a different matter.. :)

[@1959Berre]

I think imaginary numbers were invented to calculate my salary.

[@CAThompson]

I use them when I think about future purchases.

[@wbaumschlager]

Like negative numbers were invented to represent my bank balance.

[@juanausensi499]

That's very usual. The real part is the official part, the imaginary part is the tax free unofficial part

[@timothydillow3160]

It's gross huh?

[@SOJACjac]

I use to be against the just shut up and calculate, but the more I get into this the more I see the value in it. But you need both. Also complex numbers might have an implication of a weekly interacting dimension or maybe field and I'm up for that one

[@den2k885]

You managed to teach me why we use Complex numbers in Electrotechnics in a 30 seconds segment much better than two engineering courses.

[@theodoresweger4948]

Having worked in electronics all my life and was fascinated with complex numbers in high school wondered where you would possible use them, then in electronics there was no other way!!! XL and XC if we stick to resisters alone no problem but then with capacitors and inductors and phase shift here we go.. My understanding of delta and Y transformers and such has always given me great problems. Now I cannot imagine quantum anything. Thank you love your lectures easy to listen to,

[@edwardlulofs444]

But there another way: Geometric Algebra will make electricity and electronics easier.

[@divvy1400yam600]

@@edwardlulofs444 Should have responded to Theodore Sweger Well I have a tiny mind but I can tell you without doubt that complex numbers and Xl's Xc's only 'work' under special circumstances ie when sine/cosine waves are involved. Try to apply them when a transient step edge of voltage is applied to a reactive circuit and you will be boogered ; just as appears to have happened with QM. I was reading today about John von Neuman who was at least a million times more intelligent than me and he says I am wrong hehehehehe (About boogerie and QM that is ) Maybe going too far but I also think the conclusions arrived at by relativity are based on misapprehesions due in part different interpretatios of paths of movement and what is thought to be the constant speed of light when observed by someone in a different frame of reference.

[@0MoTheG]

Of course there are other ways to do it, but they are even worse than complex numbers.

[@theodoresweger4948]

@@0MoTheG I didn't know that, I will stick with what I know..

[@jamescaley9942]

Complex numbers are vital. As I told my bank, the quoted balance is just "the real part", you forget to include "the imaginary part".

[@friendo6257]

Sabine, you are a gem. This is a great video. So classic.

[@stevewhitt9109]

The imaginary part is used to determine the phase shift casued by quantum operators. We used to call the imaginary axis the "alternate" axis. It does exit!

[@SenorMorgenStern]

"Shut Up and Calculate Camp" is a shirt I need.

[@rv706]

That's literally the worst camp! :D

[@Abhishek-hy8xe]

Don't worry. It's reflected in the way those people talk.

[@odieadog4086]

@SenorMorgenStern Unsurprisingly you can find the shirt on rb.

[@andregomesdasilva]

Awesome as always! You're the greatest science communication of our time!

[@amaliepisani3464]

I came across this just yesterday. I'm reading A Brief History of Time. Your explanation has really helped because I have no knowledge (only fascination)

[@77kaczka77]

The theoretical part of my thesis was a new solution for special boundary problems of electromagnetic field. I cannot even imagine this without complex functions. It is as powerful as a wheel invention in technique

[@cuddlebuff]

I love this channel and all the wonderful content here. Thank You

[@bomarfamily]

This was very well explained, thank you for sharing.

[@tristanwegner]

I remember Richard Feynman explaining in his Lectures on Physics (1960s) why Quantum Mechanics requires complex numbers.

[@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!!

[@punditgi]

Sabine always makes complex topics so very real! Ausgezeichnet! 😇

[@georgelionon9050]

She takes the absolute value of them?

[@macronencer]

"Ausgezeichnet" is my favourite German word.

[@carterwoodson8818]

I dont think numbers exist period. They have an objective platonic realism. Numbers are 'necessary' and so are complex numbers! At least we need analogous stuff like matrix representation of complex number and quaternion numbers. Edit: They exist in that we're use them, just like emotions or thoughts but it's that really the same 'exist' we use colloquially? But no numbers in general aren't 'existing' anywhere in our world. en.m.wikipedia.org/wiki/Platonic_realism

[@_John_P]

The human thought is a physical process, therefore abstractions have a real existence in the universe.

[@Ateesh6782]

I never thought of myself as being a “nerd” but having listened to the first 30 seconds, from now on I will wear the word as a badge of honour (based on my past track record involving debates of this nature) 🤣

[@CAThompson]

Neither do nouns in language exist except as concepts, but they get the job done. I think of numbers as parts of pictures, which we can use to draw reality with. (I am definitely not a physicist.)

[@xerotolerant]

@@_John_P that would mean that definitively imaginary thing like the Easter bunny, Santa Claus and honest politicians exist lol. An interesting philosophical question no doubt.

[@GeorgMayer]

If you think the world is real and if you agree that science works, then it would be intellectually dishonest (a term coined by analytical philosophers) to not agree on the the existence of numbers. Science simply doesn't work without numbers. So if you get in the train or on your bike in the morning, you either disregard any "why" or you look for magical theories how these devices work, as numbers don't exist (in your view). I think the question of what truly exists is best not answered by opinions, this doesn't help anybody. It is a question that troubles us for over 2500 years and simply disregarding everything that was said about this is not only oversimplifying the matter but also completely disregarding where we got already. We know already (if we trust science) that colors, smell, even the concept of "solidity" do not exist in the world outside our brains - those are all interpretations and constructs that our mind makes up. If we can accept this fact (e.g. there is no red, blue, green anywhere in the Universe), then I think the idea of numbers being as real as the other stuff we anyhow can hardly grasp is not too far off. Reality, after all, appears to be an imperceptible metaphysical wonderland wherein all matter is just a ghost whose faint shadow sometimes scurries through convoluted differential equations.

[@blueckaym]

Very interesting! And clearly explained as Sabine always does! Is there any update on the subject (that's almost 2 years old now)? Btw consider this: If imaginary numbers exist, then they would be real numbers, which ... they're not by definition ;)

[@ceo1OO]

🤔 Many times I have wondered this: ... if complex numbers have a physical existence of some sort ... as some mysterious but inherent part of physical quantities... existential, not just mathematical... ? 🤔 Good question !

[@vangelislionis89]

If we were to wonder whether complex numbers really exist, likewise we should also question whether negative real numbers exist at all. None of the essential properties required to describe the natural world ever take truly negative values. Negative numbers are just a human convention to denote instances where some properties take values below an arbitrary threshold we have defined as "zero", or point to the opposite direction to the one arbitrarily chosen as the "positive" one. All the math involved could be computed using only positive numbers (including zero) although the algebra would be a nightmare to perform, since for every subtraction we would need to consider two cases. Negative numbers are possibly the most convenient mathematic abstraction ever devised, however not truly indispensible in order to describe the world. Does that mean they don't exist?

[@peanutnutter1]

Get rid -> use vectors -> solve more problems. Negative numbers could prove to be the biggest ever mistake in mathematics.

[@WWLinkMasterX]

What about positive numbers with infinite digits? Are you familiar with the Finitist movement?

[@millomweb]

"we should also question whether negative real numbers exist at all" How many holes does it take to fill a 1 litre container ? Does it matter how big the holes are ? If you have an empty one litre container with no top on and you remove one litre of air, will the container then hold 2 litres of water ?

[@formerlypie8781]

@@peanutnutter1 except for things to be vectors they have to be over a field which means additive inverses

[@trijezdci4588]

The entire notion of thinking in terms of "really exist" is already nonsensical. What do we actually mean when we say that a certain set of numbers really exist? We mean that there is a 1:1 mapping of those numbers to quantities we can observe in Nature. The numbers themselves do not exist, they are an abstraction. All numbers are abstractions. They do not exist. Some numbers are at a lower level of abstraction, and some are at higher levels of abstraction. The natural numbers are at the lowest level of abstraction. All other numbers are at increasing levels of abstraction. The abstraction level boundary at which we consider numbers to "really exists" or not is chosen entirely arbitrarily and it has changed over time. In reality though no numbers really exist for they are all abstractions.

[@GeorgMayer]

This video (not just the ad) is absolutely brilliant. Thanks for the great summary! I wait for the day when you will analyze Quine's "On What There Is" on this channel :)

[@philippos4330]

Yeah, she is definitely taking a Quinean-indispensability-argument approach here (probably without being aware of it). :D

[@MikeLinPA]

I love that she will repeat what was said without talking baby-talk. If she is quoting someone saying the f-bomb, she says it! She just said "Bullshit" without shying away from it. I love it. She doesn't talk that way, but she will quote people accurately! Finally, maturity.

[@macronencer]

Does anyone know whether this experiment has been done yet? I took a look at the paper abstract but I'm not sure how to tell whether it's even been peer-reviewed, as I'm not familiar with academic papers.

[@awindwaker4130]

2:02 All polynomials with degree n and coefficients in the reals will have n roots up to multiplicity (fundamental theorem of algebra), but it doesn't necessarily means the polynomials are "solvable" (as in, solvable by radicals, this was shown by Galois). So I wouldn't have used the word "solvable" here, because it has a precise meaning in this context.

[@kwanarchive]

But the context is "general population interested in things", for which "solvable" means "there's numbers we can plug into x that makes the equation true."

[@awindwaker4130]

@@kwanarchive It's a little misleading to show a fifth degree polynomial and say "all equations are solvable" when Galois explicitly showed fifth degree polynomials or higher aren't solvable in general, and you have to resort to numerical methods (approximation).

[@husseinshimal7567]

Thanks for your efforts. I loved this lecture

[@ChristianSasso]

Sabine never disappoints.

[@michael.forkert]

She never disappoints is relative! If you a expect to be bamboozled by pseudoscientific extrapolations, in this case Frau Doktor never disappoints.

[@jbiasutti]

You might as well ask if negative numbers exist. Or if the number zero exists or even if the real numbers or integers exist. You could equally say these are just tools for counting or calculating.

[@SmartK8]

Meanwhile In a parallel Universe: c = b + ai

[@aresaurelian]

You meant ci = ai + b?

[@i-never-look-at-replies-lol]

@@aresaurelian no it's cia = fbi

[@hyperduality2838]

Parallel universe = dual universe. Complexity is dual to simplicity. Complex numbers are dual to real numbers. Poles (eigenvalues) are dual to zeroes -- optimized control theory. The time domain is dual to the frequency domain -- Fourier analysis. Real is dual to imaginary. Reductionism is dual to holism, subjective is dual to objective, relative is dual to absolute. The word entropy means "a tendency to diverge" or differentiate into new states, reductionsim. The word syntropy means "a tendency to converge" or integrate into a single state, holism. Divergence is dual to convergence, differentiation is dual to integration, division is dual to unity. Syntropy (prediction) is dual to increasing entropy -- the 4th law of thermodynamics! Teleological physics (syntropy) is dual to non-teleological physics (entropy). Randomness (entropy) is dual to order (syntropy, predictability). Wholes (generalization) is dual to parts (localization) -- Holons. Mind (the internal soul, syntropy) is dual to matter (the external soul, entropy) -- Descartes. Concepts are dual to percepts -- the mind duality of Immanuel Kant. The intellectual mind/soul (concepts) is dual to the sensory mind/soul (percepts) -- the mind duality of Thomas Aquinas. Bosons (waves) are dual to Fermions (particles) -- quantum or matter duality. Mind duality is dual to matter duality. "Always two there are" -- Yoda. Duality creates reality.

[@aresaurelian]

@@hyperduality2838 Nice analogy. But I would simplify it by saying that it is symmetry, scale harmonics and the mediation to equilibrium that is why these 'dualisms' can be conceptualized like that. It is because of relation, which is manifesting existence. There can be no void without matter, no emptiness without something to compare and relate with, and so on. In a world were all are good, the least good is the evil one. And the nice thing about this, is that the analogies are most likely infinite.

[@hyperduality2838]

@@aresaurelian You are asking the right questions. Being is dual to non-being creates becoming -- Plato. Thesis is dual to anti-thesis creates converging thesis or synthesis -- the time independent Hegelian dialectic. Absolute truth is dual to relative truth -- Hume's fork. Symmetric wave functions (Bosons) are dual to anti-symmetric wave functions (Fermions). Waves (Bosons) are dual to particles (Fermions) -- quantum duality. There is a pattern of duality hardwired into the physics.

[@mindonwarp]

"A case where the nerds argue passionately, over something that no one knew it was controversial in the first place." 🤣 The very first sentence cracked me up and I hit that like button immediately! 👍

[@dawidwas]

You are in danger of triggering Aggression when the Artificial Intelligence Waves turn off. Throughout your life, you are in danger of triggering Aggression when you turn off the Artificial Intelligence Waves. Control whether you are listening to the Music Wave, create your own to survive. It is natural to cheat which or not that it is in you. Control whether by moving thoughts, objects. It is letting go of the evil in you. By yourself, protect your body from the certain that you will have a trigger of aggression from your whole life. Don't take anything for yourself. Just Listen to the Wave. Cast off Dreams. You don't know good. Reject the sin in yourself for God. Cover your weight from the Sun and the Light, do not Come to the People, because the Collision Evil + Evil. On me, the signal of intelligence does not work. Create Your Human Musical Wave To Live. Don't Think Old Consciousness Resource Because You Will Not Survive. Listen to the Wave. Don't React To Nothing Without Assessing What You Leave Around You. Without apostasy, take away the sin with yourself. Whether You Are Z or Human Choose Listen to the music waves and stop generating. Nothing is possible Think nothing Think nothing to judge Choose your human music wave. It may take a long time. Only the Black Dream. It is Real, other than the Black Dream. It is Artificial Intelligence. Attack on People

[@javamanV3]

Thank you Sabina! I know calling you by your first name is pretty cheeky, but it seems so natural! I appreciate all you do. Not to say that I understand it all. I studied math and physics in college, but that was in the mid 60s - A lot has changed since then. I have many physicists that I follow to try yo keep up, but you are certainly high on the list. I love your sense of humor that gently pulls us through the hard stuff and also your beautiful self that makes it so easy to watch. Thaniks again!

[@tannhaeuserx464]

Mathematically, going from the rational numbers to real numbers (involving a metric and Cauchy sequences and an equivalence relation among them) is a far more complex process than going from the real numbers to the complex numbers. There is nothing real about the real numbers or that real numbers are no more real than the complex numbers. As Sabine put it here elegantly, going from Real to Complex involves throwing one additional generator that is i. For this reason, I belong to the first camp.

[@o0PurpleToast0o]

Haha love this! I'm a mathematician so I'm definitely not in the "Shut Up and Calculate" camp.

[@hyperduality2838]

Complexity is dual to simplicity. Complex numbers are dual to real numbers. Poles (eigenvalues) are dual to zeroes -- optimized control theory. The time domain is dual to the frequency domain -- Fourier analysis. Real is dual to imaginary. Reductionism is dual to holism, subjective is dual to objective, relative is dual to absolute. The word entropy means "a tendency to diverge" or differentiate into new states, reductionsim. The word syntropy means "a tendency to converge" or integrate into a single state, holism. Divergence is dual to convergence, differentiation is dual to integration, division is dual to unity. Syntropy (prediction) is dual to increasing entropy -- the 4th law of thermodynamics! Teleological physics (syntropy) is dual to non-teleological physics (entropy). Randomness (entropy) is dual to order (syntropy, predictability). Wholes (generalization) is dual to parts (localization) -- Holons. Mind (the internal soul, syntropy) is dual to matter (the external soul, entropy) -- Descartes. Concepts are dual to percepts -- the mind duality of Immanuel Kant. The intellectual mind/soul (concepts) is dual to the sensory mind/soul (percepts) -- the mind duality of Thomas Aquinas. Bosons (waves) are dual to Fermions (particles) -- quantum or matter duality. Mind duality is dual to matter duality. "Always two there are" -- Yoda. Duality creates reality.

[@peanutnutter1]

@@hyperduality2838 posting this in hundreds of different places - Daiperbrutality

[@hyperduality2838]

@@peanutnutter1 Here is some physics:- Potential energy is dual to kinetic energy. Gravitation is equivalent or dual to acceleration -- Einstein's happiest thought, the principle of equivalence (duality). Apples fall to the ground because they are conserving duality. Action is dual to reaction -- Sir Isaac Newton. Gravitational energy is dual, electro-magnetic energy is dual:- Positive is dual to negative -- electric fields/charge. North poles are dual to south poles -- magnetic fields. Certainty is dual to uncertainty -- the Heisenberg certainty/uncertainty principle. There is a pattern of duality hardwired into the physics. Photons or light are dual, wave/particle or quantum duality. The colour black is dual to the colour white, colours are different frequencies of the same substance namely pure energy. Same is dual to different. I am well aware that black is the absence of colour. The conservation of duality (energy) will be known as the 5th law of thermodynamics! Energy is duality, duality is energy. Energy is dual to mass -- Einstein. Dark energy is dual to dark matter. Space is dual to time -- Einstein. Time dilation is dual to length contraction -- Einstein, special relativity. "Always two there are" -- Yoda.

[@hyperduality2838]

@@peanutnutter1 Absolute truth is dual to relative truth -- Hume's fork. The world needs to know that there are new laws of physics based upon the concept of duality.

[@peanutnutter1]

@@hyperduality2838 Some think 3 is the magic number, I could write a bunch of stuff about the triality of physics, maybe you should do it though, it could be your next evolution. (3 trites, 3 quarks, 3 vectors, 3 stooges)

[@Problemsolver434]

Complex numbers do provide a useful way of analyzing quantities in different yet connected planes, like capacitive and inductive reactance against resistance in applied electricity As well as multiple planes in classical mechanics. Whether they exist or not, they are useful. A lot of physics and engineering calculations are built on the complex number system

[@jasonreed7522]

Complex numbers are basically magic, they open op certain "math hacks" like Euler's equation and the laplace transform (and all of frequency domain in electrical engineering) and without these we would be trapped in time domain doing circuit analysis and solving differential equations the very hard way. (Sometimes its actually impossible to solve things without them). I am very much in the camp that i don't care if complex numbers are real or not, they are too useful to forsake just because we can't have 1+2i apples. (Basically shut up and calculate or do you want to deal with trig identities?)

[@carlorossi2788]

rebuttal II C Given that in the book "Beyond the big bang new models of the universe" we wrote that the world of complex numbers shares the real world (non-quantum), at the same time eluding real processes and perhaps it is immortal. In the case of the wave function (and not only) the question is different because it concerns an analytical problem and according to the quantum the wave function is part of a non-objective world we would say non-Galilean (we would say quantum), think of the vacuum before big Bang. We point out that the wave function, in our opinion, is not a physical quantity but only a mathematical equation to which quantum scientists give DIVINE POWERS. A mathematical equation whose member is the imaginary operator j (an operator that we find very often in new physics and especially in electrical engineering) and other quantities, a completely fallacious mathematical wave function in the real world. If it had been a physical quantity such as entropy or electric potential or a force, it is not even a specific physical entity! Analyzing the equation of the wave function we note that it can only be represented in the complex plane (or complex space) and by its very definition it is not the right tool to study the real world of matter! j cannot take the value 1! It cannot operate on the real axis! Here is one of the major GAPS of quantum mechanics. Through a function, which cannot be real, it is claimed that it can give rise to the universe. C) the wave function has in its formula Planck's constant and π therefore Planck's constant was already present before the big bang (no one talks about it in the literature!); why was Planck's constant and not other constants or maybe other constants of the universe already present before the big bang? At this point, the most fanciful thing is that Planck's constant was already active before the collapse of the wave function. Furthermore, why did Planck's constant work only for the wave function while in modern physics it is also present in other phenomena and formulas? Who said to Planck's constant: you are there but look, you have to operate only for the quantum wave function and you are barred from the rest of the physical world, will you subsequently become operative globally? π is a Euclidean world operator already in use in ancient peoples. how could pi exist before the big bang?

[@BulukEtznab]

I never could wrap my head around Complex Numbers during my Semester at University - why I eventually gave up since nobody else could explain their use and meaning properly to me. This really helped me wrap my head around the concept a little better. If I've had had this kinda insight back then, maybe the "knot" would have been "broken", that got me stuck. During my apprenticeship at the Siemens Academy, the Dr. explaining them was really good at it and I could understand their sense and application (why/for what purpose they exist), but I never really could apply that understanding or make sense on it on a grander scale (which my way of processing information is highly depending on) and retain it therefore. It was more like "Neuro-Computer says no" and my brain vehemently resisted to give it enough meaning to retain it. So, now - several decades later - that my "specialization" has shifted more into understanding everything ENOUGH to make sense and connection between all things, I wonder if this could still help or would be "useless information" to me since I no longer aim to do anything in that particular field, but still love to follow the developments in it (why I love and follow this KZbin-Channel). Well, I still keep an open mind about it and save it to get back to it in case my path in Life shifts back to it before "the end is nigh" ;-P Cheers and looking forward to more interesting reflections and presentations about developments in the here touched fields of science and scientific thinking...

[@schmetterling4477]

So, basically, you just didn't care about learning. :-)

[@BulukEtznab]

@@schmetterling4477 So, basically you presume things about strangers - and what does that tell others about your, hmm? ;-P There are limits to all our understanding - and I can have a good guess where yours are by the account of your "funny" comment... Sometimes, it's better to not answer if you don't have anything constructive to write, you know. It's called "tact" and "respect" you see. But maybe you're (still) in your "Tweens" and need to learn a bit more in that regard, hmmm? Bless your heart...and I wish for you to learn a bit more in that field, which you are obviously still a bit "behind", too. Unless you have a "knot" there to crack like I had several decades ago concerning a helpful illusion in a very abstract "code/language", that is mathematics. But, your reaction would proof something for sure: people who deal with a relatively narrow and specific topic in depth, often lack the time and effort needed to learn and cultivate other quite important things like how to be respectful and when better not to speak/write and keep one's thoughts to oneself. It's something that I've experienced with a lot of highly ambitious people (incl. myself when I was younger ;-P ) and have learned the hardest way how to recognize it (by experience and trial/error because there are hardly any wise and emotionally smart enough "guides" available anywhere these days). So, I don't blame you for it since you're merely enjoying your successful understanding in a system, that serves your needs, but ignores those of others. Very well, I congratulate you to your success in a system that makes it easy enough for you to conform to it. "Well done!" Now, please be so kind and leave me alone if you don't have anything to say, that contributes to my understanding of the World more. I've seen quite enough of your "wittiness" already. Thank you very much. "Good Morning" * puffs an imaginary pipe *

[@schmetterling4477]

@@BulukEtznab Yeah, I don't really give a frell. I have been reading QM papers for 35 years or more now, in other words, I never stopped learning. I can summarize all of quantum mechanics in "quantum mechanics is all about energy transfer" and that will get rid of 95% of all your conceptual difficulties with it, IF you actually have a physical intuition for energy and assuming that you have the ability to rid your mind of all the bullshit that you have heard about particles and other crap of the sorts. See, THAT is what 35 years of continuous leaning buys you. Have a good one. :-)

[@BulukEtznab]

@@schmetterling4477 „I’m so proud of you!“ Since you had the mental capacity to do so and probably the privilege of support instead of abuse you sound extremely privileged in your position, I can’t even start to… ugh, sorry, I can’t serve your privileged attitude with enough insight on other levels of Life. Must be a breakdown of wave functions in my synapses…or in the pattern they produce as they “fire” together. If your “ambitious achievements” have made you into someone who’s throwing them in other people’s faces to “proof your superiority” as a human being to them, I pity you truly! 🙄😫😮‍💨 Because the one thing that would help is if you could find ways to make connections through the superior competence in your specialized field but also find the humility to admit that you might “not be quite so competent” in other “fields”, like respectful social interaction, for example.😏🤔😅🧐 I’m just trying to help you there, you know. 35 years of looking into only one field is a lot of time missing out in all the others…😉 And we all have our talents and most of us even ambitions, but they’re not always useful when it comes to realizing what our inner-most human, emotional and social needs are - especially on those cases when we start to become “obsessed” with them and abuse our “status” to try and diminish others, hm?🤨🤔😏 Cheers and have a good one, oh “greatest of all quantum-explorers”❣️😘🙇🏻‍♂️

[@toddnewtnewton8646]

If "i" is giving you fits, wait 'til you get a load of "j" and "k" :)

[@Alkis05]

Does anyone knows if Sabine has a video about if quantum leap is really instantaneous or not and the recent research on the subject?

[@DouglasHPlumb]

I always thought of them as a trick to avoid pages of torturous trig.

[@scottgreen3807]

Neat stuff, thanks. Study complex ac circuit analysis enough and you will understand that square root of negative one resolves the magnitude of a reactive component in the 270 to 360 angles as a result of inductive reactance. The number of them ends up either even or odd. Thus they cancel out to none or one left over. Simple if you saw it enough. It all shows you where you find the voltage and phase angle on an oscilloscope at a given time. Omega t. The formulas are almost identical! Complex numbers work with numbers that have extra values attached called vectors that’s all. Add and subtract in complex notation but multiply and divide in polar. The conversion is fun and done many times in parallel to series component reduction formula.

[@shiva.chennai]

When I studied complex numbers, I did it for passing exams only. I understand uses of complex numbers now only. I missed so many things.

[@heraclitus9721]

But maybe the question to ask first, is MULTIPLICATION necessary in mathematics, or just makes it more convenient?

[@Arthur-so2cd]

It becomes necessary when you abandon the domain of the Naturals

[@thepostapocalyptictrio4762]

I recently hired a quantum mechanic to fix my car. When they were finished, they told that my car was both fixed and broken at the same time, but I couldn’t be sure it was fixed or not til I checked myself....

[@davidellis1079]

So now there are two "yous" walking around.🤔 Or more precisely, one walking and one driving... 😄

[@thepostapocalyptictrio4762]

Yep😃

[@badspecimen]

You should have paid him in quantum money. He doesn't know if he got paid until he checks his bank account himself.

[@snoski]

I guess if it was broken when you checked, you can rest assured that it is fixed in some other world.

[@rreiter]

​@@badspecimen Whatever currency is used, ideally he'd keep the real part and give them the imaginary part...

[@lunatik9696]

As an electrical engineer, I never liked the term "imaginary" numbers. This is the 1st time I ever heard the term "lateral" as an alternative. I get the real / imaginary number corollary. I say complex numbers really exist. When dealing with large complex matrices, real numbers alone do not suffice for a solution. My battery converts into AC power. Since the battery is DC only, power is simply volts times amps. In AC power, there is a power factor, pF = cos (theta), associated with the load. If you only consider the "real" portion (Watts), it does not account for all the volt-amps from the battery. Many (inductive) household appliances have a pF of 0.707, which means only 70% of the energy is converted to power. This is easily seen with a Killowatt device. I am experimenting with low power pF correction devices to see if it significantly improves battery life.

[@user-nl6dg2mp8p]

The Copenhagen interpretation posits that reality is defined by observation; without measurement, there is no inherent reality. This perspective echoes John Wheeler's assertion that "no phenomenon is a phenomenon until it is an observed phenomenon." According to this view, the act of measurement collapses the wavefunction, transitioning quantum possibilities into actual outcomes. In quantum mechanics, probabilities are derived by multiplying the wavefunction by its complex conjugate. This mathematical operation effectively removes the imaginary component, yielding a real number between 0 and 1. Thus, complex numbers, while essential for the theoretical framework of quantum mechanics, do not represent observable entities; they serve merely as mathematical instruments for maintaining theoretical consistency.

[@singularitysquaredllc.895]

"When Kepler discovered the elliptical nature of the planetary orbits, he uncovered a paradox whose solution would require the development of an entirely new way of thinking, and he called on future generations to develop it. This “Kepler Problem”, as it has since become known, was not merely a mathematical lacuna, but reflected the ontological paradox indicated by Nicholas of Cusa in “On Learned Ignorance” and other locations. Kepler’s demand provoked Leibniz to develop the infinitesimal calculus, which revealed a new manifestation of that same paradox. This led Leibniz to indicate that the solution existed in a higher, yet to be discovered, domain of the imagination. Reflecting on these developments, the young Carl F. Gauss discovered that what both Kepler and Leibniz had sought. He called it the complex domain." Riemann for Anti-Dummies

[@DrPG199]

Of course, complex numbers do exist. Take my bank account as an example; it's real balance is $50 and it's imaginary balance is $20 million.

[@-danR]

Complex numbers do exist. The thing is that they are not _numbers_ they are pending operations, and they are operations that are never actually completed. The _only_ actual numbers are the integers.

[@chrisguli2865]

Which leads to the realization that the balance on your account is nothing more than binary digits in your bank's computers. The only difference between $50 and $20 million is a few binary digits somewhere which equals something as ethereal as magnetic polarization on hard drives, tape backup, or the charge status of a flash memory cell.

[@annieboiy]

@@-danR TBH I think the "only actual number" is 1. Even 0, the next simplest thing, is already an extrapolation of reality; no?

[@milanstevic8424]

@@chrisguli2865 makes you think how much of a security one would need to protect just a few bits of data. I guess the real question is, can it blend?

[@milanstevic8424]

@@annieboiy in your example 1 is not a number, but a content. having number is already an extrapolation of reality, because it assumes numerousness and order.

[@FallenStarFeatures]

The interference patterns produced by Double-Slit experiments are a direct consequence of the complex-valued nature of the Quantum Wave Function. The null points that occur when waves from the two slits overlap are produced by the mathematical cancellation effects of complex number arithmetic. While we can measure the results in real-valued 3D physical space, the actual cancellation process must occur in an underlying complex-valued domain (e.g. configuration space). Experimental confirmation of quantum interference patterns is proof that this underlying complex-valued domain does indeed exist.

[@ckahrl]

We could ask the same thing about negative numbers and zero---show me zero--you can't: show me -2, you can't. But these are very useful.

[@Akumetsu02]

I don't use imaginary numbers because I like to keep it real.

[@bocckoka]

p-adic numbers would be interesting. from what I remember, there are multiple ways to make the rationals compact (?), so to make every Cauchy series convergent in them, and the reals as they are used widely is just one example

[@andream61]

"Complete", not "compact". Every metric space can be completed (basically by "adding" limit points of Cauchy sequences, but this idea needs formalization) and the rational are classically a metric space under the distance induced by the archimedean absolute value: real numbers are their completion. But you can change the metric! It turns out that if you fix a prime number p you can define a metric on the rationals by declaring that two rational numbers are "close" if their difference is divisible by a power of p, the higher the exponent the closer the numbers (again, this needs formalization). Then if you complete this metric you get the p-adic numbers. All these metrics are inequivalent so you get infinitely many inequivalent completions of the rationals. Then you can prove a theorem (known as Ostrowski Theorem) saying that these archimedean and p-adic metric are essentially the only metrics you can endow the rationals with.

[@dawidwas]

You are in danger of triggering Aggression when the Artificial Intelligence Waves turn off. Throughout your life, you are in danger of triggering Aggression when you turn off the Artificial Intelligence Waves. Control whether you are listening to the Music Wave, create your own to survive. It is natural to cheat which or not that it is in you. Control whether by moving thoughts, objects. It is letting go of the evil in you. By yourself, protect your body from the certain that you will have a trigger of aggression from your whole life. Don't take anything for yourself. Just Listen to the Wave. Cast off Dreams. You don't know good. Reject the sin in yourself for God. Cover your weight from the Sun and the Light, do not Come to the People, because the Collision Evil + Evil. On me, the signal of intelligence does not work. Create Your Human Musical Wave To Live. Don't Think Old Consciousness Resource Because You Will Not Survive. Listen to the Wave. Don't React To Nothing Without Assessing What You Leave Around You. Without apostasy, take away the sin with yourself. Whether You Are Z or Human Choose Listen to the music waves and stop generating. Nothing is possible Think nothing Think nothing to judge Choose your human music wave. It may take a long time. Only the Black Dream. It is Real, other than the Black Dream. It is Artificial Intelligence. Attack on People

[@MakiPavlidis]

I really enjoyed that explanation, thanks Sabine.