This video makes me want to do math. It’s inspiring in the best way

emotional damage!

@Interstellar d a sad fff

I’m a physics major. I’ve always had trouble understanding complex numbers and why they exist in equations. It’s like my professors were just handing out the equations like the Schrodinger equation without really explaining what they mean. As I went on throughout college I gathered an understanding, but this video gave me that “aha!” moment. Thank you Veritasium, Your videos are something special and I appreciate every single one that gives me more insight on how the universe works.

@Haine Sensei What concepts are you refering to as "useful for Physic". If it was the two other types of complex numbers, OH yes it does! Not only for Physic but Mechanics, CAO, computer graphics, gamers, robotics, etc, Geometry, Analysis, and in fact all mathematics. It's a revolution started by Grassman and Hamilton, and boosted and unified by Clifford and David Hestenes in what generalises to n dimensions what the three types of complex numbers and the nilpotent gives in 2D. It's called Geometric Algebra. It corrects in 2D and 3D the defects of Euclidian Geometry which is quite clumsy to use, in any geometric kinematic, Dynamic way. It upgrade Geometry to full invariant form, out of the heavy mud of coordinates and réferentials. It unifies linear and angular momenta in a single geometric object that is invariant and independant of réference torsor point. And it led to a Theory of Gravitation in flat metric, centered in gauge symmetry. The nilpotent gives an alternative way to build Calculs without réference to"Real numbers", "limits", "infinite processes". And is a huge step up in rigor, bringing back Calculus to Lagrange discovery, that it was indeed a translation operator giving not only thé first differential, but all at once, without any "limit" arguments. It gives furthermore a very concrete, rigorous and efficient definition to the old concept of "infinitesimals" by the cutoff that it provides. Hyperbolic complex numbers are as important, if not more, than circular usual complex numbers. They are naturally relativistics with null vectors on the "Light cone ". On rigor stand point neither Cauchy sequences not Dedekind cuts are indeed rigorous. They are abnormaly complicated exotic constructions, pretending to solve a problem they do not actually solve. Not talking of their pratical uselessness. They are flod with hands shaking and give birth to Monsters that have little to do with numbers. Not talking of total absence of arithmetic on such Monsters. This is not Mathematics but wild Safari in wonderland. And I wonder where did you found any base to believe that ingeneers undestand "Real numbers"? They don't! They never saw one. Never use one. Never compute with such chimeric objects. Thé only thing they are dealing with are decimal numbers, which are in fact integers ! And I would't dismiss the wisdom of thé Grecs, full of deepness and rigor, as you seem to do. They did NOT believe in "infinite", for potentialy good reasons..They did not give status of numbers to irrational magnitudes, for potentialy good reasons..And they in purpose avoided to use "lenghts" for their problematic behaviors in irrationality. Prefering to stay on area and quadrance ground. And Dirac confirms their wisdom by overcoming the problematic square root, by finding an algebra where such pitfall Can bé avoided. Dirac algebra that led to the discovery of spinors and antimatter. So yes square root ARE problematic, and whenever Analysis failure can be repaired by algebraic upgrades, it's a huge progress. So in all those examples, one can easely see the general pattern. When Analysis waves hands with "limits", "square roots", "irrationals", etc it indicate that there is a deeper algebraic structure understanding to discover that brings back rigor, clarity and efficiency. That's exactly why Groethendick built Category theory, to overcome many problems of set theory, implicit function theorem, functional Analysis etc. Finaly, several Times you are making confusions between mathematic theorical rigor and ingeneers practical point of view. Ingeneers don't give a dam about Real numbers or theoretical existence of irrationals. But checking if those concepts actually make sens IS hugely important for pure Mathematics. And there is many problems. Even "infinity" is a fragile Dogma. And with his Achilles heal lots of area of Mathematics are in danger. It's an old 17th century housse that will fallu apart soon or later..Concepts of infinity, sets, proofs, decidability, have been deeply attacked since the XX th century..In the profond complicity between theoretical physics and pure Mathematics lots of old temples are being questionned and New breakthroughs are springing all over. Some fishy dogmas will fall soon. Some New flowers are raising. There is much to say and see in polynômial rings. But the Terra incognita is mainly on multivariable polynômials that is extremely vast, rich and complicate. Complex Analysis developped by Riemann gave a first taste.. Meromorph theory opens an Continent...

@Arthur if you think any of that information is useful for a modern physicist, then sure, the points are valid. I did not in any way wish to say one number system is better than another. As someone who likes rigour, I actually quite like dedekind cuts as a construction of the real numbers from the rationals. That in no way says that they are “numbers” in the senses of ℕ, ℤ, ℚ, but it does point out that they do exist as a concept. And it’s probably more of a metric/topological concept than an arithmetic concept, since the idea relies on looking at vast regions of the numbers instead of just looking at those more elementary number systems we form up to and including the rational numbers. Yes, even the complex numbers are more elementary to form (given the real numbers) than the reals. But none of these things really matter in terms of how they might be thought of. As it turns out, to most engineers, the concept of the real numbers is quite well understood: anything that feels like it should converge to a number, does indeed converge to a number. This is a very important concept for many different fields and why the construction of the real numbers is helpful (if not pleasant). I’m not measuring the benefits of the number systems by how elementary the concepts with which we form them are, I’m measuring how useful the systems are to the modern sciences. Looking back to your point on how the Greeks did things. Saying that we should start with areas to you may seem like more of an elementary concept. But who defines what is elementary? It’s completely arbitrary really. All concepts can be useful in some ways, some more than others. It doesn’t matter if a concept is complicated and difficult to set up, if it’s very useful, it’s still a very important system and one that should be thought about.

@Haine Sensei No, you are missing two points. First integers, writen often in decimal base, with positioning Hindou method, IS a polynômial in base 10. And writen in base X, they are polynômials. So your distinction is superficial and essentialy illicite. Secondly negative integers have been seen for millenium as imaginary numbers. In such way that Al Khawarismi and followers persian mathematicians, exposed 6 types of quadrics and some 28 types of cubics, juste to avoid negative coefficients that were banned from "numbers" category. So your exclusion of complex numbers is similar : naïve, superficial and abusive. Why not excluding also, After the negative integers, all the fractions, since no "fraction" existe in the "Real World". And Indeed things are what they are, in their unique integrity. Only humans invent such imaginary concept as fractions. And keep going by excluding "Real numbers" since nobody has ever seen the full face of √2 or even π in their decimal chimeric "existence". And Indeed nothing is more abstract, hypothetic, dithyrambic, artificial, imaginary than the so called "construction of Real numbers", which you can hardly find nowhere since Cauchy sequences or Dedekind cuts are problematic monstruous questionable "proofs". And Indeed Kronecker famous maxime "Good created integers, all the rest is human hand", drives the red Line there. So on arithmetic structure point of view, negative integers and rationals are as well defined as "natural numbers", polynômials and complexe numbers. Which is NOT the case for "irrationals", since nobody knows in fact how to compute even the simple sum √2+π. Showing that there is in fact no arithmetic on irrationals numbers, if even they exist. They didn't in the eyes of the Grecs. Euclide restrics carefully the category of so called numbers, to rationals, that can thus be measured as a multiple of a chosen unity, or fractions of it. And they carefully labeled geometric entities like "π", as MAGNITUDES, in an analogic sens, not an arithmetic one. So they did NOT wrote, nor say, nor believe, that "π" was the ratio of the perimeter to its circle diameter, since on the contrary it was precisely the so to speack "measure" of their incommensurability, i.e. of their impossible common measure. On the contrary, for instance the pure imaginary number i, is hugely more simple, pragmatic, rigourous to define, thus "Real", than any chimeric irrational whith supposedly infinite decimal unknown sequence! And indeed i can be perfectly represented by an integer 2×2 ultra simple Matrix : [0 1 / -1 0]. What can be more simple and real than that, exept integers themself?! You are thus arbitrarily narrowing your definition of "numbers" to a heavily artificial and illicite jail. Measuring lenghts and turns in a plane, are just two partial and incomplete measures of a common one, named "complex numbers". Setting lenghts as "more fundamental", is not only artificial and arbitrary, it's in fact more abstract and questionable than starting whith areas. The Grecs didn't see Pythagore theorem as an indentity between squares of lenghts, but between square areas. And Lebesgue integral Theory followed this old deep Idea. So "lenght ideology" is in fact what is behind your arbitrary restrictive and segregative choice. But rotations or areas are as natural and real as "lenghts", if not more!... Usual complexe numbers are just THE ARITHMETIC OF SCALING AND ROTATIONS. Absolutely central, pragmatic, useful and essential. They are in all ways "NUMBERS". They match all the criteria. Finaly, there is no interest to work with a more complicate complex number than i, as a base. Since what makes complex numbers easy to use, is precisely the fact that everything works as usual, with the simplistic addition of using i^2+1=0. What is interesting on the contrary is to work with not only usual complexe numbers, but two other types, hyperbolic, squaring to +1 and also to the nilpotent €^2=0

Kelsey Oakes's Aunt stopped living (LMAO 😂) because I upload bangers! ..,....,.

History of mathematics should be taught as early as in middle school, and this video tells exactly the reason why it would immensely help students appreciate what they are taught.

I literally figured out how to complete the square by watching this video, I couldn’t do it all of algebra one and now I can

The problem with history is that even if you'd devote the entire school curriculum to it, you still wouldn't be able to teach students everything that is important. There is just so much history behind everything. And sadly, given the limited time available for a student to learn anything in school, teaching them the history of mathematics would take away from actually teaching them mathematics. Not to mention the sad fact that many teens just don't appreciate history (or mathematics) in the first place.

Exactly!! Instead we are taught about kings who killed their own brothers for the throne🙄

🙄 just because you want to doesn’t mean everyone has to

Kelsey Oakes's Aunt stopped living (LMAO 😂) because I upload bangers! ..,...,,

"Only by abandoning math’s connection to reality could we discover reality’s true nature." I cannot shake these words from my head.

i'm just confused.

It's powerful.

sounds kind of ironic

@legitjimmyjaylight What you are saying is essentially math is important in physics in overall.

That story about Ferro, Fior, Tartaglia, and Cardano could be a movie.

Albeit one that is shown during a Mensa weekend get together.

Exactly, and I can make it work sa an action movies even

@Anurag Gupta well, you can check out about Galois yourself for proof. By the way, Galois was French.

@Indian Boy That's interesting. But I think that person who you are referring to as someone who did (atleast some part) of the proof earlier is actually the Italian mathematician Ruffini. I would like some evidence to back up your claim.

@Anurag Gupta Abel also did it independently, but Galois did it a bit earlier and at a younger age. The only reason many people know about Abel and not Galois is because after Galois sent his work to his friend and died,his works were suppressed by other mathematicians(because they themselves did not understand it). Also Galois's work was much more detailed and generalized.

Dude makes math sound absolutely riveting... Incredible

Always has been

it is riveting, to those that can grasp it. the only way i can describe it those who innately understand maths, its like using a microscope to peer deeper into the things u normally see, to see what is there, but not cannot see easily.

@Tom Rhodes The term imaginary numbers is in fact a misnomer. Complex numbers are just a number system similar to reals.a

Riveting? Math just gives me a headache. And yet, I've found that which math and quantum physics look for but will never find. Because, the logical conclusion is this: If it requires imaginary numbers to describe something that we call "reality," that which we call "reality" is imaginary, and is NOT absolute Reality. And this is indeed a fact. See "The Book of GOD" at A Course in Truth.

I was bored and checked out by the time he got to the point.

I always feel really stupid when it comes to math, but knowing that it took mathematicians 4,000 years to realize 1 - 2 = -1 makes me feel a lot better about it :)

but you were told about it , you did not invent that theory yourself

@Titan Games 2 weeks is *not* extremely necro. 2 years is.

@Constan-Anaconda Oh, that's why!

Of course. Anyone could solve that.

And that paved all of the way of us becoming a galactic civilization

Can't help but think of these scientists, matheticians, and so on as heroes - in their own way. The human mind is bloody awesome.

It’s funny that how easy he makes us understand these difficult concepts.

@imPERFEKT It's a typical situation where students all nodding their heads while the expert indeed cinvincingly explains an advanced subject in a fun intuitive way. But give that same audience pen and paper afterwards and they wouldn't know where to begin. I see this everywhere like people claiming they now understand the intricacies of string theory after a 15 minute YouTube video. Must be the times we live in..

@Blokin Good point. I think these good-faith efforts to explain complex processes often produce a false sense of understanding. People who think they understand need to test their actual knowledge with some technique such as explaining, in depth, the same concepts to someone else (demonstrate "explanatory depth") Also, just use the knowledge to solve problems themselves. BTW, even though I made good grades in math, this still left me feeling disappointed in my abilities. I didn't feel that "aha" sensation that I got when I realized how calculus worked. Math is maddeningly beautiful.

@Blokin get real, you never know what's happen inside other people head.

Get real you don't understand this at all, nice cid though.

@Danny Kanerva agree, it's like a miracle to have technologies like youtube so u can watch this type of content from amazing people like him.

I am about to complete a BA in Mathematics but honestly burned out with studying maths. However, this video reminded me why I am doing what I'm doing and helped me see the beauty of solving problems, even in abstract settings. Thank you for the inspirational and enlightening video, Veritasium. Also, I never knew realized that math history was so incredible!

Great job making this extremely interesting! Math and Science is so interesting when explained well.

Yep.

Yeah.... Maths and science are interesting. But in my school it's not interesting because they are teaching how to do numerical question.... But not the conceptual basics 🙂

Now rewrite that sentence without the word “interesting”.

This is the first time I've seen an explanation of imaginary numbers that made sense to my brain. The first time I ever had someone say that mathematics was disconnected from reality, and how it moved away from geometry. This just flipped something in my brain. I wish I had this 20 years ago when I was dealing with calculus!

More like, it's not math that's disconnected from reality, but human beings. The "reality" we construct in our heads is more real to us than actual reality.

Although I am not a math person the fact that numbers, waves, coordinate systems can describe reality is just magnificent. The geometrical solutions are also wonderful. This is how you should teach match to children. It is insanely interesting.

I’ve had poor math skills since the 4th grade, so often attributed to a lack of interest or attention… but here I am watching and rewatching your videos. It’s about how the material is presented and you are without a doubt, the best.

So just for perspective, I got my degree in Art. I've always struggled with math since I guess I'm more of a visual thinker. However, I absolutely love history, I'm just attracted to the human condition and to me history is the ultimate story! Anyway, I just wanted to say I absolutely appreciate your content. The way you visually communicate math and tie it into stories is amazing and I'm so thankful for all the information you provide. I found the history behind imaginary numbers to be so enthralling! Keep up the great work and I look forward to your videos :)

You should watch this channel's video on visual learning

I wholeheartedly believe that giving context to the history and slowly guiding students through the mindset of mathematicians is objectively better than spoon-feeding them equations.

I agree with you completely but I think the issue is in how to fit this type of learning into a regular public school curriculum so that by the time kids are 18 they're able to succeed and Keep up in University. It's troubling because in the renaissance this type of learning was basically exclusive to folks with the privilege of private tutors able to learn at their pace, or folks like Tartaglia who were "Naturally Genius/exceptional ". If you taught math like this from K-12 an hour a day 180 days a year probably wouldn't be enough time so that most are able to graduate school and attend University by the time they're 18

Bet on it. It was supposed to be like that.

Yes

So true. This was interesting as hell, putting 2 and 2 together to find out what it is you know, and more importantly WHY!!!

Well said.

This is the most amazing video I have ever seen in my life. The 23 minutes changed the way I think and changed the way I want to think. Our education system could have introduced professors like these, we would have been more like Tartaglia and less like Antonio Fior. Thank you!

This video should be mandatory for learning square equations. It helps to really understand them!

If this man taught me math in highschool even I might have done something with my life. Very eye opening

Thank you so much for giving that "i=rotation" graph, it seriously helped solidify an understanding of it that I had never really understood, even all the way up into advanced calculus.

We need a Netflix series based on Math history.

The Story of Maths

​@Tom Rhodes its more accurate to call it perspective than imaginary, and absolute reality is metaphysical reality and abstractions (like math) are a closer form of or in other words, approaching metaphysical reality. religions like Christianity is depictions of abstract concepts in phenomenal representations pointing to metaphysical reality.

A Netflix series based on Math, which will never find the ultimate answer that it seeks? (Einstein tried and failed.) No thanks. I'd rather have a series that leads people to The Answer. For I've found that which math and quantum physics look for but will never find. Because, the logical conclusion to Math is this: If it requires imaginary numbers to describe something that we call "reality," that which we call "reality" is imaginary, and is NOT absolute Reality. And this is indeed a fact. See "The Book of GOD" at A Course in Truth.

@Clogica like most of us ^-^

I have always believed that these kinds of narrative particulars, and personal histories, of STEM knowledge development constitute one of the very best routes to rebuilding in every student's head that essential motivated understanding of the subject. Education is best effected, either when preceded by having the student come to own a problem that needs the specific to be learned, or when engaged emotionally in some other's need for such knowledge. The former is the sweetest but tough to contrive every time. The latter, however, with good storytelling and materials, works almost as well, and comes with copious collateral boons. Who knew that mathematicians guarded their skills like duelling magicians and their spells! Brilliant piece BTW. Subscribed.

As an electrical engineer I work with complex numbers quite often. So it's really interesting to hear the history of them! Thanks!

Your videos are excellent. I really love to learn the history of mathematics, physics, computer science etc. Thanks for taking the effort to make these videos. I really appreciate that.

This is the best explanation of complex numbers I have ever come across. Thank you.

Brilliant research and presentation! I had no idea complex numbers arose from the solution to cubic equations. Things make so much more sense when you explain the path of their discovery.

@Nerdelbaum Frink I'd argue algebra is even wider. Algebra consists of algebraic structures and their relationships. There are very diverse meaningful fields and rings, etc.

@Kurt Nowak Now that's quite a feat. Hard to believe.

@CuriousMarc and he did it all without using any vintage HP equipment!

@Someone It is found in quadratics too. But quadratics can find solutions without the imaginary unit showing up, and for quadratics where there were no real solutions, mathematicians could just dismiss those as not being solvable. Further, and more importantly, deriving a general solution for quadratics does not utilize an imaginary number, but for cubic equations, imaginary units are a necessary intermediary step for deriving a general formula. So while imaginary numbers show up in some quadratics, they're a necessary feature for getting the general solution to cubics.

Wow just wow, I also visualised the quadratic equations of algebra in geometric proofs in my high school because of our teacher, she used to teach me all these additional things. I also once got doubt how would it to be imagine cubic and bi-quadratic but they only got more complex so I stopped but finally you gave me a explanations that's very easy to understand. Thank you so much!🥰

The way you bring these complex mathematical matters into explanations for everyone is stuff of legends.

I always wondered: is it the imaginary-ness of i that makes it useful to describe reality, or the circular-ness? I always found it to be the latter, in my electrical engineering education. The fact that i decomposed into sin and cos seemed to be the central reason for its prominence in many fundamental equations

@Dante integra stop overthinking it

@PristinePerceptions "The truth is described by imaginary-ness" is sounding like BS to me 💩💩🤠🤠 I was thinking - The only reason we are getting imaginary numbers is because our understanding of reality is only approximation and it's incomplete?

The truth is described by imaginary-ness. Circular-ness is an assumption useful for practical applications. Because real and imaginary numbers cannot directly affect each other, they are represented by two axes perpendicular to each other. Because multiplication by a constant allows us to jump between the two axes, we are able to draw a loop in the complex plane. However, there is no need for the constant to be the square root of negative one. It could just as easily be the square root of negative two. And all your equations would change to account for the new conversion factor, and your circular-ness would become ellipse-ness. The crux of the equations is that reality is "complex". The fact that the magnitude of the constant (i) is 1 is a very useful and natural assumption for practical purposes.

Super interesting video. Thanks for sharing! Kept my attention the entire time. I think learning about the little tricks mathematicians used to do to make their problems easier is hilarious. "We don't know how to solve this so let's introduce a new term that ends up canceling out the problematic pieces" It seems like such a backward approach, but it's led us so far that I'm very impressed with ancient math. It's amazing to think how far we've come to have figured out the mathematical functions that guide our world, and yet we have much more to discover.

Was expecting cool math, didn’t expect the crazy history story, but it was my favorite part:D

I don't get it How cardano finds part of square that must have area of 30 and sides of 5............???????? 14:47 I am serching that part of..... So, Please answer my question

Dr Trefor, didn't expect meeting you here. Well hello...

Logical Conclusion: If it requires imaginary numbers to describe something that we call "reality," that which we call "reality" is imaginary, and is NOT absolute Reality. And this is indeed a fact. See "The Book of GOD" at A Course in Truth.

Agreed. I love a good story. There's really no better stories than that of what has really happened.

When I first learned about imaginary numbers, I thought to myself, "No way! There can't be any such numbers!" However, the more I learned about them, the more real they seemed. To me, it seems fitting that according to quantum theory, they are required to exist in the real world. Now what seems stranger to me is that we DON'T ordinarily observe imaginary numbers in real life.

Loved this. I've always just been given equations and told to plug in numbers, it's so much easier to see this conceptualized and understand what the numbers describe in the real world.

Being an Electrical Engineering graduate student, I cannot imagine my life without Complex numbers. I am ever grateful for such a beautiful formulation of complex, rather imaginary, mathematics.

This is like watching a thriller movie, out of nowhere something outstanding is revealed. From mathematics to history to physics to chemistry to subatomic to the universe.

Imagine minding your own business as a mathematician and suddenly someone challenges you to MATH DUEL, that can make you lose your job. Man, the older times were really intense for mathematicians.

Bro we need to bring that back

Love this! You just give me an idea to my undergraduate thesis!

@Tom Rhodes imaginary numbers literally describe real things (rotation, additional plane for calculations). Stop clinching to the "imaginary" word just because a concept was named that way due to going against "intuitive" basis of their creation time (euclidian geometrical interpretation of equations).

That's because this is an irrational (limited) world. And the logical conclusion is this: If it requires imaginary numbers to describe something that we call "reality," that which we call "reality" is imaginary, and is NOT absolute Reality. And this is indeed a fact. See "The Book of GOD" at A Course in Truth.

I really wanted to see this from such a long time...this question was stuck in my mind and I was shy to ask my teachers coz many of them were just not interested in such curiousities...thanks to this channel

Let’s appreciate how much awesome work and effort he puts in his videos keep up the good work.

let's take a moment to contemplate khayam's modesty and open mindness that all scientists should have "Maybe one of those who will come after us will succeed in finding it" He considered the possibility of there being a solution that just he couldn't find Complete contrast with Luca Pacioli's "impossible" writtent next to cubic equations

Absolutely brilliant video! Thanks for explaining things for the rest of us.

If you take the time to understand the *why* in maths, it can take you a long way

I always try my best to ponder that. :)

If they teach you how, you can complete a task. If they teach you why, you can manage the task completers.

If you do in biology then you will spend your whole life 🤠🤠🤠

I'm a math teacher. I spend enormous amounts of time trying to explain the 'why'. Most students will fight you to not have to listen.

@IrokoSalei it's a bit different, actually. The 'how' is more about what is the modus operandi of the subject while 'why' is more about what causes such phenomenon to occur. Also this is mainly about math, so mentioning science is kinda irrelevant here.

I find this video very insightful. I love the way how the narrator explain the solution of quadratic and cubic equation from the geometric perspective. This video helps me a lot, for this video brings not only a whole new way of understanding those high degree equation but also shows me a whole new way about how to teach/explain quadratic and cubic question solution to students.

I've always enjoyed your content. Very inspiring and informative. I hope one day to add as much value to others as you do with your content!

I studied maths during school and remember these formulas till now. But no teacher has thought me how these equations are formed. Really great video dude. I like maths more now

I remember learning in senior year of high school how to derive the quadratic solution formula by completing the square. Amazingly, I understood how it worked, but now, 50 years later seeing this video, I finally understand why it works. What a wonderful story this Veritasium video tells!

One of your best ever videos. Wonderful stuff.

Artistic look in animations is close resemblance to what Mike Maloney used in Hidden secrets of money. Anyone know the artist/group?

Agree

Just amazing as usual ❤

ooo you are the best)

Everytime I see you comment, I think you're an @electroboom alt named medi....

Dude, I did my undergraduate in math and your math videos really put concepts into perspective in a way I never could. I've always just learned math for math's sake and practically solved the questions, even in the more advanced courses. I actually really hated it, but the way you tie these complex topics together actually makes me realize the significance of some of the seemingly pointless math I forced myself so hard to learn.

Thank you a lot! Your way of explanation is amazing! There a lot of scientists but teachers like you just a few!

I remember when that happened. They said, "We need a square root of negative one." So they imagined one. It's okay. Math is imagination.

guys he can time travel

This is perhaps one of veritasium’s most important videos..simply because it touches on key topics that every student must learn if they are to enter any STEM field!

If they taught in school about the history of math and how we use it in real world, I'm sure most of people who "hate math" will see how magnificent it is.

You don’t need to put it in parenthesis we really do hate it

I agree, this would have made learning math so much more interesting to me, my favorite classes were history and language arts.

Or they would see how limited and lacking math is. Because, the logical conclusion is this: If it requires imaginary numbers to describe something that we call "reality," that which we call "reality" is imaginary, and is NOT absolute Reality. And this is indeed a fact. See "The Book of GOD" at A Course in Truth.

Actually if math was mixed with history,then i would hate it.

After watching this video, and studying several courses at university level, I still really dislike the subject of math. It’s amazingly useful for some applications, but a lot of the math that a lot of people study is not relevant to their fields.

I wish i learned some context of the math I learned. I use all these equations daily at work, and.. just use them. This is awesome and I’m saving these videos for my kids, niece, and nephew. Verasasium gave us an invaluable gift all for free on youtube. I just LOVE this.

it was refreshing to hear the history of imaginary numbers. im an electrical engineering major and we use these all the time, so learning more about its history gave me more appreciation for it. Thank you @Veritasium

This is pure beauty. Never knew that we can actually link equations with geometry so well.

I wish someone explained like this in my high school. Literally next level video and content.

As someone who's really bad with math, these visuals have helped me realize a lot of what I didn't understand with basic algebra and trig functions from school as a kid.

@Tom Rhodes imaginary numbers are not imaginary. That's just a name that reflects past mathematicians difficulties with these kind of abstractions. We call them numbers because they share a lot of properties with real numbers (they form a field extension). Nothing unreal, just pure logic. It's like calling a knife an ''imaginary arm extension''. But the truth is it's a real thing, that may feel like an arm extension, but it's a real tool that has nothing imaginary (though we thought it's impossible to cut bread with an arm, but now we 'can' since we call it arm extension, but it's purely conventional). The impossibility of x^2=-1 for x real number has not been violated, since i is not a real number Tbh I would like to call these numbers differently

Bad with math? Me too. And yet, I've found that which math and quantum physics look for but will never find. Because, the logical conclusion is this: If it requires imaginary numbers to describe something that we call "reality," that which we call "reality" is imaginary, and is NOT absolute Reality. And this is indeed a fact. See "The Book of GOD" at A Course in Truth.

I really loved geometry and so I realized why I hate maths if its not imaginable as geometric lol. It seems my math is medieval period math 🤣

@Bubba Bong math helped science. It definitely helped medical science (see how fourier transform is used)

@apidas Math experts dont take centuries to learn notations and concepts. It takes centuries to understand them only when starting with old math understanding

Spectacular . We engineering students know all about the algebra , but we never knew about this amazing time taking geometrical visualizations that led us to algebra . Its wonderful to get into the heads of these amazing thinkers . I was completely baffled at the quality of the video. Thank you !!

This was another eye-opener for me. You are now officially my maths tutor! You put everything that anybody ever tried to teach me in regards to numbers to complete shame. Or, more specifically, how it was taught in utter disregard, or mere ignorance, of provenience or practicality.

This is gold. This video makes math much much easier to understand! 🤯

This is a truly beautiful video. Truly an amazing job putting this history together. The math is so incredible and the work of those mathematicians just unreal. It makes me wonder what amazing things we haven’t discovered yet that are affecting our world, and how people in future civilizations will look back at our progress to get them there.

All throughout grade school and college I struggled to understand the "why" portion of math beyond plug and chug. Usually professors couldn't give me an adequate explanation. Completing the square was one term that never really clicked for me. The first 3 minutes of this video are pure genius. So simple and understandable. This makes math so much more digestible.

Great video. And although it might make math "much more digestible," it should also give you indigestion. Because, the logical conclusion is this: If it requires imaginary numbers to describe something that we call "reality," that which we call "reality" is imaginary, and is NOT absolute Reality. And this is indeed a fact. See "The Book of GOD" at A Course in Truth.

I took history of math class, so I know all this. But seeing the video is still awesome because I got to see the visuals of it.

I had a similar difficulty with chemistry - it seemed like just some random things to memorize 'because it worked'. It wasn't until I got into quantum mechanics that chemistry's 'random' stuff actually made sense. This video explains why.

@bobman bob You're confused. *You* know a whole lot about nothing. Others know at least something.

I struggled to get the cover off my graphing calculator.

Now this was one helluva history lesson. Fascinating! I find that the problem itself endured for multiple generations and one guy built from the other. Incredible!

This is honestly one of the most incredible videos on overall mathematics ever made, even though it's focusing on a specific topic. Should be required watching for all students. Few appreciate math because few know how to teach it.

Rarely anyone likes maths so I have tremendous respect for them

I love it when complex equations come down to something elementary like 2+2=4

Watching this, I feel like learning math was spoiled on me in my youth. I was a sad kid, who didn't really care much about anything. I was smart enough to do well in my math classes without too much trouble, but I can't say I was in any place to enjoy them the way I might have now. I don't think I valued life and the odd sense of harmony in the universe to appreciate the beauty in these sorts of things.

Most of your work is educational yet highly entertaining but this particular video deserves an award. One of my favorite channels on the platform. Proud to have subscribed to it over 10 years ago.

This particular video is indeed special. Because, a wise man sees that its logical conclusion is this: If it requires imaginary numbers to describe something that we call "reality," that which we call "reality" is imaginary, and is NOT absolute Reality. And this is indeed a fact. See "The Book of GOD" at A Course in Truth.

@Hyperduality Two of those, at least, make no sense. Go take your meds.

@Hyperduality slayyyyy✨

@10 THOU PIECE Yes, I knew most of this maths too but still resist watching it. Mesmerizing stuff.

I just got into grade 12 so picked up the courage to see this video in its entirety, and got goosebumps along every 4-5 mins. Can't express how much better this is than literally any other way in the world rn to tell people about i :) and what its for...cause we might study this and use it for MONTHS to say the least, and this 23 min video sums up its history and brings more context than anything ever can! Thanks derek and love veritasium

Love this video. Math took what people thought were errors, and revealed that understanding them was crucial to the calculations of multiple fields of study involving 3 dimensions. Imagine what the next set of numbers will bring us.

Such a well told story! Thank you!

The amount of knowledge I got from this video is unimaginable for me. Thankyou So much ❤️

A History, Math and Science smoothie blended to perfection. Well done 👏

Yes ,you should

Add in Metaphysics for true perfection. For the logical conclusion is this: If it requires imaginary numbers to describe something that we call "reality," that which we call "reality" is imaginary, and is NOT absolute Reality. And this is indeed a fact. See "The Book of GOD" at A Course in Truth.

My name is bill gaytz and i did a horrible thing, then i did a more horrible thing and now i can't stop and i can't really think straight.

add compter science in the mix and all of them are my favourite subjects to study

my favorite math teacher always showed us how equations like the quadratic came to be. he is the biggest reason I'm studying math in college

If I had a teacher who taught me about real applications for these equations and showed examples of how to work through them I would have definitely paid more attention in class.

Very interesting stuff. All elementary kids that go "why do we have to learn math" should watch this. Math was a form of battle, and mathematicians hoarded their secrets to be able to attack others. People made a career in knowing how to solve math problems, and it solves complex questions in our explanation of everyday items.

This video was so inspiring and incredibly beautiful. I want to thank you from the bottom of my heart from encouraging me, educating me, and inspiring me. Seeing e^ikx like that was so beautiful. Thank you. I am sharing this with my physical chemistry professor. I want future students of his to see this video and be inspired. You are so great. Thank you again.

The phrase "completing the square" makes much more sense now. Holy crap my mind is blown. I really wish math was taught like this. I thought I hated math but I'm finding that isn't actually the case when I learn through mediums such as YouTube. Does anyone have any suggestions for other videos that combine math and history like this one?

If your mind is blown that easily, my suggestion might be too much. Because, here's the logical conclusion to all of this: If it requires imaginary numbers to describe something that we call "reality," that which we call "reality" is imaginary, and is NOT absolute Reality. And this is indeed a fact. See "The Book of GOD" at A Course in Truth if you seriously want your mind blown.

@Anon Me ive graduated so im mostly watching for entertainment. Interesting to hear that challenge in the psych field. In terms of encouraging students to learn, yeah why not though this method might help a bit or not at all, it needs to be tested.

I could not agree more, The moment he provided the visual representation I was just blown away by how much easier it made understanding the entire process. Truly something I wish was introduced to when I was learning the subject instead of just being told the steps without truly understanding why.

@̇ it's the whole education systems fault Elon Musk started his own school for the kids(spacex employees) for that reason

What I don't like about school (well my math subject): They teach us how to do it, but they don't teach us how it works and why it works

I did my bachelor in Venice and my MSc in Bologna. It's so nice to see them both mentioned in the video. It was an amazing journey following you. In Italy we play a lot with the Tartaglia triangle in school with some basic math quiz.

probably one of the best videos you have made. Never understood how the quadratic formula worked but in 15mins I know dude thank you!

Hearing the story behind some maths concepts really feeds our curiosity and drives us to search for more, instead of just learning formulas by heart

this got to be my fav math video of all time

Dude I love the way explained such problem using visual animation, honestly might be the very first time I was actually intrigued to relearn my math classes since back then I wasn't really interested in the subject. You officially changed my perception my good sir.

I just came across this video, and oh my god this question which has been bothering me for YEARS, finally is answered (is Youtube reading my mind?). I always wonder if the invention of imaginary number was accidental or intentional, how it become the greatest of math and helped advance science so much (especially physics). Thank you Veritasium your videos are always amazing

So I’ve always loved math and have been good at it. It was my favorite subject growing up. I graduated high school with 6 math credits even though I only needed 4 to graduate. I took AP calc my senior year. I majored in chemistry with a math minor originally (I didn’t finish the math minor though). And this video still blew my mind! Here I am at 28 years old and finally being introduced to the actual concepts that made modern math what it is. Math history is surprisingly interesting, coming from someone who isn’t a history fan in general. Amazing video!

Excellent video.! Your explanations are both enlightening and fascinating. Thank you.

Hey Derek! Just wanted to let you know I appreciate all your hard work about teaching unintuitive things in creative ways. I’ve been watching you for 6 years now and your contents only gotten better, thanks!

@RK That Indians are usually kind, I’m assuming.

@M3Z_9 So what are you trying to imply?

Would it be too random to declare my intend to recommend my fellow science-youtuber-fans some... well... more science-youtuber? I mean, in my mind, it just makes sense, but many call me B0t, so... your choice...

@RK bruh wha–

Holy mackerel! That was really amazing. You inspired me to get back to Grade 11 and find my old textbooks. I love the analysis of reality - but time is pushing me to the imaginary.

I seriously wish Derek Muller and his team writes a book on the history of science. I would definitely buy it. And I wish Mathematics was this fun at school and engineering.

Excellent history. Congratulations for showing that in a so interesting way!

I really wish I had the processing power to understand any of this. I'll settle for following along with enthusiasm

For the entirety of my higher education, I've been told to "complete the square," but 6 teachers and 4 professors have never explained this further than restating the equation. In one extremely brief visual and explanation, you've managed to answer a question I'd long since forgotten. I don't know how to describe my astonishment, nor my gratitude for your content.

Big same

My Algebra 2 teacher never really did that, in the same year where I also took an Algebra based physics class I did pretty well. Yet in my Algebra 2 I did not do so well. My physics teacher taught exactly like Veritasium and I did well. Also as a side note at my school all of the physics classes are in an alignment with the Calculus classes since a lot of the students are in calc and take physics with it. I do believe I would have done way better with that method.

It blew my mind so much it almost hurt

had a similar experience with gravity, or rather the equation to calculate gravity. it IS such a relief to realize that ur past confusion on a matter is really a higher understanding that what was taught to u was lacking, and there had to be more to it, but the others didnt see that there had to be more.

This blew my mind haha

I’m always failing with mathematics but I love it so much and it’s really amazing and so cool.

This is the first time I have appreciated the concept of imaginary numbers. What a rich and beautiful history

One can't possibly underestimate the amount of work that has gone into producing this amazing video.

This lesson is what I needed to here when I took pre algebra in high school. This was so satisfying 😌

Math teachers, please, please, show this kind of stuff during class. It would've changed my life.

@Paradox so true...cell phones cell phones

@pitthepig 🤣😂😅so true

@Kevin Bugusky thispurplefox didnt. And I assure you, the kids who legitimately found it boring would find anything you did boring. But the one or two "thispurplefox" kids that stayed silent/disagreed/lied will have their lives changed, growing up enthralled with math rather than terrified of their own struggles with it.

@Paradox I think we do, just not about imaginary numbers.

@wonderful bees Exactly, most would talk loudly ruining other students, and other would just use their Phones and ignore it

Wow! I just re-found this video watched a few months ago, and realized how amazing💫 is this explanation watching again. I am a math teacher💡, and have always found interesting, including for myself back⏳ in high school, why some students not happy with algebra, suddenly get interested in geometry, while others (like myself at the time) find geometry not as interesting as algebra. Of course in striving to be the best😇 math teacher, I did learn👍 as a teacher, that of course I will always ALSO be a student 🌞

Helpful on a difficult topic. Well presented in lovely historical and practical choices, thank you

Honestly the way you tell the stories can inspire ppl, like rn I imagined a whole action movie from this history

We have always used imaginary numbers in oscillation physics as a trick, as such the wave function has no meaning but the modulus has which is real stating that the imaginary number in Schrodinger equation is also a trick if we compare it to the oscillation physics and hence it corresponds to the wave physics, thus the most genuine and sensible part about the wave equation is the number 'i'.

It's just wonderful to see how he is explaining math, physics and chemistry with such ease

Now just add the metaphysics. Because, here's the logical conclusion to all of this: If it requires imaginary numbers to describe something that we call "reality," that which we call "reality" is imaginary, and is NOT absolute Reality. And this is indeed a fact. See "The Book of GOD" at A Course in Truth.

@Hyperduality Two of those clearly make no sense, so you're likely out of your depth.

@vikas malik If you want history-lessons-that-are-fun, try 'Oversimplified', 'CGP Grey' and 'Bluejay'. Oh, and Illuminaughtii.

History

Complex numbers are dual to real numbers. Perpendicularity or orthogonality = DUALITY! Column vectors are dual to row vectors -- group theory. Electro is dual to magnetic -- Maxwell's equations. The electric field is perpendicular (dual) to the magnetic field -- probability waves. Positive charge is dual to negative charge -- electric fields. North poles are dual to south poles -- magnetic fields. Electro-magnetic energy or photons are dual. Points are dual to lines -- the principle of duality in geometry. Group theory:- the image is a copy, equivalent or dual to the factor or quotient group. Isomorphism (absolute sameness) is dual to homomorphism (relative sameness or difference). Homo is dual to hetero, same is dual to different. Injective is dual to surjective synthesizes bijective or isomorphism. Positive curvature is dual to negative curvature -- Gauss, Riemann geometry. Curvature or gravitation is dual. Gravitation is equivalent or dual to acceleration -- Einstein's happiest thought, the principle of equivalence (duality). "Perpendicularity in hyperbolic geometry is measured in terms of duality" -- Universal hyperbolic geometry, Professor Norman J. Wildberger. Duality (energy) creates reality. Action is dual to reaction -- Sir Isaac Newton (the duality of force). Attraction is dual to repulsion, push is dual to pull -- forces are dual, e.g. the electro-magnetic force. Monads are units of force -- Gottfried Wilhelm Leibnitz. Monads are units of force which are dual -- monads are dual. Energy = force * distance. If forces are dual then energy must be dual. Potential energy is dual to kinetic energy, gravitational energy is dual. Apples fall to the ground because they are conserving duality. "May the force (duality) be with you" -- Jedi teaching. "The force (duality) is strong in this one" -- Jedi teaching. "Always two there are" -- Yoda.