How Imaginary Numbers Were Invented

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A general solution to the cubic equation was long considered impossible, until we gave up the requirement that math reflect reality. This video is sponsored by Brilliant. The first 200 people to sign up via brilliant.org/veritasium get 20% off a yearly subscription.

Thanks to Dr Amir Alexander, Dr Alexander Kontorovich, Dr Chris Ferrie, and Dr Adam Becker for the helpful advice and feedback on the earlier versions of the script.

Some great videos about the cubic:

500 years of not teaching the cubic formula. -- trvid.com/video/video-N-KXStupwsc.html

Imaginary Numbers are Real -- trvid.com/video/video-T647CGsuOVU.html

Dunham, W. (1990). Journey through genius: The great theorems of mathematics. New York. -- ve42.co/Dunham90

Toscano, F. (2020). The Secret Formula. Princeton University Press. -- ve42.co/Toscano2020

Bochner, S. (1963). The significance of some basic mathematical conceptions for physics. Isis, 54(2), 179-205. -- ve42.co/Bochner63

Muroi, K. (2019). Cubic equations of Babylonian mathematics. arXiv preprint arXiv:1905.08034. -- ve42.co/Murio21

Branson, W. Solving the cubic with Cardano, -- ve42.co/Branson2014

Rothman, T. (2013). Cardano v Tartaglia: The Great Feud Goes Supernatural. arXiv preprint arXiv:1308.2181. -- ve42.co/Rothman

Vali Siadat, M., & Tholen, A. (2021). Omar Khayyam: Geometric Algebra and Cubic Equations. Math Horizons, 28(1), 12-15. -- ve42.co/Siadat21

Merino, O. (2006). A short history of complex numbers. University of Rhode Island. -- ve42.co/Merino2006

Cardano, G (1545), Ars magna or The Rules of Algebra, Dover (published 1993), ISBN 0-486-67811-3

Bombelli, R (1579) L’Algebra ve42.co/Bombelli

The Manim Community Developers. (2021). Manim - Mathematical Animation Framework (Version v0.13.1) [Computer software]. www.manim.community/

Special thanks to Patreon supporters: Luis Felipe, Anton Ragin, Paul Peijzel, S S, Benedikt Heinen, Diffbot, Micah Mangione, Juan Benet, Ruslan Khroma, Richard Sundvall, Lee Redden, Sam Lutfi, MJP, Gnare, Nick DiCandilo, Dave Kircher, Edward Larsen, Burt Humburg, Blake Byers, Dumky, Mike Tung, Evgeny Skvortsov, Meekay, Ismail Öncü Usta, Crated Comments, Anna, Mac Malkawi, Michael Schneider, Oleksii Leonov, Jim Osmun, Tyson McDowell, Ludovic Robillard, Jim buckmaster, fanime96, Ruslan Khroma, Robert Blum, Vincent, Marinus Kuivenhoven, Alfred Wallace, Arjun Chakroborty, Joar Wandborg, Clayton Greenwell, Pindex, Michael Krugman, Cy 'kkm' K'Nelson,Ron Neal

Executive Producer: Derek Muller
Writers: Derek Muller, Alex Kontorovich, Stephen Welch, Petr Lebedev
Animators: Fabio Albertelli, Jakub Misiek, Iván Tello, Jesús Rascón
SFX: Shaun Clifford
Camerapeople: Derek Muller, Emily Zhang
Editors: Derek Muller, Petr Lebedev
Producers: Derek Muller, Petr Lebedev, Emily Zhang
Additional video supplied by Getty Images
Music from Epidemic Sound and Jonny Hyman



30 Eki 2021




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Akira Leelarapin
Akira Leelarapin 9 aylar önce
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.
HTO Gün önce
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
Inception1338 2 gün önce
Bet on it. It was supposed to be like that.
Doremy Sheep
Doremy Sheep 3 gün önce
Kamran Namdarpour
Kamran Namdarpour 3 gün önce
So true. This was interesting as hell, putting 2 and 2 together to find out what it is you know, and more importantly WHY!!!
David Nelson
David Nelson 3 gün önce
Well said.
TommyTheTsunami 4 aylar önce
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.
Arthur 16 gün önce
@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...
Haine Sensei
Haine Sensei 16 gün önce
@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.
Arthur 17 gün önce
@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
Simpson Comments
Simpson Comments 17 gün önce
Kelsey Oakes's Aunt stopped living (LMAO 😂) because I upload bangers! ..,....,.
welovfree Aylar önce
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.
wren 2 gün önce
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
JoDoFe 3 gün önce
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.
Sadia Rahman
Sadia Rahman 11 gün önce
Exactly!! Instead we are taught about kings who killed their own brothers for the throne🙄
Jumbomuffin 13
Jumbomuffin 13 16 gün önce
🙄 just because you want to doesn’t mean everyone has to
Simpson Comments
Simpson Comments 17 gün önce
Kelsey Oakes's Aunt stopped living (LMAO 😂) because I upload bangers! ..,...,,
Jeremy Ward
Jeremy Ward 3 aylar önce
"Only by abandoning math’s connection to reality could we discover reality’s true nature." I cannot shake these words from my head.
You think i'm funny?
i'm just confused.
FlyingStone Mon
FlyingStone Mon 15 gün önce
It's powerful.
hieu dang
hieu dang 15 gün önce
sounds kind of ironic
i 16 gün önce
@legitjimmyjaylight What you are saying is essentially math is important in physics in overall.
Vividh Kothari
Vividh Kothari 4 aylar önce
That story about Ferro, Fior, Tartaglia, and Cardano could be a movie.
Heart2HeartBooks 2 gün önce
Albeit one that is shown during a Mensa weekend get together.
Nasty Gamer
Nasty Gamer 3 gün önce
Exactly, and I can make it work sa an action movies even
Indian Boy
Indian Boy 8 gün önce
@Anurag Gupta well, you can check out about Galois yourself for proof. By the way, Galois was French.
Anurag Gupta
Anurag Gupta 8 gün önce
@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.
Indian Boy
Indian Boy 9 gün önce
@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.
Steven He
Steven He 9 aylar önce
This video makes me want to do math. It’s inspiring in the best way
Saurav Mishra
Saurav Mishra 2 gün önce
emotional damage!
George Washington
George Washington 2 gün önce
@Interstellar d a sad fff
BlizzardWuffy 3 aylar önce
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 :)
Saurav Mishra
Saurav Mishra 2 gün önce
but you were told about it , you did not invent that theory yourself
simar karmani
simar karmani 4 gün önce
@Titan Games 2 weeks is *not* extremely necro. 2 years is.
simar karmani
simar karmani 4 gün önce
@Constan-Anaconda Oh, that's why!
simar karmani
simar karmani 4 gün önce
Of course. Anyone could solve that.
Alok Kumar
Alok Kumar 6 gün önce
And that paved all of the way of us becoming a galactic civilization
Jinpachi Bobochan
Jinpachi Bobochan 3 aylar önce
Can't help but think of these scientists, matheticians, and so on as heroes - in their own way. The human mind is bloody awesome.
Krrish Deshpande
Krrish Deshpande 4 aylar önce
It’s funny that how easy he makes us understand these difficult concepts.
Blokin 7 gün önce
@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 TRvid video. Must be the times we live in..
imPERFEKT 7 gün önce
@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.
Nur Fatsyah
Nur Fatsyah 17 gün önce
@Blokin get real, you never know what's happen inside other people head.
Blokin 17 gün önce
Get real you don't understand this at all, nice cid though.
Nur Fatsyah
Nur Fatsyah 20 gün önce
@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.
Ian Cap
Ian Cap Aylar önce
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!
Scott Walker
Scott Walker 14 gün önce
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.
Dr. Trefor Bazett
Dr. Trefor Bazett 9 aylar önce
Was expecting cool math, didn’t expect the crazy history story, but it was my favorite part:D
Ansh Daheriya
Ansh Daheriya 7 aylar önce
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
Rizal Purnawan
Rizal Purnawan 8 aylar önce
Dr Trefor, didn't expect meeting you here. Well hello...
Tom Rhodes
Tom Rhodes 8 aylar önce
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.
Elizabeth Moore
Elizabeth Moore 8 aylar önce
Agreed. I love a good story. There's really no better stories than that of what has really happened.
Jonathan Molina
Jonathan Molina 2 aylar önce
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!
Máté Ócsai
Máté Ócsai 3 aylar önce
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.
Imzaadi 3 aylar önce
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.
JanicekTrnecka 3 aylar önce
This video should be mandatory for learning square equations. It helps to really understand them!
izzynobre 9 aylar önce
Dude makes math sound absolutely riveting... Incredible
Adam 2 aylar önce
Always has been
Ashmeed Mohammed
Ashmeed Mohammed 3 aylar önce
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.
Parker Moss
Parker Moss 8 aylar önce
@Tom Rhodes The term imaginary numbers is in fact a misnomer. Complex numbers are just a number system similar to reals.a
Tom Rhodes
Tom Rhodes 8 aylar önce
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.
ForestCity Fishing
ForestCity Fishing 9 aylar önce
I was bored and checked out by the time he got to the point.
Joshua Jimenez
Joshua Jimenez 3 aylar önce
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 :)
Ian 2 aylar önce
You should watch this channel's video on visual learning
Grey Acumen
Grey Acumen 3 aylar önce
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.
Black Spurr
Black Spurr 3 aylar önce
If this man taught me math in highschool even I might have done something with my life. Very eye opening
Tanishq Jain
Tanishq Jain 2 aylar önce
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!
Scenic Fights
Scenic Fights 9 aylar önce
Great job making this extremely interesting! Math and Science is so interesting when explained well.
Kep Spark
Kep Spark 7 aylar önce
Guna selan guna
Guna selan guna 9 aylar önce
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 🙂
MLG35U5 10/10 ign rated gr8m88/8
Now rewrite that sentence without the word “interesting”.
Oliver May
Oliver May 3 aylar önce
This is the best explanation of complex numbers I have ever come across. Thank you.
milad aghajohari
milad aghajohari 4 aylar önce
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.
Nathan Janssen
Nathan Janssen 3 aylar önce
As an electrical engineer I work with complex numbers quite often. So it's really interesting to hear the history of them! Thanks!
Atchfam77 3 aylar önce
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.
CuriousMarc 9 aylar önce
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.
Queta Arbuste
Queta Arbuste 8 aylar önce
@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.
CuriousMarc 9 aylar önce
@Kurt Nowak Now that's quite a feat. Hard to believe.
Kurt Nowak
Kurt Nowak 9 aylar önce
@CuriousMarc and he did it all without using any vintage HP equipment!
Nerdelbaum Frink
Nerdelbaum Frink 9 aylar önce
@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.
fliprim 15 gün önce
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.
KRISHNA 4 aylar önce
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!🥰
Josiah Ray
Josiah Ray 2 aylar önce
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.
Fidel Soto
Fidel Soto Aylar önce
The way you bring these complex mathematical matters into explanations for everyone is stuff of legends.
PuzzLEGO 9 aylar önce
If you take the time to understand the *why* in maths, it can take you a long way
Supertron 8 aylar önce
I always try my best to ponder that. :)
Fisharepeopletoo 8 aylar önce
If they teach you how, you can complete a task. If they teach you why, you can manage the task completers.
Chinmayee Lenka
Chinmayee Lenka 8 aylar önce
If you do in biology then you will spend your whole life 🤠🤠🤠
Totally Amateur Tom
Totally Amateur Tom 9 aylar önce
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.
Alec Xander
Alec Xander 9 aylar önce
@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.
Jing Jing Abawi
Jing Jing Abawi 3 aylar önce
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.
Earn Money Online
Earn Money Online 3 aylar önce
Let’s appreciate how much awesome work and effort he puts in his videos keep up the good work.
Adam El Meziane
Adam El Meziane 9 gün önce
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
Tom Franklin
Tom Franklin 4 aylar önce
Absolutely brilliant video! Thanks for explaining things for the rest of us.
Raided Cluster
Raided Cluster 9 aylar önce
We need a Netflix series based on Math history.
Abdi Marmalade
Abdi Marmalade 5 gün önce
The Story of Maths
Good Goyim
Good Goyim 8 aylar önce
​@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.
Tom Rhodes
Tom Rhodes 8 aylar önce
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.
Krishna Soni
Krishna Soni 8 aylar önce
@Clogica like most of us ^-^
Linda Dee
Linda Dee 3 gün önce
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!
AM D 3 aylar önce
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!
Mridul Mishra
Mridul Mishra 3 aylar önce
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.
Isha Kawade
Isha Kawade Aylar önce
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
Dr. Mudarris د. بشير المدرس
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.
Tim H
Tim H 8 gün önce
You don’t need to put it in parenthesis we really do hate it
R0KURU 28 gün önce
I agree, this would have made learning math so much more interesting to me, my favorite classes were history and language arts.
Tom Rhodes
Tom Rhodes 8 aylar önce
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.
Robert Fox
Robert Fox 9 aylar önce
Actually if math was mixed with history,then i would hate it.
Martin Gutsch
Martin Gutsch 9 aylar önce
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.
Ricardo Guardiola
Ricardo Guardiola 3 aylar önce
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!
David Terr
David Terr 28 gün önce
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.
Ralf Kleemann – Harpist
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.
Terry Keggereis
Terry Keggereis Aylar önce
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.
FinFET 9 aylar önce
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.
HkayakH 19 gün önce
Bro we need to bring that back
Nur Fatsyah
Nur Fatsyah 20 gün önce
Love this! You just give me an idea to my undergraduate thesis!
GrayFoxHound9 8 aylar önce
@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).
Tom Rhodes
Tom Rhodes 8 aylar önce
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.
Maqsudsho Nemati
Maqsudsho Nemati 2 aylar önce
Thank you a lot! Your way of explanation is amazing! There a lot of scientists but teachers like you just a few!
Sparro 3 aylar önce
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
Chaithanya Can
Chaithanya Can 3 aylar önce
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.
Bhavesh Kakwani
Bhavesh Kakwani 3 aylar önce
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
Kevin Wu
Kevin Wu Aylar önce
@Dante integra stop overthinking it
Dante integra
Dante integra Aylar önce
@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?
PristinePerceptions 2 aylar önce
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.
Josh 9 aylar önce
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.
Queta Arbuste
Queta Arbuste 8 aylar önce
@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
Tom Rhodes
Tom Rhodes 8 aylar önce
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.
Arthuria Pendragon
Arthuria Pendragon 8 aylar önce
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 🤣
Queta Arbuste
Queta Arbuste 8 aylar önce
@Bubba Bong math helped science. It definitely helped medical science (see how fourier transform is used)
Queta Arbuste
Queta Arbuste 8 aylar önce
@apidas Math experts dont take centuries to learn notations and concepts. It takes centuries to understand them only when starting with old math understanding
Anna-Maria Liu
Anna-Maria Liu 2 aylar önce
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.
Viplove Gupta
Viplove Gupta 15 gün önce
This is pure beauty. Never knew that we can actually link equations with geometry so well.
Shaurya kansal
Shaurya kansal 2 aylar önce
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
Harald König
Harald König 6 gün önce
One can't possibly underestimate the amount of work that has gone into producing this amazing video.
Medlife Crisis
Medlife Crisis 9 aylar önce
One of your best ever videos. Wonderful stuff.
Cubert Miso
Cubert Miso 9 aylar önce
Artistic look in animations is close resemblance to what Mike Maloney used in Hidden secrets of money. Anyone know the artist/group?
Antoine Olivier
Antoine Olivier 9 aylar önce
Mohamed Flaifel
Mohamed Flaifel 9 aylar önce
Just amazing as usual ❤
Максим 9 aylar önce
ooo you are the best)
Just Woody's Cuck
Just Woody's Cuck 9 aylar önce
Everytime I see you comment, I think you're an @electroboom alt named medi....
hippieparents420 Gün önce
16:25 - 17:35 When performing the equation visually with "blocks" it gives me really strong vibes to "dark matter" . It's not there, but its mass is there in the equation.
dan725 3 aylar önce
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.
Sid 4 aylar önce
This is gold. This video makes math much much easier to understand! 🤯
Dr10Jeeps 3 aylar önce
Excellent video.! Your explanations are both enlightening and fascinating. Thank you.
TheFinalRevelation 9 aylar önce
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.
Tom Rhodes
Tom Rhodes 8 aylar önce
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.
Nerdelbaum Frink
Nerdelbaum Frink 9 aylar önce
@Hyperduality Two of those, at least, make no sense. Go take your meds.
Indorap -YT
Indorap -YT 9 aylar önce
@Hyperduality slayyyyy✨
TheFinalRevelation 9 aylar önce
@10 THOU PIECE Yes, I knew most of this maths too but still resist watching it. Mesmerizing stuff.
Nick Latino
Nick Latino 3 aylar önce
Such a well told story! Thank you!
Lr Im
Lr Im 4 aylar önce
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.
Gibran Bedra
Gibran Bedra 3 aylar önce
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!
Yaseen Hussain
Yaseen Hussain Aylar önce
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
ren egade
ren egade 9 aylar önce
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 TRvid. Does anyone have any suggestions for other videos that combine math and history like this one?
Tom Rhodes
Tom Rhodes 8 aylar önce
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.
Work count
Work count 9 aylar önce
@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.
Bryan Williams
Bryan Williams 9 aylar önce
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.
NothingsGood 9 aylar önce
@̇ it's the whole education systems fault Elon Musk started his own school for the kids(spacex employees) for that reason
̇ 9 aylar önce
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
Your Best Beautiful
Your Best Beautiful 4 aylar önce
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 !!
Bboy Network
Bboy Network 3 aylar önce
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.
Dudley Boy
Dudley Boy 26 gün önce
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
Sidharth Singh
Sidharth Singh 5 gün önce
I wish someone explained like this in my high school. Literally next level video and content.
Justin 9 aylar önce
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.
Tom Rhodes
Tom Rhodes 8 aylar önce
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.
Andromeda Messier 31
Andromeda Messier 31 9 aylar önce
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.
John Bjorkman
John Bjorkman 9 aylar önce
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.
Nerdelbaum Frink
Nerdelbaum Frink 9 aylar önce
@bobman bob You're confused. *You* know a whole lot about nothing. Others know at least something.
Mikey P
Mikey P 9 aylar önce
I struggled to get the cover off my graphing calculator.
Cristal Laanstra
Cristal Laanstra 27 gün önce
This is such a great video. I agree with many of the comments - I've tried to wrap my mind around this so many times and watching this video was definitely an 'aha' moment. I feel like almost every video you do is the equivalent of a scientific paper. So much knowledge and energy. I can only imagine how different my Complex Mathematics course in university would have been with this information hah!
Billy 3 aylar önce
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.
Nasty Gamer
Nasty Gamer 3 gün önce
Honestly the way you tell the stories can inspire ppl, like rn I imagined a whole action movie from this history
MysteriousPsychology 2 aylar önce
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
Kaleb Johnson - CompByte
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.
Samaya Erickson
Samaya Erickson 2 aylar önce
Big same
Jake Sullivan
Jake Sullivan 2 aylar önce
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.
Animelytical 3 aylar önce
It blew my mind so much it almost hurt
Ashmeed Mohammed
Ashmeed Mohammed 3 aylar önce
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.
Benjamin Allred
Benjamin Allred 3 aylar önce
This blew my mind haha
blah blah
blah blah Aylar önce
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!
CrimsonDragon 3 aylar önce
probably one of the best videos you have made. Never understood how the quadratic formula worked but in 15mins I know dude thank you!
Paper Aylar önce
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.
Mukesh Rana
Mukesh Rana 14 gün önce
The amount of knowledge I got from this video is unimaginable for me. Thankyou So much ❤️
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