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.

History should be a fundamental block for anything in life (ie. we learn from life experiences). But the history of how we have come to know and understand basic engineering and mathematic fundamentals must never be lost. Our understanding of the vastness around us is only achieved by understanding the smallest of building blocks that we can prove to be constant. If we only look to the practical everyday solutions derived by the rules we have created to simplify the process to get there then we will miss out on the ideas and discoveries that make instrumental changes.

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

@pitthepig You wouldn't even pay attention to the video

You would not have paid attention

@Kevin Bugusky Trust me, the ones that said it was boring don't appreciate math yet but the ones who said nothing appreciated it the most. You probably inspired the quiet kids about the subject more than you think.

Unfortunately, the society can't afford to employ a doctor in math doubled by a psychologist, for every math lesson in school. You can do it for your children if your name is Bill Gates.

According to critical race theory, math is racist.

This -video- treasure should be present in every Maths class when introducing imaginary numbers (for me it was in 12th grade). Understanding the story behind the theorems is a brilliant complement to learning the uses of those same theorems. Maths would be even more interesting this way. But, anyway, I'm biased because I love history.

@XXX The teachers that just put on a video and clock out are the worst lmao don't kid yourself

@Masketta Man - The time invested in this video results in fundamental understanding that unlocks many blocks that students encounter. As a former math teacher decades ago, I would assign this video; the next class we would discuss it. I would then be nominated by my class as the greatest math teacher that ever lived. Thank you Brilliant!

I’m was never an amazing math student (I got through basic calculus) but I had no idea the quadratic equation had that visualization attached to it. Or the square root of 5 are the two side of a square that has an area of 5. It’s mad to me we go through algebra and geometry with almost no mention of how the two relates. This makes me wish I could retake algebra with the history in mind.

@John Wright Love that. All the best with your classes!

As said to another respondent : I am a math teacher and am teaching this topic this month. These videos did not exist eleven years ago when I last taught this topic. When i was young, we had it hammered in our heads that one not take the square root of a negative. But I love history too and am getting caught up in the idea that a person said: "well, let's work with the square root of -1 and see where it takes us" . My students don't get my enthusiasm about how incredible it is that someone (And then many after) kept building the ideas...connections to trigonometry and algebraic expansions. One of my students rolled his eyes and I said:"You're doing physics too, right? Well, wait until you see how this ties to your electricity unit." That got his attention.

The history of science/math is, to me, even more fascinating than the science and math itself. What we have learned is stunning. How we learned it is the greatest story ever told.

@Nosty Fripples Relativity has been proven many times,Tesla said that 100years ago,there was no technology to tell the difference between 1 second and 1.000000001 second

God must be a very complex man.

Same here, it amazes me how we went from mixing coloured liquids in a bottle, to knowing what is happening in it WITHOUT actually knowing what is happening, chemical equations

Tesla on the THEORY of relativity: “The theory, wraps all these errors & fallacies and clothes them in magnificent mathematical garb which fascinates, dazzles and makes people BLIND to the underlying errors. The theory is like a beggar clothed in purple WHOM IGNORANT PEOPLE take for a king. It’s exponents are very brilliant men, but they are meta-physicists rather than SCIENTISTS. ***NOT A SINGLE ONE OF THE RELATIVITY PROPOSITIONS HAS BEEN PROVED***”

@Sibler Inkognito E

19:48 "So when you're multiplying by i, what you're really doing is rotating by ninety degrees on the complex plane." Oh my gosh that was brilliant. So very well done. I had to stop right there to leave this compliment on a great job!

Well.. welcome to 11,12

@GaussianVectorⁿ because you can represent waves as e^ix it's really handy to work with because of the rules of exponentiation, i.e. a^b * a^c = a^(b+c)

There's no i in mathematics. Wait.... :o)

@anon video that's what I keep saying.

@Achtsekundenfurz I just see it as fancy labels for a 2 dimensional vector space. Exponentials are just a 4x4 matrix (first introduced with a zero row, but without one it takes you to a new space.) eⁱ•ᵖⁱ=-1 is equivalent to multiplication on the left by the following matrix with the unit "y" vector. | -1 0 | | 0 1 | Not as fancy, but it's all the same. I really don't know why people why physicist and others are to drawn to attaching the "equation form" instead of the vector form .

This was a fascinating insight into the origins of the mathematics that's so familiar. Wonderful. Thanks Derek!

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@West Explains Best if you want to explain the math, you might want to look at mathologer's video. Derek has more production quality but mathloger gives more mathematical insights.

I skipped four grades in school. I thought I was smart. Then I watched this video.

@kNautyyEspicyyy! Sounds like you're the one who's broken.

I've learned more from YouTubers like Mathologer, Vsauce, and Veritassium that I did through high school (specializing in Mathematics with Complex Numbers abs Physics) and a diploma of engineering (specializing in computers, electronics, and electrical).

I've always wondered why are we using "i" in math and physics. This makes lot more sense and cleared out several questions in my mind right from high school days. Was expecting some complex talk but this video is very clean and explains the history behind in a more cohesive way. Kudos to the team!

It's simply using two Real axes and using "i" to rotate in the constituted plane. For instance e^ix is just using sin on one axis and cos on the other, so you're using sin and cos at the same time. It has great usage in the Fourier transformation, because you wouldn't detect all the wave if it had inconvenient phase with only sin or cos, but while using both sin and cos at the same time, you don't have to worry about the phase of the analyzed wave anymore.

I've taken some semi-advanced math courses (from a non-math and physics standpoint) and I've dealt with a lot of these things, but never understood any of it. I just accepted that the math is supposed to make sense and this is how I solve it. I love that this paints a picture that makes a lot more sense to me. I could never do mathematical theories and proofs. Numerical analysis confused me as is.

@gbob99 that made things even worse somehow

@gbob99 You should definitely make a YouTube video to demonstrate how NOT to explain something. 🤣

I recommend you read the book Synergetics 1 and 2. It is a geomery that uses the tetrahedron as volumetric unity (1) and the cube is a volume 3. The number "i" is used because the cube is a 90 degree based system. The tetrahedron is a 60 degree cosmic neutral system. The edge length of the tetrahedron is the [sixth root of (9/8)] (synergetics 3D to 4D conversion constant) and is the hypotenuse of the cube.

I went through school and university not really understanding the 'why' of imaginary numbers - just really using them as a 'magic spell' that happens to work. I wish I'd been able to see this explanation twenty five years ago, it's brilliant.

Everyone talking about the history and math, meanwhile I am amazed by all the visuals and illustrations to make this clear to non-mathematicians! The art style and illustrations was absolutely amazing and fit perfectly with the storytelling! Overall, an amazing video! Thanks Youtube Algorithm! :D

That's veritasium for you. I really recommend you his other video like this "Math has a fatal flaw" I'm absolutely sure that you'll love it. More stories, visuals, inspirations, and aw, like this one video (i like it even more)

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.

@Nen Master5 ttyóyyytoy piggies 9ui9ityoti9y9i899⅞tt9ttuttititií9r7

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

I'm currently studying for the first time imaginary numbers in high school, our teacher is going really fast and I wasn't grasping the concept of imaginary numbers really well, but, thanks to this video I now understand them much more and I'm really exited to go to class to keep working with them :)

This is the type of math education that I personally love. In college, I took a Math for Liberal Arts class because I never did well with more complex math. It taught math by connecting it to its history and explaining it visually as you have done here. It truly taught me what math IS and not just numbers and equations. I LOVED that class and this video took that style of teaching to the next level. It genuinely makes me want to learn more complex math just to fully understand its connection to the visuals in the latter half of this video. Well done. Like truly, Bravissimo.

Tesla on the THEORY of relativity: “The theory, wraps all these errors & fallacies and clothes them in magnificent mathematical garb which fascinates, dazzles and makes people BLIND to the underlying errors. The theory is like a beggar clothed in purple WHOM IGNORANT PEOPLE take for a king. It’s exponents are very brilliant men, but they are meta-physicists rather than SCIENTISTS. ***NOT A SINGLE ONE OF THE RELATIVITY PROPOSITIONS HAS BEEN PROVED***”

If math isn't ideas primarily, then it is a truly awful topic. I am a math teacher of 27 years and my wife is a former primary teacher and now principal and curriculum leader. She always reminds me about my topic: "The whole thing has to be grounded in reality, every lesson."

Excellent video. Math was my favorite topic, all the way through engineering school. I have to say, I’m ashamed it took me almost 40 years and this video to finally understand the term “completing the square”. I was never taught about math in the early days. I always visualized early mathematicians writing papers, but using symbolic math. This is probably the best video on mathematics I’ve ever seen. Thank you.

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

This also made it click for me why it's called "completing the square"!

Mathematics and history were never so interesting. I was and am the dumbest student in my school on these two subjects. However couldn’t even dared to skip a second while watching this video. Absolutely loved it.

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.

@Chucky chuck which country

I absolutely despise the way I am being taught. When I learn something myself, I like to do that. It works better. But when my school "teaches" me, they just tell me equations that I don't even understand the essence of. It's completely idiotic

I'm literally crying right now remembering my 8th grade teacher. He had this same idea you described. And his methods where frowned upon by every teacher, student an parent. I just found it superb. From that year on I have grow to love mathematics, and I remember him every time I see a video like this, he has undoubtedly changed my mind forever. And for that I'm thankful. Veritasium should make this a video serie, guiding us through the history of math. It would be delightful.

This is so true

Well said. Applies to all school taught subjects. They expect people to just understand without knowing the origin

I wish it was presented to me this way when I first encountered it at math classes in EE uni. It would’ve made so much more sense back then! Great video!

I think this whole "imaginary numbers are weird and imaginary" thing makes them way more difficult to get your head around when you're first learning about them. It's not true - you can totally measure real world quantities and get complex numbers out of them, they map really well to quantities with a phase and an amplitude like waves and alternating currents in a circuit

Such a perfect example of the necessity of studying math. Something that begins as a purely mathematical concept winds up being an unreasonably effective explanation of physics. Just think of all the amazing connections we've yet to make.

I saw this video recommended by YT for a long time and I wasn't convinced to watch it because it's a bit too long. However, I'm glad that I did. It's an excellent video that explains how imaginary numbers lurked on math for centuries, why they had to be invented, how strange and counterintuitive they seem and how beautiful and necessary they really are.

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

but understanding is pretty confusing mr emotional damage

emotional damage

YOU FAILIURE, you should have learned this when you were not born yet. when I was your age, I was 27 years Old.

Math is RACIST according to BLM

when i was your age, I had to go up a mountain both ways to go to school

learning about acoustics made imaginary numbers a necessity. To my surprise I was able to watch university level lectures from MIT on quantum mechanics without batting an eyelash, which I do like all the time. It's is nice to see others feel the same as I do about imaginary numbers. Watching this video started me thinking about how to introduce these ideas to my son, who is a constant source of bewilderment. I hadn't thought of using negative numbers as a stepping stone to generalize out to the complex plane, thank you. Using the concept of negative area is fantastic.

I remember having to get over the existance of imaginary numbers when I was shown that the overdamped solution to a damped oscillator was solved using complex numbers. Maddening.

Derek, this is awesome. It almost brought me to tears. The history of humanity of discovering a mental construct of negative numbers, combine with someone (you) who has the passion to give it the thoughtful care and storytelling it deserves. Amazing. I was actually in the middle of writing a personal essay about how negative numbers must be a mental construct and not "exist" in the real world, but at the same time have gotten us to the moon. Really amazing video.

Best video yet! I wish they still taught math this way. context is everything, leaving it out is why people don't get math.

I've always been a maths freak myself and this is absolute treasure. A perfect veritasium video. With Love Regards Sharyar

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 🙂

@Miles Doyle please get a hobby...

Now rewrite that sentence without the word “interesting”.

Knowledge hiding to obtain Wealth and to secure survival is a well known practice even today. It is called Capitalism.

Understood ~20% but still captivated. Fascinating stuff, excellent production!

This is one of the most beautiful and delightful videos I've ever watched on YouTube in my life! Great script, great storytelling, great production and true love into it! The stories behind the brilliance of men and how they've contributed each and every one towards us getting what we have today is phenomenal! Every single person watching a YouTube video , perhaps, can't realise that even this that we take for granted today, is the hard ingenious work of great people that came before us. I feel myself blessed I live in a time and circumstance where I can so freely and easily have access to such great content like this video! Cheers and thank you so very much and keep up the good work!

I love one thing the engineering professors repeatedly told me about equations. Don’t memorize equations, but understand how the equations function.

Honestly this is the most engaging and most interesting video I’ve seen on mathematics. While I love math and find it so beautiful, I understand why others don’t like it. But that’s mostly because the teachers that teach math don’t really understand the beauty behind it, like the quadratic equation for example. They just give and say “this is what you do”, leaving very little room to question and freely think, something that is most fundamental to mathematicians, known as logicians before

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

Dude makes math sound absolutely riveting... Incredible

@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.

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He does - with help from the fact that - it is.

Like a lot of people on here, when I took advanced maths in school and was shown imaginary numbers, the course material made no attempt to describe the fundamentals of how imaginary numbers work, or why. This video in 15 mins made more sense than a year of schooling. Having this globally available on youtube is a gift to humanity

Brilliant ! I am impressed by the quality of each veritasium video I see, the clarity and depth of explanations, on different subjects rather far away from each other, the devices created for the explanations, such as here as the geometric figures on paper, but also the animations for volumes. Great quality videos. And I think a lot of work behind the scene. Thnaks !

Just made my day ! Brillant explanation ! I still have a smile in my face. When you know where problems come from, then you can understand the solution others found for them

Amazing video, Derek! This is, by far, the most engaging clip I've seen on your channel, and that's saying something given I have loved every single of your YT masterpieces! Thanks, and keep up the great work! 👏

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!

@hamidreza mahmoudian False. Complex numbers are the fundamental number system of algebra. To understand this process, you need to first look at how we get integers from real numbers and addition by way of looking at additive inverses. Then look at the way we get rational numbers by look at multiplicative inverses of integers. Then look at how we get algebraic numbers from those, and you'll come to see that in the end, complex numbers are where we end up from the natural operations and their inverses. Complex numbers can be represented through other means (but so can any number system), but complex numbers are both real and fundamental as much as any other number is. The fundamental theorem of algebra should be the first thing you look into. Yes, they are planar vectors, but they also possess a unique behavior under multiplication, and it's this behavior that shows they're more fundamental than the biased view of them you're presenting. In order to represent them as vectors, you need to introduce a special operation for multiplication that's derived from the behavior of how sqrt(-1) acts under our typical understanding of multiplication. This new vector multiplication, distinct from any other multiplicative operations on vectors, cannot be derived from pairs of real numbers alone. It can only be derived from observing how the behavior of sqrt(-1) behaves under our traditional concept of multiplication. Since you cannot derive the behavior of complex numbers simply from pairs of real numbers, it follows that they are not "basically nothing but planar vectors that make two variable real calculations more compact" because it dismisses the entire field of complex dynamics, which only follows from the how sqrt(-1) behaves under multiplication.

@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.

Maths should be taught as a story not as a language. I literally had my attention all towards what he was telling, it was an amazing experience. Thanks Veritasium!

Thanks for your hard work and quality content ! This is a type of material that should be tough in universities combining math, history and physics !

A daft question from someone who failed high school maths: are imaginary and unreal numbers the same thing?

I've been studying chemical process engineering and this video is just insane! I loved it and I hope you'll do more of those! Cheers!

If math had been explained to me like this in school I would have actually remembered it

If math had been explained to me like this in school, I would have forgotten math and physics and would have gone straight to metaphysics. 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.

@asder alt the problem isn’t learning it, the problem is trying to absorb it into memory

​@Fraken COB's The way you described it and explained it, I was completely lost. If teachers bring in fancy words, kids will definitely lose it. The best way to start is with pictures and objects. In lower grades, every concept should be taught in 3~8 different ways in math. Not everyone makes the connection in the same way. It's not going to be the end of the world if by 4th grade the child only knows how to do addition, subtraction, multiplication and division. I was going through grade 5 stuff on Khan Academy, there is a fraction topic. The student always gets it right, I always get it wrong. We just understand and see the answer differently. That is why one way is not enough, if only that way was taught, I would run away from math. I think your idea about STEM subjects is valid. People are introducing complex details in lower grades and repeating them in higher grades. Most children intrinsically understand microeconomics. Just add the terms in a fun way, and not suck the joy out of it. I love designing on computers because I suck at drawing by hand. But I can use equations and use lines to make fascinating designs. Sadly teachers do not encourage such things or teach it in an interesting way. We just need to provide the tools, a bit of understanding and kids these days will fill in the blanks themselves if you get them interested. Otherwise, we just create robots sadly, who get a rude awakening when they join the job market. They will feel very inadequate and ill prepared.

​@asder alt Yup. :)

@Fraken COB's Totally lost on the Arabic numbering system, does this mean in the very beginning when we learn to add and subtract, carry things over once they cross a 10 value? Like how 11 + 9 = 20, 9 + 1 = 10, we place a zero there, carry the one over, 1+1 = 2, that is why we get 20??

Loved the presentation!! Wish I'd been introduced to quadratic and cubic equations this way in grade 8-9 ... I didn't start getting interested in math until grade 11 (until then I saw it just as a chore... to pass and move onto the next grade.)

The "old way" to look at imaginary numbers, was very informative. Thank you for sharing.

The interesting thing is that all numbers are actually imaginary, at least in the sense of being abstract and man-made and thus not part of tangible reality but related to it as all abstractions are not tangible. Numbers are conceptio-abstractions which have become so common that we consider them to be tangible realities, if we take a moment to reflect we will realize all numbers are mental creations.

This is freaking pure gold. I love this guy, can’t get enough of him.

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

This is the most awesome video ever! I took several quantum mechanic courses over 20 years ago. Never encountered a presentation so clear and impactful as this.

Had I been taught this way in my school days, I would have been at peace with myself till now, but seems it's never too late🌼 Thanks a ton😊

I don't remember too much about Math, and my studies in school. You always explain everything in such a simple way, and with such a pleasant voice. Thank you for your content.

I remember working with complex numbers for the first time in school and yes, I was stunned at first too cause taking a square root of any negative number is calculator will result in an error but when I found out that the same negative square root can be solved on paper with the help of "i", i was blown away. :D

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.

@Good Goyim Lol what is stopping the "free market" from hiring people like Tartaglia ? You're a jobless conspiracist and hate people with a degree because you can't take responsability for being a failure. Your language "cringe" "woke" sets your intellectual prowesses.

@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.

@N.O.S.T.I.A Oh! Here comes the irrational intolerance of an other ignoramous who thinks she knows it all!

@GrayFoxHound9 One needs a solution once you have the key and the cypher. You still need to unscramble the code.

This was so amazing (as usual). That is why at school when the teacher taught about equations just as a template to plug in values and everybody seemed to be satisfied with no visual and intuitive explanation mentioning of geometry at all, I always felt something fundamental is missing and became frustrated. It was posed as if these concepts dawned upon' mathematicians, but in reality it is an achievable step wise process! I just had a thought ,can AI ever exceed the creativity of mathematicians ?? Like the most advanced AI possible, could it have ever discovered imaginary numbers as a bridge to complete the task in its utility function ? Thank You !

I did years of math including PDE's and never paused to appreciate any of it like I did this. Makes me want redo again with a fresh perspective. Brilliant. Thank you 🙏

These lectures are simply “brilliant” not only in content but, supremely, in presentation!

This is an excellent video on this topic. For those interested in a more detailed foray into this subject I highly recommend the following book. An Imaginary Tale: The Story of the Square Roof of -1; Princeton University Press, 1998 I was surprised the book wasn't listed in the references, because it not only presents the history, but extensively delves into the background, theory, development and uses of the math. However, it's written in a way anyone with a basice high school math background can understand and follow. If the video has whetted your appetite on this subject the book will serve as a main course.

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.

This is incredible. I wish we were taught this when we learned about imaginary numbers back in high school. Amazing!

By explaining the origins of imaginary numbers you have made math more real for ordinary people like me. You sir are legendary.

Wish I had this to show my HS students before I retired. Absolutely awesome video. Thanks.

Beautiful mathematics and even physics, presented beautifully! Thank you so much! (Dr. Michael W. Ecker, PhD mathematician, retired PSU mathematics professor)

Derek is about to take over the YouTube history scene, isn’t he? 😅

@Vikram Singh Lol, I wouldn’t be surprised at all

Better than girls twerking

@Josiah the NF and if he doesn't he can always make a video on 'Youtube algo has a fatal flaw.'

No joke, there should be a science history channel

He’s capable enough, that’s a given 👌

This is amazing. I always thought complex numbers were merely a theoretical mind game fancied by mathematicians. But they Do have everything to do with reality as we know it!

I understood part of this, having gone through MATH 101 in college many years ago (getting a 4.0), but actually enjoyed watching this even without really getting it all! Haha.

Thanks for showing me how these 'names' I've heard about from mathematics relate to each other. You make the history of math come alive. Thanks

14:50 imagine drawing the square with part of it outside the piece of paper. as in, if you have you X and Y axis, draw the square on it and move the axis around. the square is 5/5 but the positive part of the diagram only shows 4/4 so there is a side that is there, it's just in the negative part of the axis. You can see it and you can draw it, its just that you measure it not from the corner but offset by a fifth on both axis.

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.

@NEprimo check out Welch lab's Imaginary Numbers Are Real playlist (Derrick has it in the description). It goes even further into Imaginary Numbers and it's some of the best mathematical educational content I have seen

@Hyperduality slayyyyy✨

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

I absolutely love your stuff. This is how maths should be taught. Thank you so much.

I study physics and never thought about it, it's really interesting. I love it, thank you!

I successfully found my most favorite video on YouTube. Blessed to be here by the curiosity to know more about the science of complex numbers (imaginary numbers) and it's so satisfying when all your questions are perfectly answered. Thanks a ton Veritasium for such a helpful and amazing content. ❤️

A description of the concepts as brilliant as the concepts themselves. I simply cannot get enough of these videos, and have made them a native function of our family movie/YouTube nights.

Wow, this is such an incredible presentation of imaginary numbers. I am surprised that you didn't mention it's use by Einstein in landing Special Relativity, so indeed it seems to be a fundamental aspect of reality, not just in quantum mechanics.

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

@Charly Taylor Not in any resource with an example I've seen. But clearly this is pointless then. I saw one article claiming it and a dozen saying no, and the one claiming it has no examples, so odds are just one stupid writer tbh.

@Xavier Magnus I did look it up. It said a number can be a noun.

@Charly Taylor Yes... it's not something on it's own. It's a description of other properties. Save yourself time and just look it up. This is exactly why I said schools approach it wrong. They try to remove it from abstract, and thus lose why imaginary numbers etc work. Because you can describe things in odd ways, and come back to reality after. Try this, can you have - 4? Nope, you can owe 4 of something. But without context it has no more meaning than green, which is a property of something else.

@Xavier Magnus OK,,, but if someone says 'what's 2 + 2' and they say '4', are they saying an adjective?

Thanks for pointing out the cubicles as means of cubes. Then imaginaries are of course just a new dimension. This was eye opening!

Solving algebraic equations using geometry seems fascinating. I know the video is about how we moved algebra away from geometry, but I feel students must be taught solving equations by drawing shapes. This will help them understand and then later they will be comfortable in handling complex equations. I feel this will ease many students fear of maths.

Great video, since I am studying electrical engineering, I have to deal with imaginary numbers a lot (btw we use j instead of i to avoid confusion, since i is used for time-dependent currents). I always accepted imaginary numbers as sth I just have to deal with. Never understood where it came from and why it was needed and especially why it works out, for me the definition of i=sqrt(-1) always seemed quite arbitrary. But your video shed some light into this, thx for that.

Mathematics is such an exquisitely beautiful subject…. I never get tired of playing with numbers, shapes and formulas.

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.

O my god. What a class it is. I never received such a lecture as before. Love you sir.

Fantastic video Derek! Engaging and every informative both technically and historically, love it!

Thanks so much on making this video, it is so impressive and helps me a lot for understanding the meaning of complex solutions of wave function.

Great information! I had forgotten so very much from my days at university! Thank you!

Woooow that was amazing, to learn the story of imaginary numbers and actually understanding Euler's formula. Amazing video!

PERO MADRE MIA WILLY QUE HACES AQUI COMPAÑERO

@Hyperduality Isomorphisms are literally homomorphisms. Real numbers are entirely contained in the complex numbers. You seem like you came up with an idea and started combining words you don't understand to try and pretend you have some profound insight.

@タメル c'est pas ouf de généraliser tout un peuple à une seule chose, la nuance c'est bien... Aussi j'ai beacoup ri en voyant ton pseudo mdr

@Selling Soul 5 Bucks oh yeah, and so... what was dumb in it...? By the way, not every comment under a video is the most interesting, and that's normal, youtubers can obviously be part of those comments, you would ignore classic comments, but you don't with checkmarks, the difference just is that Youtube put those comments higher in everyone's reccomendations so they tend to get more popular even if they are not extremely interesting. And that's not really the Youtuber's fault, they're just casually commenting like us and not really expecting much, yet YT might give them attention. Doesn't apply to this comment though, nothing dumb in here.

So many reject Science, it makes me sad. They will never know what they missed out.

This deserves being taught in schools Congratulations on putting this into a descriptive video!

After this wonderful thought provoking video , I immediately signed up for more Brilliant presentations! Bravo!!

The ancient (practical) way of solving mathematical equations was (is) interesting but also very convincing. I think today also the same way of explaining mathematics should be adapted in schools for a much better understanding of mathematics and making it easier to grasp.

One of the very best Veritasium videos in a long time. This was wonderful.

We need a Netflix series based on Math history.

​@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 ^-^

@Dexter Westin no you cannot steal knowledge, intellectual property doesnt exist.

​@Good Goyim How do you steal the dissemination of knowledge? Was algebra stolen from the Arabs? Were Newton's laws stolen? What about Descartes' Meditationes de Prima Philosophia?

Hands down man for the explanation, it seems I am reading a short introduction book, so detailed!

Wow, if all of my math classes were like this at school i guess i would have understood a lot more of it.

I remember an epiphany I had in university on an equation of electromagnetic waves in different mediums, where (if I recall correctly) and certain parameter (k?) was positive in air, water, but negative in metals. That parameter was in a square root in a wave equation. Plug that into the e^i*sqrt(k)*x equation and it would generate a wave in air/water because it had an imaginary exponential, but an exponential decrease in metal because it was real (e^-x). So an EM wave cannot exist in metals (or materials with negative k) and incoming light rays have to be reflected. WHICH IS WHY METALS ARE SHINY! Completely blew my mind. But it has been too long for me to find that equation and parameter again. If this makes sense - please make a video to explain why metals are shiny and water is translucent, as derived from the wave equation!

For exploring nature, often a change in perspective may be enough to lead to great discoveries as you had shown in your brilliant video. When it comes to physics and complex or irrational numbers, more wonders await. One example is: some guy also looked from a different perspective on physics and found a formalism to calculate (yes: calculate) particle masses. Since its publication back in the 1980's, the deviations between theoretical and measured results (e.g. electron mass) are still below the per mill range (and smaller), which is fantastic. In this derivation, he found an approach to calculate the fine structure constant on the base of the number pi alone. So: a fundamental aspect of reality which quantifies the strength of the electromagnetic interaction between elementary charged particles is based on Pi alone. How strange is that? ;)

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.

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.

As one who absolutely hated math in school, I can very much agree to your statement, this, was infinitely more interesting than anything that was thaught to me in school.

My teacher actually did this last week and i hated every part of it lol

Thank you for making this effort to feed our minds and inspire our souls, this channel is magnificent

Could this be applied to solving a tetrahedral root? There must be some formula that could extract a root from a tetrahedral number like 1, 4, 10, 20, etc;

the efforts behind making this video is truly appreciable to greater levels

I never grasp the imaginary numbers till now. It's a beautiful explaination and will help students appreciate the value of i.

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....

Fantastic video. Much appreciate knowing the why behind things then just memorizing an equation. Do more! :)

Was this told at introduction of Complex numbers, I would have loved mathematics much more. I have always searched for physical interpretation of most math equations and tried to tie them to basic geometrical understanding. I am speechless on how much work was done, how many lives have been spent to reach this point.

Excellent explanation. Thanks, I am going through complex numbers with my children and this video makes it easier.

Derek, very possibly one of the best videos on this or any mathematical subject. This should be shown in every Algebra class. And it should win some sort of award.