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

Vor 2 Monate

Ansh Daheriya

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

Vor 25 Tage

Rizal Purnawan

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

Vor Monat

Tom Rhodes

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.

Vor Monat

Elizabeth Moore

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

Vor Monat

Jacob McAfee

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.

Vor Monat

thispurplefox

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

Vor Monat

Padoru

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

Vor 3 Tage

Padoru

You would not have paid attention

Vor 3 Tage

Bows

@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.

Vor 5 Tage

Andreas Klein

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.

Vor 7 Tage

Matt Augusto

According to critical race theory, math is racist.

Vor 8 Tage

João Soares

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.

Vor Monat

Masketta Man

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

Vor 8 Tage

XXX

@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!

Vor 8 Tage

Adam Trotter

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.

Vor Monat

João Soares

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

Vor Monat

John Wright

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.

Vor Monat

Russell Cohen

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.

Vor Monat

Pranab Mallick

@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

Vor Tag

Kain_niaK

God must be a very complex man.

Vor 2 Tage

Hardaat Singh Baath

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

Vor 5 Tage

Nosty Fripples

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***”

Vor 23 Tage

Erhan 12

@Sibler Inkognito E

Vor 27 Tage

c.w. B O G U S

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!

Vor Monat

Surendra I

Well.. welcome to 11,12

Vor Tag

Jorian Weststrate

@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)

Vor 2 Tage

Trinka Buszczuk

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

Vor 2 Tage

GaussianVectorⁿ

@anon video that's what I keep saying.

Vor 5 Tage

GaussianVectorⁿ

@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 .

Vor 5 Tage

DoS - Domain of Science

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

Vor 2 Monate

OjoRojo40

FAKE NEWS CHANNEL! BE AWARE, IT"S SPONSORED BY BIG CORP.

Vor 2 Monate

TNB1

@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.

Vor 2 Monate

Jeff Ward

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

Vor 2 Monate

Nerdelbaum Frink

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

Vor 2 Monate

Brendon Green

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).

Vor 2 Monate

Gangadharan Esakki

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!

Vor Monat

Zu Fex

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.

Vor 28 Tage

John Le

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.

Vor Monat

StarBurst Flame

@gbob99 that made things even worse somehow

Vor 5 Stunden

Glen MacDonald

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

Vor Monat

gbob99

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.

Vor Monat

Chris Sugg

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.

Vor Monat

Ash U

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

Vor Monat

Carl Alfredo

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)

Vor 18 Tage

Factnomenal

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

Vor 2 Monate

Ioe Aspirant

Yes ,you should

Vor Monat

Tom Rhodes

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.

Vor Monat

cfcreative

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

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

Vor 2 Monate

PolyRaptor

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 :)

Vor Monat

David

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.

Vor Monat

Nosty Fripples

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***”

Vor 23 Tage

John Wright

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."

Vor Monat

Charles Crawford

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.

Vor Monat

Ansh Daheriya

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

Vor 25 Tage

pagw2002

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

Vor Monat

gaurav kumar

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.

Vor Monat

Akira Leelarapin

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.

Vor 2 Monate

dude knowledge

@Chucky chuck which country

Vor 3 Tage

Westle the Great

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

Vor 8 Tage

Erik Brito Magalhães

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.

Vor 8 Tage

Payton J

This is so true

Vor 10 Tage

harry

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

Vor 11 Tage

Sophoclis K

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!

Vor Tag

Chronosaur

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

Vor Monat

TorA

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.

Vor Monat

Mariano P

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.

Vor Monat

Steven He

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

Vor 2 Monate

PIYUSH TEACHER

but understanding is pretty confusing mr emotional damage

Vor 8 Stunden

Brian Quin

emotional damage

Vor 5 Tage

Nonsense games

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

Vor 8 Tage

Matt Augusto

Math is RACIST according to BLM

Vor 8 Tage

Nonsense games

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

Vor 9 Tage

Git Yer Git

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.

Vor Monat

Louis Penn

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.

Vor Monat

James Younger, DDS

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.

Vor 8 Tage

Anthony b

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

Vor Monat

Sheena Abdul Basit

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

Vor Monat

Scenic Fights

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

Vor 2 Monate

Kep Spark

Yep.

Vor 13 Tage

Guna selan guna

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 🙂

Vor 2 Monate

V?bez

@Miles Doyle please get a hobby...

Vor 2 Monate

MLG35U5 10/10 ign rated gr8m88/8

Now rewrite that sentence without the word “interesting”.

Vor 2 Monate

Reasoner Enlightened

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

Vor 2 Monate

James Fleming

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

Vor Monat

nsfa19

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!

Vor Monat

Isai Ayala

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

Vor Monat

Lou Marz

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

Vor Monat

Ansh Daheriya

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

Vor 25 Tage

izzynobre

Dude makes math sound absolutely riveting... Incredible

Vor 2 Monate

Parker Moss

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

Vor Monat

Tom Rhodes

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.

Vor Monat

ForestCity Fishing

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

Vor 2 Monate

OjoRojo40

FAKE NEWS CHANNEL! BE AWARE, IT"S SPONSORED BY BIG CORP.

Vor 2 Monate

Adrian Bartholomew

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

Vor 2 Monate

Andy Badke

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

Vor Monat

Alain Péan

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 !

Vor Monat

Angel Luis Villanueva

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

Vor Monat

Ali Mojebi

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! 👏

Vor Monat

CuriousMarc

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.

Vor 2 Monate

Queta Arbuste

@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.

Vor Monat

CuriousMarc

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

Vor 2 Monate

Kurt Nowak

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

Vor 2 Monate

Nerdelbaum Frink

@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.

Vor 2 Monate

Nerdelbaum Frink

@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.

Vor 2 Monate

MetalSolid

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!

Vor Monat

Levon Poghosyan

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 !

Vor Monat

Peter Laubscher

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

Vor 4 Stunden

Jérémy FAVERJON

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!

Vor Monat

Jabroney

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

Vor 2 Monate

Tom Rhodes

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.

Vor Monat

Ramphys

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

Vor Monat

asder alt

@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.

Vor Monat

Fraken COB's

@asder alt Yup. :)

Vor Monat

asder alt

@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??

Vor Monat

David Henningson

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.)

Vor Monat

Aage152

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

Vor Monat

Andrew Switzer

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.

Vor 2 Tage

mofu mofu

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

Vor Monat

Josh Hampton

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.

Vor 2 Monate

Queta Arbuste

@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

Vor Monat

Tom Rhodes

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.

Vor Monat

Arthuria Pendragon

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 🤣

Vor Monat

Queta Arbuste

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

Vor Monat

Queta Arbuste

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

Vor Monat

Kenneth Brown

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.

Vor Monat

wittySpace

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😊

Vor 24 Tage

Zeek

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.

Vor Monat

Kamil Khan Kakar

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

Vor Monat

FinFET

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.

Vor 2 Monate

IrokoSalei

@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.

Vor Monat

GrayFoxHound9

@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).

Vor Monat

Tom Rhodes

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.

Vor Monat

88marome

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

Vor Monat

Jesus Christ

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

Vor Monat

Vedant Kohli

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 !

Vor 25 Tage

frank

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 🙏

Vor Monat

matthew fisher

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

Vor 5 Tage

Jabari Zakiya

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.

Vor Monat

PuzzLEGO

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

Vor 2 Monate

Supertron

I always try my best to ponder that. :)

Vor Monat

Numbzie

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

Vor Monat

Chinmayee Lenka

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

Vor Monat

Tom Gilson

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.

Vor 2 Monate

Alec Xander

@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.

Vor 2 Monate

James N

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

Vor Monat

Sanjith David

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

Vor 2 Tage

MrSimmies

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

Vor Monat

Dr Mike Ecker

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

Vor Monat

History Dose

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

Vor 2 Monate

Josiah the NF

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

Vor 2 Monate

tofucube

Better than girls twerking

Vor 2 Monate

Vikram Singh

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

Vor 2 Monate

Supreet Sahu

No joke, there should be a science history channel

Vor 2 Monate

Josiah the NF

He’s capable enough, that’s a given 👌

Vor 2 Monate

Jehag2

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!

Vor Monat

Stephen DeVore sadsongsishere

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.

Vor Monat

Max. Sanches

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

Vor Monat

Diva_4720

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.

Vor Monat

TheFinalRevelation

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.

Vor 2 Monate

Tom Rhodes

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.

Vor Monat

Nerdelbaum Frink

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

Vor 2 Monate

C Huang

@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

Vor 2 Monate

Indorap -YT

@Hyperduality slayyyyy✨

Vor 2 Monate

TheFinalRevelation

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

Vor 2 Monate

Mark Newman

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

Vor 20 Tage

Marie Sourroubille

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

Vor Monat

Awais Qureshi

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. ❤️

Vor 28 Tage

KJ

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.

Vor Monat

Mose

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.

Vor 2 Monate

Ansh Daheriya

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

Vor 25 Tage

Xavier Magnus

@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.

Vor Monat

Charly Taylor

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

Vor Monat

Xavier Magnus

@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.

Vor Monat

Charly Taylor

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

Vor Monat

Julien Beck

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

Vor Monat

Avinash Heral

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.

Vor 5 Tage

Robin Khn

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.

Vor 26 Tage

Tristram Sidey

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

Vor Monat

Justin

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.

Vor 2 Monate

Tom Rhodes

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.

Vor Monat

Andromeda Galaxy

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.

Vor 2 Monate

John Bjorkman

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.

Vor 2 Monate

Nerdelbaum Frink

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

Vor 2 Monate

Mikey P

I struggled to get the cover off my graphing calculator.

Vor 2 Monate

Chinmoy Mondal

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

Vor Monat

metalhead2550

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

Vor Monat

Bully Bored

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.

Vor 5 Tage

thesagefoxbat

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

Vor Tag

Shadoune666

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

Vor 2 Monate

Ezequiel Angelucci

PERO MADRE MIA WILLY QUE HACES AQUI COMPAÑERO

Vor 2 Monate

Nerdelbaum Frink

@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.

Vor 2 Monate

GoldEEr

@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

Vor 2 Monate

GoldEEr

@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.

Vor 2 Monate

Nen Master5

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

Vor 2 Monate

Andrei on Bikes

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

Vor Monat

Thomas Schwarz

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

Vor Monat

Shripad Kulkarni

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.

Vor Monat

Whisky Guzzler

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

Vor Monat

Raided Cluster

We need a Netflix series based on Math history.

Vor 2 Monate

Good Goyim

@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.

Vor Monat

Tom Rhodes

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.

Vor Monat

Krishna Soni

@Clogica like most of us ^-^

Vor Monat

Good Goyim

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

Vor Monat

Dexter Westin

@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?

Vor Monat

Shyam Patadia

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

Vor 12 Tage

TomatoLove

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

Vor Monat

Jasper Klewer

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!

Vor 16 Tage

sledge721

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? ;)

Vor 20 Tage

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.

Vor 2 Monate

Tom Rhodes

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.

Vor Monat

Robert Fox

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

Vor 2 Monate

Martin Gutsch

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.

Vor 2 Monate

JonzuihGaming

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.

Vor 2 Monate

Carlos Goytia

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

Vor 2 Monate

MrPeak0

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

Vor 25 Tage

Randolph Hall

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;

Vor 3 Tage

Satish Kannaiyan

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

Vor Monat

joeboxervr6

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

Vor Monat

Medlife Crisis

One of your best ever videos. Wonderful stuff.

Vor 2 Monate

Cubert Miso

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

Vor 2 Monate

Antoine Olivier

Agree

Vor 2 Monate

Mohamed Flaifel

Just amazing as usual ❤

Vor 2 Monate

Affectionate ASMR

ooo you are the best)

Vor 2 Monate

K K

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

Vor 2 Monate

dasfx

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

Vor 4 Tage

Venkatesh Pulakhandam

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.

Vor Monat

Marcos

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

Vor Monat

Matthew Harvell

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.

## KOMMENTARE

## Dr. Trefor Bazett

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

Vor 2 Monate## Ansh Daheriya

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

Vor 25 Tage## Rizal Purnawan

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

Vor Monat## Tom Rhodes

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.

Vor Monat## Elizabeth Moore

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

Vor Monat## Jacob McAfee

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.

Vor Monat## thispurplefox

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

Vor Monat## Padoru

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

Vor 3 Tage## Padoru

You would not have paid attention

Vor 3 Tage## Bows

@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.

Vor 5 Tage## Andreas Klein

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.

Vor 7 Tage## Matt Augusto

According to critical race theory, math is racist.

Vor 8 Tage## João Soares

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.

Vor Monat## Masketta Man

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

Vor 8 Tage## XXX

@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!

Vor 8 Tage## Adam Trotter

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.

Vor Monat## João Soares

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

Vor Monat## John Wright

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.

Vor Monat## Russell Cohen

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.

Vor Monat## Pranab Mallick

@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

Vor Tag## Kain_niaK

God must be a very complex man.

Vor 2 Tage## Hardaat Singh Baath

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

Vor 5 Tage## Nosty Fripples

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***”

Vor 23 Tage## Erhan 12

@Sibler Inkognito E

Vor 27 Tage## c.w. B O G U S

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!

Vor Monat## Surendra I

Well.. welcome to 11,12

Vor Tag## Jorian Weststrate

@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)

Vor 2 Tage## Trinka Buszczuk

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

Vor 2 Tage## GaussianVectorⁿ

@anon video that's what I keep saying.

Vor 5 Tage## GaussianVectorⁿ

@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 .

Vor 5 Tage## DoS - Domain of Science

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

Vor 2 Monate## OjoRojo40

FAKE NEWS CHANNEL! BE AWARE, IT"S SPONSORED BY BIG CORP.

Vor 2 Monate## TNB1

@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.

Vor 2 Monate## Jeff Ward

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

Vor 2 Monate## Nerdelbaum Frink

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

Vor 2 Monate## Brendon Green

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).

Vor 2 Monate## Gangadharan Esakki

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!

Vor Monat## Zu Fex

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.

Vor 28 Tage## John Le

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.

Vor Monat## StarBurst Flame

@gbob99 that made things even worse somehow

Vor 5 Stunden## Glen MacDonald

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

Vor Monat## gbob99

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.

Vor Monat## Chris Sugg

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.

Vor Monat## Ash U

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

Vor Monat## Carl Alfredo

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)

Vor 18 Tage## Factnomenal

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

Vor 2 Monate## Ioe Aspirant

Yes ,you should

Vor Monat## Tom Rhodes

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.

Vor Monat## cfcreative

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.

Vor Monat## Parry 255

@Nen Master5 ttyóyyytoy piggies 9ui9ityoti9y9i899⅞tt9ttuttititií9r7

Vor Monat## Bôrgïr

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

Vor 2 Monate## PolyRaptor

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 :)

Vor Monat## David

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.

Vor Monat## Nosty Fripples

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***”

Vor 23 Tage## John Wright

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."

Vor Monat## Charles Crawford

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.

Vor Monat## Ansh Daheriya

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

Vor 25 Tage## pagw2002

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

Vor Monat## gaurav kumar

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.

Vor Monat## Akira Leelarapin

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.

Vor 2 Monate## dude knowledge

@Chucky chuck which country

Vor 3 Tage## Westle the Great

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

Vor 8 Tage## Erik Brito Magalhães

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.

Vor 8 Tage## Payton J

This is so true

Vor 10 Tage## harry

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

Vor 11 Tage## Sophoclis K

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!

Vor Tag## Chronosaur

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

Vor Monat## TorA

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.

Vor Monat## Mariano P

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.

Vor Monat## Steven He

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

Vor 2 Monate## PIYUSH TEACHER

but understanding is pretty confusing mr emotional damage

Vor 8 Stunden## Brian Quin

emotional damage

Vor 5 Tage## Nonsense games

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

Vor 8 Tage## Matt Augusto

Math is RACIST according to BLM

Vor 8 Tage## Nonsense games

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

Vor 9 Tage## Git Yer Git

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.

Vor Monat## Louis Penn

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.

Vor Monat## James Younger, DDS

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.

Vor 8 Tage## Anthony b

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

Vor Monat## Sheena Abdul Basit

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

Vor Monat## Scenic Fights

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

Vor 2 Monate## Kep Spark

Yep.

Vor 13 Tage## Guna selan guna

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 🙂

Vor 2 Monate## V?bez

@Miles Doyle please get a hobby...

Vor 2 Monate## MLG35U5 10/10 ign rated gr8m88/8

Now rewrite that sentence without the word “interesting”.

Vor 2 Monate## Reasoner Enlightened

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

Vor 2 Monate## James Fleming

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

Vor Monat## nsfa19

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!

Vor Monat## Isai Ayala

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

Vor Monat## Lou Marz

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

Vor Monat## Ansh Daheriya

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

Vor 25 Tage## izzynobre

Dude makes math sound absolutely riveting... Incredible

Vor 2 Monate## Parker Moss

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

Vor Monat## Tom Rhodes

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.

Vor Monat## ForestCity Fishing

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

Vor 2 Monate## OjoRojo40

FAKE NEWS CHANNEL! BE AWARE, IT"S SPONSORED BY BIG CORP.

Vor 2 Monate## Adrian Bartholomew

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

Vor 2 Monate## Andy Badke

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

Vor Monat## Alain Péan

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 !

Vor Monat## Angel Luis Villanueva

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

Vor Monat## Ali Mojebi

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! 👏

Vor Monat## CuriousMarc

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.

Vor 2 Monate## Queta Arbuste

@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.

Vor Monat## CuriousMarc

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

Vor 2 Monate## Kurt Nowak

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

Vor 2 Monate## Nerdelbaum Frink

@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.

Vor 2 Monate## Nerdelbaum Frink

@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.

Vor 2 Monate## MetalSolid

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!

Vor Monat## Levon Poghosyan

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 !

Vor Monat## Peter Laubscher

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

Vor 4 Stunden## Jérémy FAVERJON

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!

Vor Monat## Jabroney

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

Vor 2 Monate## Tom Rhodes

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.

Vor Monat## Ramphys

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

Vor Monat## asder alt

@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.

Vor Monat## Fraken COB's

@asder alt Yup. :)

Vor Monat## asder alt

@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??

Vor Monat## David Henningson

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.)

Vor Monat## Aage152

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

Vor Monat## Andrew Switzer

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.

Vor 2 Tage## mofu mofu

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

Vor Monat## Josh Hampton

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.

Vor 2 Monate## Queta Arbuste

@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

Vor Monat## Tom Rhodes

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.

Vor Monat## Arthuria Pendragon

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 🤣

Vor Monat## Queta Arbuste

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

Vor Monat## Queta Arbuste

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

Vor Monat## Kenneth Brown

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.

Vor Monat## wittySpace

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😊

Vor 24 Tage## Zeek

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.

Vor Monat## Kamil Khan Kakar

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

Vor Monat## FinFET

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.

Vor 2 Monate## IrokoSalei

@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.

Vor Monat## GrayFoxHound9

@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).

Vor Monat## Tom Rhodes

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.

Vor Monat## 88marome

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

Vor Monat## Jesus Christ

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

Vor Monat## Vedant Kohli

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 !

Vor 25 Tage## frank

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 🙏

Vor Monat## matthew fisher

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

Vor 5 Tage## Jabari Zakiya

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.

Vor Monat## PuzzLEGO

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

Vor 2 Monate## Supertron

I always try my best to ponder that. :)

Vor Monat## Numbzie

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

Vor Monat## Chinmayee Lenka

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

Vor Monat## Tom Gilson

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.

Vor 2 Monate## Alec Xander

@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.

Vor 2 Monate## James N

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

Vor Monat## Sanjith David

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

Vor 2 Tage## MrSimmies

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

Vor Monat## Dr Mike Ecker

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

Vor Monat## History Dose

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

Vor 2 Monate## Josiah the NF

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

Vor 2 Monate## tofucube

Better than girls twerking

Vor 2 Monate## Vikram Singh

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

Vor 2 Monate## Supreet Sahu

No joke, there should be a science history channel

Vor 2 Monate## Josiah the NF

He’s capable enough, that’s a given 👌

Vor 2 Monate## Jehag2

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!

Vor Monat## Stephen DeVore sadsongsishere

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.

Vor Monat## Max. Sanches

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

Vor Monat## Diva_4720

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.

Vor Monat## TheFinalRevelation

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.

Vor 2 Monate## Tom Rhodes

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.

Vor Monat## Nerdelbaum Frink

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

Vor 2 Monate## C Huang

@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

Vor 2 Monate## Indorap -YT

@Hyperduality slayyyyy✨

Vor 2 Monate## TheFinalRevelation

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

Vor 2 Monate## Mark Newman

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

Vor 20 Tage## Marie Sourroubille

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

Vor Monat## Awais Qureshi

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. ❤️

Vor 28 Tage## KJ

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.

Vor Monat## Mose

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.

Vor 2 Monate## Ansh Daheriya

## Xavier Magnus

@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.

Vor Monat## Charly Taylor

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

Vor Monat## Xavier Magnus

@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.

Vor Monat## Charly Taylor

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

Vor Monat## Julien Beck

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

Vor Monat## Avinash Heral

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.

Vor 5 Tage## Robin Khn

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.

Vor 26 Tage## Tristram Sidey

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

Vor Monat## Justin

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.

Vor 2 Monate## Tom Rhodes

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.

Vor Monat## Andromeda Galaxy

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.

Vor 2 Monate## John Bjorkman

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.

Vor 2 Monate## Nerdelbaum Frink

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

Vor 2 Monate## Mikey P

I struggled to get the cover off my graphing calculator.

Vor 2 Monate## Chinmoy Mondal

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

Vor Monat## metalhead2550

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

Vor Monat## Bully Bored

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.

Vor 5 Tage## thesagefoxbat

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

Vor Tag## Shadoune666

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

Vor 2 Monate## Ezequiel Angelucci

PERO MADRE MIA WILLY QUE HACES AQUI COMPAÑERO

Vor 2 Monate## Nerdelbaum Frink

@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.

Vor 2 Monate## GoldEEr

@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

Vor 2 Monate## GoldEEr

@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.

Vor 2 Monate## Nen Master5

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

Vor 2 Monate## Andrei on Bikes

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

Vor Monat## Thomas Schwarz

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

Vor Monat## Shripad Kulkarni

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.

Vor Monat## Whisky Guzzler

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

Vor Monat## Raided Cluster

We need a Netflix series based on Math history.

Vor 2 Monate## Good Goyim

@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.

Vor Monat## Tom Rhodes

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.

Vor Monat## Krishna Soni

@Clogica like most of us ^-^

Vor Monat## Good Goyim

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

Vor Monat## Dexter Westin

@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?

Vor Monat## Shyam Patadia

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

Vor 12 Tage## TomatoLove

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

Vor Monat## Jasper Klewer

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!

Vor 16 Tage## sledge721

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? ;)

Vor 20 Tage## 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.

Vor 2 Monate## Tom Rhodes

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.

Vor Monat## Robert Fox

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

Vor 2 Monate## Martin Gutsch

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.

Vor 2 Monate## JonzuihGaming

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.

Vor 2 Monate## Carlos Goytia

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

Vor 2 Monate## MrPeak0

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

Vor 25 Tage## Randolph Hall

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;

Vor 3 Tage## Satish Kannaiyan

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

Vor Monat## joeboxervr6

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

Vor Monat## Medlife Crisis

One of your best ever videos. Wonderful stuff.

Vor 2 Monate## Cubert Miso

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

Vor 2 Monate## Antoine Olivier

Agree

Vor 2 Monate## Mohamed Flaifel

Just amazing as usual ❤

Vor 2 Monate## Affectionate ASMR

ooo you are the best)

Vor 2 Monate## K K

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

Vor 2 Monate## dasfx

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

Vor 4 Tage## Venkatesh Pulakhandam

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.

Vor Monat## Marcos

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

Vor Monat## Matthew Harvell

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.

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