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

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

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References:

Some great videos about the cubic:

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

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

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

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

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

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

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

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

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

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

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

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

▀▀▀

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

Written by Derek Muller, Alex Kontorovich, Stephen Welch and Petr Lebedev

Animation by Fabio Albertelli, Jakub Misiek, Iván Tello and Jesús Rascón

Mathematical animations done with Manim -- thanks Grant Sanderson and the Manim community!

SFX by Shaun Clifford

Filmed by Derek Muller and Emily Zhang

Edited by Derek Muller and Petr Lebedev

Additional video supplied by Getty Images

Music from Epidemic Sound

Additional Music By Jonny Hyman

Produced by Derek Muller, Petr Lebedev and Emily Zhang

30 Eki 2021

Çalma listem

Daha sonra izle

Scenic Fights 2 aylar önce

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

Kep Spark 14 gün önce

Yep.

Guna selan guna 2 aylar önce

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

V?bez 2 aylar önce

@Miles Doyle please get a hobby...

MLG35U5 10/10 ign rated gr8m88/8 2 aylar önce

Now rewrite that sentence without the word “interesting”.

Reasoner Enlightened 2 aylar önce

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

thispurplefox Aylar önce

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

Laura Finley Gün önce

@Fynzmirs Unfortunately, no. Thankfully I found Montessori and the teaching system does. 4-year-olds build a multiplication but instead of using numerals they use squares and rectangles with areas that equal the respective numeral. Its called a "Pythagoras Square."

Padoru 5 gün önce

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

Padoru 5 gün önce

You would not have paid attention

Bows 6 gün önce

@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.

Andreas Klein 8 gün önce

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.

c.w. B O G U S Aylar önce

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!

Surendra I 2 gün önce

Well.. welcome to 11,12

Jorian Weststrate 3 gün önce

@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)

Trinka Buszczuk 3 gün önce

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

GaussianVectorⁿ 7 gün önce

@anon video that's what I keep saying.

GaussianVectorⁿ 7 gün önce

@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 .

Russell Cohen Aylar önce

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.

Pranab Mallick 3 gün önce

@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

Kain_niaK 3 gün önce

God must be a very complex man.

Hardaat Singh Baath 6 gün önce

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

Nosty Fripples 24 gün önce

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***”

Erhan 12 28 gün önce

@Sibler Inkognito E

Gangadharan Esakki Aylar önce

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!

Zu Fex 29 gün önce

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.

Factnomenal 2 aylar önce

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

Ioe Aspirant Aylar önce

Yes ,you should

Tom Rhodes Aylar önce

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.

cfcreative Aylar önce

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.

Parry 255 Aylar önce

@Nen Master5 ttyóyyytoy piggies 9ui9ityoti9y9i899⅞tt9ttuttititií9r7

Bôrgïr 2 aylar önce

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

Ash U Aylar önce

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 TRvid Algorithm! :D

Carl Alfredo 19 gün önce

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)

Chris Sugg Aylar önce

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.

David Aylar önce

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.

Nosty Fripples 24 gün önce

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***”

John Wright Aylar önce

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."

PolyRaptor Aylar önce

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 :)

DoS - Domain of Science 2 aylar önce

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

OjoRojo40 2 aylar önce

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

TNB1 2 aylar önce

@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.

Jeff Ward 2 aylar önce

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

Nerdelbaum Frink 2 aylar önce

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

Brendon Green 2 aylar önce

I've learned more from TRvidrs 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).

Charles Crawford Aylar önce

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.

Ansh Daheriya 26 gün önce

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

pagw2002 Aylar önce

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

James Younger, DDS 9 gün önce

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.

Mariano P Aylar önce

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.

gaurav kumar Aylar önce

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.

Akira Leelarapin 2 aylar önce

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

dude knowledge 5 gün önce

@Chucky chuck which country

Westle the Great 9 gün önce

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

Erik Brito Magalhães 9 gün önce

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.

Payton J 12 gün önce

This is so true

harry 12 gün önce

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

TorA Aylar önce

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.

Jabari Zakiya Aylar önce

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.

Lou Marz Aylar önce

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

Ansh Daheriya 26 gün önce

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

Chronosaur Aylar önce

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

John Jordan Saatler önce

Agreed. It's a very well produced video, but I dislike this "we had to abandon mathematics's connection to the real world, to discover deeper truths about the universe" stuff. All a bit mystical. We can interpret 'i' as right-angle rotations in a plane, a unit of perpendicular movement, which can be used to describe the cyclical behaviors that you mentioned. No mystic mumbo-jumbo needed.

izzynobre 2 aylar önce

Dude makes math sound absolutely riveting... Incredible

Parker Moss Aylar önce

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

Tom Rhodes Aylar önce

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

ForestCity Fishing 2 aylar önce

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

OjoRojo40 2 aylar önce

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

Adrian Bartholomew 2 aylar önce

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

Anthony b Aylar önce

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

James Fleming Aylar önce

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

Andy Badke Aylar önce

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

nsfa19 Aylar önce

This is one of the most beautiful and delightful videos I've ever watched on TRvid 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 TRvid 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!

Steven He 2 aylar önce

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

PIYUSH TEACHER Gün önce

but understanding is pretty confusing mr emotional damage

Brian Quin 6 gün önce

emotional damage

Nonsense games 9 gün önce

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

Matt Augusto 9 gün önce

Math is RACIST according to BLM

Nonsense games 10 gün önce

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

Sheena Abdul Basit Aylar önce

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

Levon Poghosyan Aylar önce

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 !

Angel Luis Villanueva Aylar önce

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

MetalSolid Aylar önce

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!

Dr. Trefor Bazett 2 aylar önce

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

Ansh Daheriya 26 gün önce

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

Rizal Purnawan Aylar önce

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

Tom Rhodes Aylar önce

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

Elizabeth Moore Aylar önce

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

Jacob McAfee Aylar önce

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.

Zeek Aylar önce

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.

Ali Mojebi Aylar önce

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! 👏

Sophoclis K 2 gün önce

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!

Vedant Kohli 27 gün önce

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 !

CuriousMarc 2 aylar önce

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

Queta Arbuste Aylar önce

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

CuriousMarc 2 aylar önce

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

Kurt Nowak 2 aylar önce

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

Nerdelbaum Frink 2 aylar önce

@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.

Nerdelbaum Frink 2 aylar önce

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

Kamil Khan Kakar Aylar önce

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

mofu mofu Aylar önce

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

frank Aylar önce

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 🙏

Jérémy FAVERJON Aylar önce

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!

Jabroney 2 aylar önce

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

Tom Rhodes Aylar önce

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.

Ramphys Aylar önce

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

asder alt Aylar önce

@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.

Fraken COB's Aylar önce

@asder alt Yup. :)

asder alt Aylar önce

@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??

Stephen DeVore sadsongsishere Aylar önce

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.

Kenneth Brown Aylar önce

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.

Alain Péan Aylar önce

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 !

David Henningson Aylar önce

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.)

PuzzLEGO 2 aylar önce

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

Supertron Aylar önce

I always try my best to ponder that. :)

Numbzie Aylar önce

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

Chinmayee Lenka Aylar önce

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

Tom Gilson 2 aylar önce

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

Alec Xander 2 aylar önce

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

Max. Sanches Aylar önce

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

Isai Ayala Aylar önce

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

Shripad Kulkarni Aylar önce

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.

Aage152 Aylar önce

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

History Dose 2 aylar önce

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

Josiah the NF 2 aylar önce

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

tofucube 2 aylar önce

Better than girls twerking

Vikram Singh 2 aylar önce

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

Supreet Sahu 2 aylar önce

No joke, there should be a science history channel

Josiah the NF 2 aylar önce

He’s capable enough, that’s a given 👌

Diva_4720 Aylar önce

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.

James N Aylar önce

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

wittySpace 26 gün önce

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😊

Jehag2 Aylar önce

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!

Mose 2 aylar önce

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.

Ansh Daheriya 26 gün önce

Xavier Magnus Aylar önce

@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.

Charly Taylor Aylar önce

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

Xavier Magnus Aylar önce

@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.

Charly Taylor Aylar önce

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

KJ Aylar önce

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/TRvid nights.

Awais Qureshi 29 gün önce

I successfully found my most favorite video on TRvid. 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. ❤️

Dr Mike Ecker Aylar önce

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

Tristram Sidey Aylar önce

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

Dr. Mudarris د. بشير المدرس 2 aylar önce

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.

Tom Rhodes Aylar önce

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

Robert Fox 2 aylar önce

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

Martin Gutsch 2 aylar önce

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

JonzuihGaming 2 aylar önce

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.

Carlos Goytia 2 aylar önce

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

Git Yer Git Aylar önce

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.

Louis Penn Aylar önce

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.

metalhead2550 Aylar önce

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

Venkatesh Pulakhandam Aylar önce

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.

Avinash Heral 7 gün önce

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.

Josh Hampton 2 aylar önce

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

Queta Arbuste Aylar önce

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

Tom Rhodes Aylar önce

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

Arthuria Pendragon Aylar önce

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

Queta Arbuste Aylar önce

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

Queta Arbuste Aylar önce

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

MrSimmies Aylar önce

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

Thomas Schwarz Aylar önce

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

Mark Newman 21 gün önce

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

Ali Hammad Aylar önce

I remember encountered e^ix in furrier transform and thinking how does this lead to winding original signal in the complex plane. Thank to your explanation it helped a lot.

FinFET 2 aylar önce

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

IrokoSalei Aylar önce

@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.

GrayFoxHound9 Aylar önce

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

Tom Rhodes Aylar önce

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

88marome Aylar önce

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

Jesus Christ Aylar önce

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

Robin Khn 27 gün önce

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.

प्रबुद्ध माथुर 29 gün önce

This whole video was one heck of a journey. Truly a worth watch for any student. I don't know why it just gave me goosebumps for some reason. Thank you for making this! Truly.

Ansh Daheriya 26 gün önce

Bully Bored 6 gün önce

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.

Matthew Harvell Aylar önce

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.

Justin 2 aylar önce

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

Tom Rhodes Aylar önce

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

Andromeda Galaxy 2 aylar önce

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

John Bjorkman 2 aylar önce

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

Nerdelbaum Frink 2 aylar önce

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

Mikey P 2 aylar önce

I struggled to get the cover off my graphing calculator.

spaceship raceship Aylar önce

this is amazing. my country has a really bad education system so i felt very ashamed that I don't know anything about quadratic equations, but now that i have context around the idea i feel more inspired to learn. it's hard to self-teach math when you have never learned a lot about it. but it's very exciting.

CRISTOBAL GARICA POSADAS Aylar önce

I ABSOLUTELY LOVE YOUR WORK!!! We should have more people like you in the world!!

Marie Sourroubille Aylar önce

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

SAGAR SETH Aylar önce

I always wondered if I would be born in a previous era would it have been easier to go down in history by doing some basic maths. This video has shown me what kind of genius was needed even in those times. I can now truly appreciate standing on the shoulder of Giants!

GalliadII Aylar önce

like inventing minecraft?

ren egade 2 aylar önce

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

Tom Rhodes Aylar önce

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

Work count 2 aylar önce

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

Bryan Williams 2 aylar önce

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

NothingsGood 2 aylar önce

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

̇ 2 aylar önce

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

Roboto C. Aylar önce

It's really hard to say you hate math after watching something like this. Amazing work, man!

Josué Aylar önce

what a great video about imaginary number! Could be a introduction to that subject in schools and university! Students would get much more excited to learn that and definetly make much more sense!

Jasper Klewer 17 gün önce

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!

Andrei on Bikes Aylar önce

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

Raided Cluster 2 aylar önce

We need a Netflix series based on Math history.

Good Goyim Aylar önce

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

Tom Rhodes Aylar önce

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

Krishna Soni Aylar önce

@Clogica like most of us ^-^

Good Goyim Aylar önce

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

Dexter Westin Aylar önce

@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?

Jeremy Black Aylar önce

About 12 minutes in I checked how much time remained because I was worried it would end soon, and I really wanted more! Very well done, and super informative.

Julien Beck Aylar önce

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

sledge721 22 gün önce

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? ;)

Whisky Guzzler Aylar önce

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

Scott McNaughton 2 aylar önce

This REALLY brought an eye opener to my "how the heck did they figure this out" during math classes. Awesome explanation. Thanks

DON'T 2 aylar önce

WOWW YOU DON'T READ MY PROFILE PICTURE 😶😶❌❌....

Lem ele 2 aylar önce

well said!

matthew fisher 6 gün önce

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

Carlos Javier Viñas Velazquez 19 gün önce

I cannot describe how thrilled I was while watching this masterpiece. I always had this problem with math where I did not understand why every step on solving a problem, using a certain formula, seemed like a magic trick or something. I was always trying to connect the abstraction of math to reality in order to understand them better. Nowadays I understand a little bit better and I love physics and math, but this video took that to a whole new level. Thanks for this inspiring content! 🙌🏻💙

Satish Kannaiyan Aylar önce

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

KING SLAYER - AMV / EDITS Aylar önce

The way they explained the 'i' symbol in sodienger's wave eq. , that how they work on graphical plain...... gave me goosebumps.....

Philip ROSE 2 aylar önce

This is a faultless presentation of one of the most inspiring naratives in history, maths and physics. Congratulations! You have set a new paradigm YT. Could you do the same for Dirac's equation?

Philip ROSE 2 aylar önce

@Vikraal s`right, but beauty of following through to Dirac is bringing matrices into the story and thus antimatter! How maths reveals the world. After that, string theory?

Vikraal 2 aylar önce

Dirac and Schrodinger were both genius. Although Dirac's equation gets the fame, Schrodinger was the one who built the mathematical framework for Quantum particles.

Ata Toraman 2 aylar önce

Dirac pls

Linzzz 2 aylar önce

@West Explains Best I was just saying to someone that I sorely wish the history of all of this had been taught to me back when I was learning it. I went on to study math in college, but I still wish that someone earlier on had showed us the humanity in math, the bickering scientists and the disbelief/hope that a solution would ever exist. I think it would be awesome for you to show it

West Explains Best 2 aylar önce

I am seriously contemplating showing this video to my Alg 2 class. Visual demonstration of completing the square and math history? Too good!

gal2659 Aylar önce

Great, interesting, thank you very much. Do we know what was Del Ferro's method of solving the depressed cubic ? was it the same idea as Tartaglia's ?

Ruaraidh McDonald-Walker Aylar önce

This might be the best Maths lecture I’ve ever watched (watching and commenting again!) and I’ve used complex maths for 40 years as an electrical engineer. Love the fusion of History, Maths and Physics. Quite literally Brilliant (yes been enjoying that for a while). Great video, more please!

Aidan Homewood Aylar önce

this video was just incredible. it makes me glad i am doing a maths degree at university, and continuing to study this amazing topic.

Oneoneone Aylar önce

I just listen to you speaking with Destin and Matt on their Podcast like a few days ago saying that you get chills when you teach about how electron equations are acting as particle and wave. I got chills too when you speak about it, THIS ONE TOO!! Thank you so much for teaching us this super cool math with super cool visualizations!

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