How Imaginary Numbers Were Invented
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- Опубликовано: 13 май 2024
- 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. -- • 500 years of NOT teach...
Imaginary Numbers are Real -- • Imaginary Numbers Are ...
Dunham, W. (1990). Journey through genius: The great theorems of mathematics. New York. -- ve42.co/Dunham90
Toscano, F. (2020). The Secret Formula. Princeton University Press. -- ve42.co/Toscano2020
Bochner, S. (1963). The significance of some basic mathematical conceptions for physics. Isis, 54(2), 179-205. -- ve42.co/Bochner63
Muroi, K. (2019). Cubic equations of Babylonian mathematics. arXiv preprint arXiv:1905.08034. -- ve42.co/Murio21
Branson, W. Solving the cubic with Cardano, -- ve42.co/Branson2014
Rothman, T. (2013). Cardano v Tartaglia: The Great Feud Goes Supernatural. arXiv preprint arXiv:1308.2181. -- ve42.co/Rothman
Vali Siadat, M., & Tholen, A. (2021). Omar Khayyam: Geometric Algebra and Cubic Equations. Math Horizons, 28(1), 12-15. -- ve42.co/Siadat21
Merino, O. (2006). A short history of complex numbers. University of Rhode Island. -- ve42.co/Merino2006
Cardano, G (1545), Ars magna or The Rules of Algebra, Dover (published 1993), ISBN 0-486-67811-3
Bombelli, R (1579) L’Algebra ve42.co/Bombelli
The Manim Community Developers. (2021). Manim - Mathematical Animation Framework (Version v0.13.1) [Computer software]. www.manim.community/
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Special thanks to Patreon supporters: Luis Felipe, Anton Ragin, Paul Peijzel, S S, Benedikt Heinen, Diffbot, Micah Mangione, Juan Benet, Ruslan Khroma, Richard Sundvall, Lee Redden, Sam Lutfi, MJP, Gnare, Nick DiCandilo, Dave Kircher, Edward Larsen, Burt Humburg, Blake Byers, Dumky, Mike Tung, Evgeny Skvortsov, Meekay, Ismail Öncü Usta, Crated Comments, Anna, Mac Malkawi, Michael Schneider, Oleksii Leonov, Jim Osmun, Tyson McDowell, Ludovic Robillard, Jim buckmaster, fanime96, Ruslan Khroma, Robert Blum, Vincent, Marinus Kuivenhoven, Alfred Wallace, Arjun Chakroborty, Joar Wandborg, Clayton Greenwell, Pindex, Michael Krugman, Cy 'kkm' K'Nelson,Ron Neal
Executive Producer: Derek Muller
Writers: Derek Muller, Alex Kontorovich, Stephen Welch, Petr Lebedev
Animators: Fabio Albertelli, Jakub Misiek, Ivy Tello, Jesús Rascón
SFX: Shaun Clifford
Camerapeople: Derek Muller, Emily Zhang
Editors: Derek Muller, Petr Lebedev
Producers: Derek Muller, Petr Lebedev, Emily Zhang
Additional video supplied by Getty Images
Music from Epidemic Sound and Jonny Hyman
Man, change "depressed quadratic" to an obscure magic spell and you literally get a fantasy duel story, complete with a sage old mentor, an underdog protagonist, an enchantment and a boastful proud villain wtf
It is magic to people of the era. Guys with beards fight in a duel to the death. They use secret formulas as spells.
Frfr
Stories of wizards and stuff are probably inspired directly by mathematicians.
So of all those T-shirts comparing some profession to wizardry (you see it for engineers or IT a lot), the most legitimate claim is for the mathematician imo.
no idea how on the same page we are, I was coming up with a whole revolutionary way to look at insanity based magic systems too
All without eating a single pizza. Incredible.
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.
😂😂😂
Blue Eyed W- erm, sorry wrong series 😂😂😂😂👏
Intense for everyone tbh
Imagine minding your own business as the Burger King and someone knees you in the stomach.
Its time to DUEEEEEEELLLLL
I can't believe that now, a decade after struggling to understand it, I finally know what "completing the square" means.
congrats!
Not just you. Even some teachers don't know what it means they just memorize the process.
I don't think the teachers knew either, they were just cutting off the tip of the ham because everybody else had done it before them
my thoughts exactly. I always wondered why it was called that
It’s always funny that the answer to why something is called what it is called, is often right in front of you, but (in the case of completing the square) due to how math is taught in school it often comes off as nonsense.
It's shocking how thoroughly you managed to deceive me into thinking I almost understood this topic. You, sir, are phenomenal!
lmaoo same
He was like "even kid can do this" and pull into abyss call math
@@b0mby1 Although your word made me feel insulted, guess it cant be helped. Well you see. First of all,I dont talk english. Second, I know what he talk about when he talk about that cubic stuff. Third, I understand all of this except that last part where he start do some reality bending edit, turning 2d into 3d before turning it into 2d again. The only thing I need to watch back is the part the -5 cube being introduce. That all.
@@unknownman5090did the comment get deleted?
@@ayuballena8217 I think so. He said something like im not good at math and something like that, which I agree. Im not math genius. However, his sentence feel like trying to say that im dumb, which is true, but I dont think he have good meaning behind it
I wholeheartedly believe that giving context to the history and slowly guiding students through the mindset of mathematicians is objectively better than spoon-feeding them equations.
exactly
Not just mathematics
My Algebra teacher in college used to tell us stories like that and I remember him telling us this one too. He later went on to become the minister of education in my country.
plus 1
We had this in every chapter's explanation in our books, a large one ~one page paragraph explaining the history or the person behind the concept, unfortunately it was never used
"Anyone who's passed 8th grade knows the general solution."
Yes yes, of course, heh... *starts sweating*.
"Knew at some point" would be more accurate
“anyone who’s been accepted at harvard” should be more acceptable
We don't have that in the UK, so I'm fine... Stupid, but fine...
@@derptyderp5287 or in canada (nova scotia)
I'm in the second year of high school in Sweden and we haven't even touched upon the subject yet.. big facepalm there lol
Visualizing "i" as describing a value that cancels itself out and seeing the CG of e^ix described as a spiraling 3-dimensional waveform with the X & Y functions 90° out of phase may have contributed more to my understanding of physics and mathematics than the entirety of my college calculus courses.
That’s how I know you found the actual Easter egg in the video. It literally made me gasp and shout as soon as I saw why Euler formula used e. Integrating and differentiating e is always going to be the same and the way they reflect sin and cos and their interwind is simply mind blowing 19:57
The fallacy of your comment is that you assume that your "understanding" back then is equal to your understanding now, after accepting "not understanding" and dedicating your attention to your individual set-up. Being patient, curious and brave (having FAITH) - i.e. staying HUMBLE - are the only ingredients of LIFE.
The best reference to LIFE being THE BIBLE!
@@ACuriousChildthe bible ain't all that
What is this place
I have a masters degree in physics, so I'm confident in saying I'm pretty good at maths. You describing the completing the square method of solving a quadratic just genuinely blew my mind. I never understood where any of it was coming from and opted instead just to use the quadratic formula and ignore completing the square. I just thought it was entirely irrelevant. But holy wow it makes so much sense now, I see where the steps all come from, and it's actually extraordinarily elegant. It makes soo much more sense now!
Just goes to show how much influence a teacher has on their students and why so many people think they're bad at maths. I hope more teachers start teaching things they way you did there. Thank you!
Maybe they should incorporate more math in physics classes. It is kinda the basic for introducing complex numbers by starting with the kwadratic equations.
It's hard to imagine a world without algebraic notation, but when you understand that ancient mathematicians were using visuals to do math, it makes sense that all the terms we get from them are a lot more literal than one might think.
Stop trolling man
are you really a masters in physics
"I did not deem him capable of finding such a rule on his own." Savage 😂
I know, right?
Tartalia was a beast. He had no chill. Ended the guy's career in 2 HOURS for something that was supposed to take 1.5 months.
Savage i tell you! Savage!
lol
though anyone can reach the level of a genius, it' only a matter of decades
We need a Netflix series based on Math history.
I will watch it even if i dont even understand this video
agree
@@Arvl. After watching you would understand, i guess :)
Netflix doesn't have great history documentaries. It would be better to do it on Curiosity Stream.
Check out Marcus du Sautoys series.
It just feels like I've uncovered some chunk of fundamental knowledge of absolute purity. Thank you for letting us fools taste the beauty of maths in a 23-minute video.
dont say "fools" lol
You worded this so beautifully
I'd say I'm very average at math. I think I learned more from this 23-minute video than in the past 20 years of my life.
I was one of the worst math graduates in my highschool class but recently I had a spark of love for maths and reteach myself everything. This video is nothing short of amazing. Its just mindblowing!!
Next brilliant video: "Your Daily Equation #2: Time Dilation" by Brian Greene. Only pythagoras and basic algebra needed. But, for bonus, you can try to find the unit circle that links time dilation with speed
Today I learned that I've never passed the 8th grade.
:'(
in my country its in the 9th grade , not 8th...
@@henriquepereira2811 It would depend on the country/region. Curricula differ from place to place
Today I learned that Bologna is a place and not just a food.
@@henriquepereira2811 I’ve never heard of it and I’m graduated lol
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.
It's hard to teach math concepts in a memorable way. But that's what Derek does best!
Schools convert man to a learning engine so that he will grasp everything that makes humans worse than google . But human brain is much more than just a fact storage device.
me too! its actually simple to see that this way
@@chanderparkash4537 not a "learning engine" at all
@@austinhernandez2716 to some extent
"Only by giving up maths' connection to reality could it guide us to a deeper truth about how the universe works." Bravo!
A thoroughly professional presentation from algebraic dependence on visual geometry through Mediterranean ego vignettes segueing into physics, with remarkable insights along the way, culminating in the quote above.
This video was incredible, I cannot put into words the fantastic journey I experienced in these last few minutes, thinking about the realities of mathematicians, how problems that have been considered to be impossible for thousands of years are solved, and how we naturalize the legacy of these incredible minds. Thanks my friend
Just had my mind blown learning that "complete the square" is literal.
Even after learning it in high school, it still sometimes blow my mind with how much sense it makes
I wish geometry was focused on more in schooling.
@@Trowa71 I hope schools now a days show these videos. I didn't pay attention at all in school, but now, I find myself deeply enthralled by it.
Lol I was thinking the same thing.
@@tyzxcj34 You commented this while I was watching the video.
I love it when complex equations come down to something elementary like 2+2=4
I don't know. It usually means I have chosen the wrong approach and lost the x.
same
Sums up my higher school years with math
That is what its supposed to be in the first place we are doing a top down calculation to simplify it to our understanding. The complex structure is still the answer regardless of whether it is solved (simplified) or not so we are not trying to solve it but make it understandable for ourselves, which means making it elementary
@Pradeep Singh I think your a key is dying
I thought I was just going to browse the video but here am i going through it all and even rewinding. Thanks it was very engaging and brilliantly undertaken.
I rewinded too. It was a great video.
Superb in everyway. This is how mathematics should be taught. You deserve a prestigious award.
History of mathematics should be taught as early as in middle school, and this video tells exactly the reason why it would immensely help students appreciate what they are taught.
History of everything should be taught.. otherwise the new student must do what literary fiction does.. cause suspension of disbelief. In other words.. believe in magic..
@@Sierrahtl This video is about mathematics, hence the my comment.
My 8 year old (still 3 years away from middle school) understood *just* enough of this video that I'd have to agree.
It should, but it isn't and it won't. Schools are more concerned about your kid's dress code violations and football than to teach them anything useful. And if they are close to flunking, there's always the Army to set them straight ( in all manners).
Borrrrriiiinnnnnggggggg. History is boring to teens, they do not care about the past, more about the future, or even more the present, for most anyway.
Math teachers, please, please, show this kind of stuff during class. It would've changed my life.
While I agree that this video is very well done and engaging, the moment that your teachers would have made you start solving equations even after showing you this video, you would've get bored of maths again.
@@pitthepig good point.
I did and they said.... it was boring.
@@kevinbugusky7416 Kids don't want to learn or think anymore.
Wait, they don't teach that everywhere?
This video is so impressively well made! The storytelling, the animation, the music, the drama, the education!! What triumph!
This is my favorite math history episode ever! I love the "cubic battle" and the invention of "imaginary numbers" so much, thanks for making it even more interesting with your narrative.
Dude makes math sound absolutely riveting... Incredible
Math is riveting if you don't know math. If you actually learned at least high school math and some history behind it (like how was calculus developed), you'd know that it is riveting.
I dont understand the Mathias he is doing but im interested
( colé izzy o/ ) the thing that i take from this is just how insane being a mathematician this day and age problaby is when you're needed to know all prior knowledge at the same time that you need to challenge it to find new questions and keep progressing mathematics towards the future.
Good job on getting 600 k subs
Yeah, I was going to say, what’s sad 😞 is that you didn’t find it riveting to begin with as it literally is the language we use to describe the reality of our universe. But to each his own and I don’t blame you.
This video makes me want to do math. It’s inspiring in the best way
Totally agreed!
It hurts my brain
Math
@Faizan & Stuff wdym jk it’s not a joke he is a failure he didn’t know even in preschool at most
Legends know the original title was "This problem broke math(and led to quantum science)
This is probably my favourite video on YT. It is the best maths one by a mile, I really, really enjoyed it - I learned so much. The enthusiasm of the presenter was tangible!
The concept of "Math duels" is hilarious to me
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.
@@bobmanbob341 meaning?
@@Siso_Mnguni bruh if u look around it’s full of absolutely random comments. I think there must be some kind of bot at work
Exactly the same for me. Incredible video.
3blue1brown did a video series on calculus in an equally visual way that helped me understand it better than any teacher in high school ever could. This has been the pattern of my learning after school in general tbh, the internet has been the most valuable resource in my learning journey.
If you'd like a book that focuses on visuals, I would recommend "Proof without words" - my professor recommended it to me recently & gotta say, it's a fun gift for those who like picture books.
A History, Math and Science smoothie blended to perfection. Well done 👏
Would it be too random to declare my intend to recommend
my fellow science-youtuber-fans some... well... more science-youtuber?
I mean, in my mind, it just makes sense, but many call me B0t, so... your choice...
Complex numbers are dual to real numbers.
Perpendicularity or orthogonality = DUALITY!
Column vectors are dual to row vectors -- group theory.
Electro is dual to magnetic -- Maxwell's equations.
The electric field is perpendicular (dual) to the magnetic field -- probability waves.
Positive charge is dual to negative charge -- electric fields.
North poles are dual to south poles -- magnetic fields.
Electro-magnetic energy or photons are dual.
Points are dual to lines -- the principle of duality in geometry.
Group theory:- the image is a copy, equivalent or dual to the factor or quotient group.
Isomorphism (absolute sameness) is dual to homomorphism (relative sameness or difference).
Homo is dual to hetero, same is dual to different.
Injective is dual to surjective synthesizes bijective or isomorphism.
Positive curvature is dual to negative curvature -- Gauss, Riemann geometry.
Curvature or gravitation is dual.
Gravitation is equivalent or dual to acceleration -- Einstein's happiest thought, the principle of equivalence (duality).
"Perpendicularity in hyperbolic geometry is measured in terms of duality" -- Universal hyperbolic geometry, Professor Norman J. Wildberger.
Duality (energy) creates reality.
Action is dual to reaction -- Sir Isaac Newton (the duality of force).
Attraction is dual to repulsion, push is dual to pull -- forces are dual, e.g. the electro-magnetic force.
Monads are units of force -- Gottfried Wilhelm Leibnitz.
Monads are units of force which are dual -- monads are dual.
Energy = force * distance.
If forces are dual then energy must be dual. Potential energy is dual to kinetic energy, gravitational energy is dual.
Apples fall to the ground because they are conserving duality.
"May the force (duality) be with you" -- Jedi teaching.
"The force (duality) is strong in this one" -- Jedi teaching.
"Always two there are" -- Yoda.
Yummy
@@hyperduality2838 stroke
@@nathanlevesque7812 Asinine!
Duality allows you to create new laws of physics:-
Syntropy (prediction, projection) is dual to increasing entropy -- the 4th law of thermodynamics!
Teleological physics (syntropy) is dual to non teleological physics (entropy).
Making predictions to track targets and goals (objectives) is a syntropic process -- teleological.
Complex numbers are actually dual numbers -- the complex plane.
Poles (eigenvalues) are dual to zeros -- optimized control theory.
Duality is everywhere if you look for it, male is dual to female.
The effort that went into this was not unnoticed (by me, as much as I could lol) Thank you so much on educating me so effectively on this fascinating topic
This channel is awesome. Both in terms of video production and, more importantly, selection of inspiring and informative topics for each video. Thank you!
You are so correct. Glad to be subscrbed!
This level of animation deserves appreciation.
True
Is this manim like @3blue1brown?
and after that you have to plug the red wire into the socket to make sure the engine boots at launch. Wrap the green wire around it's coil that sits directly beside the A button. After you put the back shell on, place the battery in the slot. Screw the Vr26 Jeeper back up and press the reset button. If everything worked according to plan you're device should show a thumbs up sprite. Plug the HDMI port into a monitor and wait three seconds. If it boots up on TV your in the good side. If it doesn't boot in less then 5 seconds quickly unplug. This can severely damage your TV and possibly start a fire
@@pattyryopotybuttongamer3063 why… are you trying to teach us how to hotwire a car?
bro imagine going back in time with like HS calculus on your belt you would be an oracle
Ikr. It's a common Sci Fi trope but I love the thought of these guys throughout history being aliens or something changing our whole civilisation with what is high school maths to them 🤣
As soon as they say "now proof it" you are done
Yea and they're gonna ruin your dreams with a simple: "And why is that?"🙂
@@agam9085 "Let me explain the next few hundred years of math to you"
or burned at the stake...
Brilliant way of explaing it. Math used to be extremely hard in high school and teachers didnt know how to explain it to us.
I WISH i had access to this sort of resource when i was in school. I never had any interest in maths because i didnt understand what it was for, what it explained, how it was applied. I just learned the equations the teacher wrote on the board by heart. If my maths education had been filled out with this kind of fascinating information about the broader context of what i was being taught, it would have been a totally different experience for me. I hope teachers everywhere are making use of this kind of online content!
Instead of letter grades A through D, 8th graders should get a grade placement based on which century of Italian mathematics they most closely align with.
Guys, we can reform standardized testing now! We found the perfect scoring system!
Congratulations! You have scored "Roman Republic" in Math!
If you think 8th graders are learning about imaginary numbers, solving cubic equations, or quadratics for that matter you either don't remember primary school or were an exceptionally gifted child. My guess is the concept of a variables is introduced in 7 or 8th grade, probably putting 8th grades some where in the dark ages. Probably where they belong from what I've seen , haha.
@@901blitz Variables are taught in 5th grade my guy. We live in a very different world than the one we grew up in.
@@901blitz yeah variables are taught in the middle of primary school and in 8th you learn algebra 1 which definitely has quadratics, imaginary numbers, etc. im in 12th grade right now taking calculus using the prior knowledge from middle school with graphs and algebra needed for complicated derivatives
For the entirety of my higher education, I've been told to "complete the square," but 6 teachers and 4 professors have never explained this further than restating the equation. In one extremely brief visual and explanation, you've managed to answer a question I'd long since forgotten. I don't know how to describe my astonishment, nor my gratitude for your content.
No kidding! I recall that math was usually taught by coaches reading from a book. I had the same reaction as you to the "complete the square" part of the video. I could have been so much smarter had I only had math teachers with this gentleman's style...
Mathematics needs a James Burke. This video is totally on that path... talking about secrets, jobs, politics, challenges, motivation, compromises and re-derivation of formulas.
This comment right there.
@@allmotion_video_channel5434 whether would it make you smarter depends on what aspect are you talking about. if we are only talking about taking test and exam, explaining the equation to you wouldn't make you smarter in doing in math classes. It mostly depends on how much practise questions you've done. The same philosophy can also apply to college.
Surely, however, explaining the meaning behind mathematical equations and practices would help you have a better understanding of the world. Though it have little use in real life since mathematics are mainly used in real world as a tool to solve, not a tool to understand.
@@jimmyli319 interesting outlook on the subject you have. I am a flight instructor and have seen how different people learn in different ways. For me, I like to understand the underlying theory. That helps me judge whether the results of a “solution”makes sense. I also have beginning engineers that I work with that will just plug numbers in to a program like “MATLAB” but do not understand the fundamentals well enough to judge whether the “answer” is reasonable enough to be correct.
But, in your context, it is valid that one does not need to know how an internal combustion engine works just to drive a car…
This was the first video of Veritasium i watch and now I can't stop. Learning is really fun in this way.
I'm so thankful to my math teacher, because this was how she taught completing the square. I see the comments about schools that didn't teach it this way, and it just shows me how great that teacher was.
This is exactly how you teach people the aesthetics of something.
The beauty, the thing which motivates people.
I agree; however, I feel that you can never teach others: you can only motivate others to dive deeper into the material themselves, and this is a great video that fires up people’s interest to do exactly that.
@@MrFrazerz Yeah, especially with pure math being mostly proofs a lot of them non-constructive, this medium would mostly be unavailable.
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.
Facts
Chances are they might hate maths even more because now they have to learn about history of maths lmao
Teaching at school will never teach you mathematics, it is more the influence of parents who have been engaged in your development since the cradle.
na they wil still hate math because they hate doing maths, having a interesting backstory doesn't make it suddenly fun to do.
@@yuseisatouissuffering they shouldn't be supposed to memorize the history . It should just be told to inform them on how maths is actually done, i.e. how new stuff is actually figured out. The focus on memorizing formulae to solve most math problems is what i think is stopping math education from being fun.
I didn't understand a single word said here, but I found the video to be enjoyable 👍
Please separate these videos from others and make a separate playlist for mathematics. It is fun understanding maths in this fashion ❤.
Was expecting cool math, didn’t expect the crazy history story, but it was my favorite part:D
Meanwhile 10 million people die from preventable cancer every single year and not a peep from the press or any politician. Solve that unsolvable problem.
@@TheFirstBubbaBong population control. They don’t want the USA to turn into India.
@@TheFirstBubbaBong Not the place to discuss that, you will attract extremely biased opinions
i loved the animations with it too!
@@TheFirstBubbaBong What have you done in that regard?
This REALLY brought an eye opener to my "how the heck did they figure this out" during math classes. Awesome explanation. Thanks
well said!
So interesting!!! I would have loved to have known this when I was teaching maths all those years ago. Thank you for sharing!
i just thought some mathematician went
"yo guys, imagine..."
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 RUclips. Does anyone have any suggestions for other videos that combine math and history like this one?
Try searching for "History of Science". A guy in my college dormitory actually goat a degree in this field.
3blue2brown
@@cachecollin6984 4blue1brown?
@@camgere goat a degree?
@@fatitankeris6327 History has many fields. History of England, History of the 16th Century, History of Agriculture and yes, History of Science. Did you say GOAT? Shemakhinskaya Bayaderka Festival / Yana Kremneva / 201. Science GOAT. James Burke Connections, Ep. 4 "Faith in Numbers". From 1978, pre-internet. I actually used Hollerith cards (punch cards) to write my first computer programs in Algol inn 1973.
"Only by abandoning math’s connection to reality could we discover reality’s true nature." I cannot shake these words from my head.
That s because it is a poor statement to begin with.... it is the arrogance that blinds us.... in our quest for knowledge with each step forward, we stop to admire ourselves and claim that we now have acquired foundation for reality. Until we learn it is still out of our grasp.
@@francoisiannacci2615 go back to your fairy tale about a ghost that literally claims to be the greatest thing in the universe
@@Flaystray go back to your fairy tale world with triangle and infinite number?
we need to move on from old rules so we can seek further newer better ones.
we need to move on from our prior definition and understanding of what reality is, to seek new better definitions of reality!
in the end, it's OUR perception of reality.
@@francoisiannacci2615 I agree, there's always further to seek. standing on one discovery and thinking it's the end of the line, is such a toxic narcissistic thing to do.
Please keep on making more math videos like this
Specially about the history of group theory.
The way you present the videos is amazing. It takes a lot of work. Good luck!!
This is how students should be taught in schools and colleges. You are a perfect teacher.
Nahh i assure you, dumb and lazy students will always be like that. They would dismiss the historical story and ignore the illustration of solving an equation using geometry.
@@vandalm9311 it's about those who are really curious to know about the core concepts and essance of science. Here neither the dumb nor the intelligent has access to this quality of education.
He probably makes 10X a teacher's salary as a RUclips celebrity.
Exactly! Motivate the students with the lore rather than just throwing numbers at them and expecting well performance.
Ah yes, just spend hundreds of hours on a video for a 1 hour lesson. I'm sure that will be highly efficient. Tell me you've never taught before without telling me.
If math had been explained to me like this in school I would have actually remembered it
They didn’t because teachers didn’t know , they were just looking to get paid on the 15th and 30th of each month, rarely you will find a true math teacher.
@@aaronovski9949 math teacher here. Reality is, we often cannot go too far beyond the math curriculum given to us. I plan on showing this to my students little by little, because if we did this instead of class I'd be in trouble
Math is still math tho
Not sure why YT has recommended me this vid. I am too dumb to understand this shite. Just gonna casually scroll pass and listen to CardiB WAP and watch Logan Paul.
Bye Ngl.
@@MrUssy101 I SCROLLED THROUGHT IT FOR 15 TIMES IT CANT BE GONE
A couple idiot coworkers and I have a game where we all see who can keep up with veritasium videos the longest. It's a fun way to pass the time. It was a 3 way draw literally a minute in.
Cool game! I suggest you try “How to count past infinity” by Vsauce. It’s not a Veritasium video, but the contents is just as interesting
This video has simplified my years studying Astrophysics, quantum mechanic and relativistic physics in particular. Powerful stuff through the lens of analyzing: Why. Two big thumbs up!!!
Imagine being a mathematician walking around town and out of nowhere, a guy jumps out of a bush and challenges you to a math duel
*jumps out of bush* math duel NOW
a wild pokemon has appeared
Italians..
That's almost literally what happened to Tycho Brahe. However, instead of pulling out their quills to solve the problems, the insult of "I'm a better mathematician than you" led to pulling of swords and the other guy showed that he was a better swordsman, at least by cutting off Brahe's nose.
He used proof by induction! It’s super effective!
My 4 year old daughter sat though this whole video and then said, "That was a fun video!" When asked what she liked about it, "I liked watching the blocks. Let's watch another block video." :)
^ this is great ^
4-year-olds will be 4-year-olds.
Good! That means she'll be interested in watching it again, and learning its content a few years from now.
Still sounds like a good start 🤷♂️
When I was 4 years old I didn't even know how to speak lmao
This remains my favourite video of all time. The educational content herein is so phenomenal and at the same time entertaining.
Incredible lesson in mathematics. Wish I had known this 40 years ago. Thank you!
It's just wonderful to see how he is explaining math, physics and chemistry with such ease
I have to rewind the video a couple of time to get it
@@tonyng3285 at least you are taking an effort and finally coming to the understanding. :)
SUCH EASE????
Pretty easy for me, though I lack high knowledge I get the spirit.
Absolutely. But he should definitely get some more sleep. ;-)
"you can't have a root of a negative number; but let's imagine we can" - Degree level maths
Several centuries later: "you can't have a number greater than any natural number; but let's imagine we can"
Essentially "Plug it in and see what happens" for mathematicians.
"Let's suppose" - My undergraduate exams' most used phrase.
That's exactly what advanced maths is! "This is a rule...but what if it wasn't?" I love it.
Obviously you simply just _can_ by moving from real numbers to imaginary ones. You need to be careful though, because after the switch not all of the arithmetic is the same. Especially when dealing with roots of arbitrary fractions, order of operation between powers and roots, and when dealing with (semi)inner products.
bro this video is actually so good, I am yet to deal with stuff like cubic equations and complex numbers rn but you made it easy to understand
I love how simple he makes this concept, just everything was interesting and understandable even to a 14 year old, me.
blows my mind how these guys figured out math. Studying math today it feels like everything is pretty much figured out. I guess you need to be at an incredible level to figure out what does not yet exist.
I'm sure they thought the same way back then hahaha
On the contrary Victor; you can quickly catch up as the hard work has been done leaving you with advancement.
The only hiccup is when an axiom turns out to be false; which is only discovered when it is pushed to the limits.
Yeah I can't imagine that level of comprehension to the subject.
Somewhere out there in this world.
Someone again made a huge mathematical, or physics discovery, but didn't think much about it because they thought other physicist already considered it but didn't publish because it's wrong.
If back in the past, people kept great discoveries secret, now it could be possible that someone out there made a discovery but isn't confident about their own ability to explain it.
I mean, we already got the internet.
If a person thought of something, they could search the internet for answers, or to confirm something. If it doesn't show up, it may be because it was a stupid idea that noone ever considered it, or it may have not existed considering them to be the first to come up with it.
It is possible that the latter could happen. It's not such a bad situation though because some other people could just come up with the same idea, until one actually is confident enough to go public with it. Same with the story in this vid.
Related to complex numbers is a mathematical problem called the Reimann Hypothesis....
It's widely considered to be the hardest mathematical problem.
It's one of the 7 millennium problems, of which only one has been solved to date.
Solving any of these problems would be revolutionary and would win the solver a prize of a million dollars
Some poeple like to joke saying "There are much easier ways to earn a million dolllars than to solve the reimann hypothesis"
Very intresting stuff IMO
That story about Ferro, Fior, Tartaglia, and Cardano could be a movie.
Quick reminder that the solution to fifth-degree equations was discovered by a political revolutionary who died in a possibly unrelated duel. The history of this part of mathematics has been almost excessively dramatic.
Kelsey Oakes's Aunt stopped living (LMAO 😂) because I upload bangers!
..,...
@@btf_flotsam478 I see you are talking about Evariste Galois and the Galois theory,but he did not find a solution to fifth degree equations,he just proved that they don't have a general solution
@@indianalphazero I think that was Abel
@@78anurag Abel also did it independently, but Galois did it a bit earlier and at a younger age. The only reason many people know about Abel and not Galois is because after Galois sent his work to his friend and died,his works were suppressed by other mathematicians(because they themselves did not understand it). Also Galois's work was much more detailed and generalized.
Thank you so much. As someone who's taken classes in Quantum Mechanics, I now have a fundamental understanding of the implications of using Euler's formula. ❤
This is really brilliant. Thank you so much for this.. I've been studying 1st year algebra, graphs and equations for two years part-time now, and I just never understood what the heaven it was all about. This has opened up a whole new perspective on what the quadratic equations really mean. What a brilliant video, thank you again.
I swear, if more of my math classes were like this and explained the "why" behind the concepts, the content would've been much easier to grasp.
That would require _good_ math teachers, though. You only get those at the graduate-school level, because that's where they all end up. The teachers who end up teaching algebra, geometry, and trigonometry in grade-school are the people who just barely graduated with their math degrees -- they aren't qualified to teach anything harder than "x + 5 = 10" or "sohcahtoa". Whereas anything _simpler_ than algebra, geometry, and trigonometry doesn't require a math degree to teach it _at all,_ so there's nowhere lower for those bottom-tier math teachers to go. So _every single kid's_ first introduction to math that requires _actual thinking_ (instead of memorizing tables) is with a math teacher who, objectively, sucks at math.
If so then we wouldn't have enough teachers
@Hypnotize: Honestly most grade-school teachers are too burned out from grading homework until midnight and paying for classroom supplies with their own money, and aren't looking for even more ways to make their lives more complicated.
If you have a good real-world example and explanation for a specific math concept, pass it along to your kid's math teacher. It will help them.
not "easier" " interesting" is the correct word
I mean... I've had math classes where we had to read about the history behind things like this, and I found it uttermost boring and useless. I've also had classes where we had to read some of the proof for the new concept/rule we were learning. This was sometimes boring, sometimes helpful. But, sadly, sometimes there isn't a way to showcase the proof, or at least not on a high school level, so you just have to accept it. Which sometimes suck, but you get used to it. XD
(damn, I said sometimes a lot.)
One can't possibly overestimate the amount of work that has gone into producing this amazing video.
@Melon Husk Yes. Fixed.
@@hckoenig no no no, I think you mean "Understand". Overestimate means praising the capability of a person or something that doesn't even have the ability to do that. Underestimate means doubting the capability of something or someone. Understand is knowing how something works.
@Pradeep Singh Dear Mr Singh, I kindly advise you to go and find the book "My Big Toe" by Thomas Campbell. It will explain a whole lot more as well to you.
Have a nice day.
It seens you underestimated my power!
8574694746473853 universes and a singular rubber ducky probably went into the production
One can't possibly overestimate the amount of work that has gone into producing the subject mathematics
One thing that was really cool about taking abstract algebra was when we constructed the field of complex numbers entirely from real numbers. It meant that the complex numbers were given a definition in terms of a previously understood set of number, which is how all numbers get there definition. (Except the cardinals which are the sizes of sets)
Your videos are so good, thank you and all veritasium team!
I’m 31 years old and remember learning about imaginary numbers but never taught why they exist, or what examples there were in nature. I absolutely loved this video and thank you for making it. It reminds me that things you thought were useless info in school have an immense impact on the collective knowledge of humanity.
This is a common problem in school, especially math. You are not taught why, just how to apply it (and sometimes you don't even get taught when to apply it).
I lost interest in mathematics after never being told why we use such things as imaginary numbers and their applications. After years of doing math problems you get to a point of questioning why you are doing all of this with no context of application. I think this is a real problem with how math is taught, the student is never explained the relevance of what they are actually doing in terms of the real world. It is sad that math is taught like a menial task to be performed until the correct solution is found yet you don't understand why you are doing it.
@@Leaptab The problem is, the field of mathematics in the pure sense indeed is never concerned with the application of math itself. You will notice to in this video, that the "invention" of imaginary number theory far predate the application of the imaginary number itself in real life. If you find math to be menial task just because you don't know the application in the real world, chances are that will be the same feeling you'll get if you ever dig deeper into the field of math in pure math research.
@@Leaptab You're conflating the application with the subject. What you're saying is "Why do I learn a physics concept instead of an engineering concept?" and it really misses the point.
If you take the time to understand the *why* in maths, it can take you a long way
This comment will blow up soon
Maybe idk
Generations, Prolly
This comment will indeed blow up, now we shall wait…
Math is the only subject which seeks absolute truth. The WHY will indeed take most amount of time here
I was a chemistry major and tolerated math because I had to and I was fairly good at it. I might have been far more inspired and really good at math due to applied inspiration if I had seen this video 20 years ago. Every student taking algebra and calculus should have to watch this video at the beginning of those courses.
What a wonderful video! I definitely going to use this to show my students the importance and utility of complex numbers. Thanks a lot!
The part where he explains how to solve the equation with literal visuals in my opinion should be taught in schools. It helps people grasp the concept much more easier.
I too think it should be shown, but only later on. I was shown the geometric version of the Pythagorean theorem and it didn't click until it was shown to me later on
I mean, the whole idea of it makes negative numbers literally impossible, so that might not be the best way to teach math.
2d variant is taught in schools. It is called the Pythagorean theorem. And all sqare equations could be transformed into a perferct square.
Well its quite similar to algebra however alot more understandable.But anyways there is no easy way to show maths.
Much easier*. No need to add “more”.
So that's why its called "completing the square". Damn, that's cool.
my eyes widened at this part lmao
literally had that realisation too 😩😩😩😩😩
I've never actually known how to complete a square, and still don't know. It doesn't come up designing beam-column fames.
EDIT: I was thinking about it after this comment, and after seeing the visual, I realized I do get what completing the square is and how to use it. The image of that square being completed, with x+c/a sides just makes too much sense.
Ha, good comment
I love revisiting these sorts of videos, listening for the cool big words I heard on the first viewing, now recognizing them, their concepts and real world applications.
Awesome video. I am going to show this to my math students.
Keep up the good work!
"written in five years , may it last for five hundred"
It did.
HS math teacher here: thanks for showing the weird and cool the history of the equations, and visually describing how they all relate back to basic geometric shapes (even when they then veer off into the imaginary land). Definitely borrowing this for class.
Showing the geometric interpretation of completing the square is a must. Please teach this to your students.
Complex numbers are dual to real numbers.
Perpendicularity or orthogonality = DUALITY!
Column vectors are dual to row vectors -- group theory.
Electro is dual to magnetic -- Maxwell's equations.
The electric field is perpendicular (dual) to the magnetic field -- probability waves.
Positive charge is dual to negative charge -- electric fields.
North poles are dual to south poles -- magnetic fields.
Electro-magnetic energy or photons are dual.
Points are dual to lines -- the principle of duality in geometry.
Group theory:- the image is a copy, equivalent or dual to the factor or quotient group.
Isomorphism (absolute sameness) is dual to homomorphism (relative sameness or difference).
Homo is dual to hetero, same is dual to different.
Injective is dual to surjective synthesizes bijective or isomorphism.
Positive curvature is dual to negative curvature -- Gauss, Riemann geometry.
Curvature or gravitation is dual.
Gravitation is equivalent or dual to acceleration -- Einstein's happiest thought, the principle of equivalence (duality).
"Perpendicularity in hyperbolic geometry is measured in terms of duality" -- Universal hyperbolic geometry, Professor Norman J. Wildberger.
Duality (energy) creates reality.
Action is dual to reaction -- Sir Isaac Newton (the duality of force).
Attraction is dual to repulsion, push is dual to pull -- forces are dual, e.g. the electro-magnetic force.
Monads are units of force -- Gottfried Wilhelm Leibnitz.
Monads are units of force which are dual -- monads are dual.
Energy = force * distance.
If forces are dual then energy must be dual. Potential energy is dual to kinetic energy, gravitational energy is dual.
Apples fall to the ground because they are conserving duality.
"May the force (duality) be with you" -- Jedi teaching.
"The force (duality) is strong in this one" -- Jedi teaching.
"Always two there are" -- Yoda.
I wish I had this, I failed math one, then it built on this until geometry. I failed all three classes, it might not have been as bad if I understood what I was looking at, and this did that for me, at least better than I had it before. Hopefully your students get that out of it too
@@hyperduality2838 drugs
@@hyperduality2838 The first statement is completely false. The real numbers are contained in the complex numbers. You're thinking of imaginary numbers. Complex numbers and imaginary numbers are different things.
Thank you so much for merging math and history so beautifully. Absolutely delightful to watch ❤
One of your best ever videos. Wonderful stuff.
You do some wonderful stuff yourself.
your comment is going to blow up
By going into interesting history he made advanced math engaging. It’s an incredibly smart method executed well.
Im no surgeon, but I did help create life once......Does that make me God, or dumb for not wearing a condom.
Hey my other favorite RUclipsr!
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
actually I feel like this video is worth more than 3 years of high school
@@kienthanhle6230 I'm in my 3rd year of Electrical engineering. One of the fields that uses imaginary number the most. Now I actually understand what it means. I finally get why e^x and cos(x) and sin(x) are related.
@@kienthanhle6230 That's BC highschool only prepares us to take and pass tests. Fundamentals and true understandings are never required. BC schools only worry about average grades and test results for funding.
@@kienthanhle6230 Wow you actually said something this cringy
@@BygoneT I don't know how good your teacher at high school is, but mine is pretty bad.
I am simply awestruck by this amazing trip through mathematical history that brings us all the way to the Euler's equation. The graphic explaining e^ix = cos x + i sin x is the best I have seen. Congratulations and thank-you, Veritasium! BTW, is that an iron ring on your finger?
This was the best video I've seen from you guys. Thank you!!
This REALLY feels like a lost episode of the new Cosmos, from the pacing, the language used, the explanation strategy, the animations, to the deep dive into the history, making it all concrete. This is a masterpiece, and it might be the best video you've ever made.
I got that same exact feeling. This is TV quality.
This seems really over the top, Cosmos has extremely high production values for an educational program, and is a lot more accessible to general audiences.
This is great. Cosmos is garbage. Don't compare the two.
@@aguywithanopinion8912 cosmos is really good and so is this video. Wtf are you on about?
@@SahilP2648 what is cosmos?
I love how this video displays how difficult advancing mathematics can actually be, and the sheer imagination required to conceptualize another dimension and it’s properties when you contemporary mathematics has no answers for what’s going on and will treat the discovery as fiction. Some people think math is all logic and that the ancients were fools for not knowing what HS freshmen know today. They fail to understand what the process of advancing a field of knowledge is actually like.
Breakthroughs can take an unreasonable amount of time. But once it occurs, it can be passed on. I completely agree with you and this is why we have to honor the greats. They made it easier to push the limits just a tad bit further. It just takes that one eureka moment.
Crazy to think it takes one hell of an imagination to see and understand reality... lol
Anyone who thinks that way is not that intelligent. Why would someone think that people before us, with less access to knowledge, had less problem solving skills? It would be like claiming the person who figured out that rubbing two sticks together to make fire, was stupid. Sure we all know that now, but i dont think there is anyone in the comment section, that could figure this out, if they were raised by wolves. Knowledge is nurture not nature.
Math is all logic. Logic is what tells you that something is a mathematical statement or just conjecture. Also, it's not so much sheer imagination to come to work with imaginary numbers. Imaginary numbers and complex numbers are fundamentally the number system of algebra, and inherently come out of polynomial solutions. All that was needed was a pen, paper and the ability to not reject something simply because it doesn't immediately make sense.
This is a captivating story and should be required viewing for all math students
I wish math classes would've been presented this way, I'd probably loved it to learn more.
Fior: **challenges Tartaglia**
Tartaglia:
"This opportunity is hard to come by. Well then, amuse me. Surrender is a valid option"
A fellow player I see
Yep, I expected this some where down in the comment
Noice!
Fior is the real villain in this story
Yeah. Had to fi d this comment haha
Tar Tartaglia lover of Snezhnayan queen, he was called in to a math duel.
I’m a physics major. I’ve always had trouble understanding complex numbers and why they exist in equations. It’s like my professors were just handing out the equations like the Schrodinger equation without really explaining what they mean. As I went on throughout college I gathered an understanding, but this video gave me that “aha!” moment. Thank you Veritasium, Your videos are something special and I appreciate every single one that gives me more insight on how the universe works.
yes he is a good "teacher", knows how to explain and visualize things. In math one would learn that "things" exist as parts of other things, real numbers are a subset of complex numbers. And then there's quaternions ... and one would think, does it ever stop... and yes it does, but that means walking into set theory and such, everything is kinda 'connected'. I worked a lot with physicists, they always came to me "to pick my brain".
I majored in math and I always got that impression from the courses taught by physics profs. They seem to think of math as a set of tools that "just work". If you ever want to unravel those tools and figure out why they work, the field of math you are looking for is called "analysis" (probably "complex analysis" and "real analysis" in most universities). Those were always my favorite courses because they helped so much to explain all of those weird formulas from calculus and differential equations.
"i" mean imaginery ≈ imatter ≈ dark matter ??
Math came from nature phenomenon and "i" explainted it.
This comment got a lot longer than I initially intended, but covers a number of things about constructions of the complex numbers and how else they can be thought of:
As a pure mathematics student, I like to think of imaginary numbers as a construction. They are not “numbers” in the same sense as real numbers, but they can be paired with real numbers to produce a helpful construction which allows people to manipulate things in ways that may not initially seem possible. In the context of ring theory, we consider general systems of numbers with addition and multiplication. You can add and multiply polynomials with coefficients from any given ring as one might expect. And it turns out that a construction basically the same as imaginary numbers appears when you “quotient” the polynomial ring of the real numbers by “the ideal generated by” X^2 + 1. If you’re not familiar with this language, that basically means that if you take this ring, but now consider that whenever X^2 + 1 appears (or any of its multiples), it is now considered to be 0. You can see that this can produce the complex numbers intuitively since we are essentially just treating X as a number which squares to -1. Which is exactly equivalent to this idea behind the complex numbers: allowing the number i to exist and square to -1. All the properties of rotation then naturally appear through all the classical studies of complex numbers.
However, a possibly interesting idea from this is that if instead someone had decided that they wanted a *different* cube root of 1, let’s call it j. Then j^3 = 1, and (-j + 1)^3 = -1 + 3j^2 - 3j + 1 = 3j^2 - 3j. But noting that j^2 = 1/j (and allowing some algebraic manipulation with some extra assumptions), we can find that (-j + 1)^3 = 1 as well (which matches what you’d get if you treated j as either of the complex cube roots of unity). I think we can all be pleased that no one did find this, because working with this number is a lot more tedious than working with complex numbers as we know them, but it does work.
From the ring theoretic perspective, for this construction we wouldn’t set X^3 + 1 = 0, but instead we’d use X^2 - X + 1 = 0 which is (X^3 + 1)/(X - 1). This is basically so that we don’t have worries about X actually being 1. Obviously this comment misses out on a lot of rigour, but is intended to provide the general intuition behind the fields mentioned and give an alternative perspective of complex numbers.
Complex numbers are pretty cool once you get really into the theory on it. Like it’s applications and stuff is cool but things like reimanns hypothesis or schrodingers equation are very interesting
Thank you, really. I want to express to you my gratitude to have stumble upon this video, and the joy it brings me.
Absolutely LOVED this one! Thanks for making math fun again ❤
I wish schools taught the geometric way of solving quadratics and cubics first, then the algebraic way would’ve made much more sense for many students.
You're country !
Definitely
true
That would require teachers to actually teach.
In the Montessori Curriculum, they do teach like this
This was a fascinating insight into the origins of the mathematics that's so familiar. Wonderful. Thanks Derek!
Cool
Hey guys this is Derek from more plates more dates
Yeah, video was really interesting
*VERITASIUM* is my inspiration!! My mom said that if I got 30k subscribers!! She definitely buy me a professional mic!! *begging you GUYS alot* literally begging.!.
Also isn't it surprising no nobel prize in mathematics and it is continuing, the members of nobel committee should announce that nobel prize should also be given to mathematicians for their work
Simply one of the best videos about mathematics on RUclips.
I believe this is the best video you've made, out of a lot of excellent ones.
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!
mindblown
Same as multiplying by -1 but with intermediary steps.
Hell yes that's cool- teaching my students that this week...to see their faces light up made my school year.
You should look into the polar coordinate system in the complex number field. It basically uses a similar system. Instead of an X/Y coordinate system (where X is the real number and Y imaginary) it uses a magnitude and some rotation.
Complex numbers are just great. You basically do a "Let's assume that i is a number, and that its square is -1" exercise. You get all the basic maths really fast. For example:
Let's assume that (a + b i) and (c + d i) are complex numbers, and a, b, c, d are all real numbers. Then,
(a + b i) + (c + d i) = (a + c) + (b i + d i) = ((a+c) + (b+d) i),
(a + b i) - (c + d i) = (a - c) + (b i - d i) = ((a-c) + (b-d) i),
(a + b i) (c + d i) = (a c) + (a d i) + (b c i) + (b d i²) = (a c - b d) + (b c + a d)i.
Division is a bit more complicated, but you start with a twist: try multiplying (a + b i) by (a - b i):
(a + b i) (a - b i) = a² - a b i + a b i - b² i² = a²+b², a real number. So,
(a + b i) (a - b i) / (a²+b²) = 1, as long as the denominator isn't zero.
In other words, the inverse of (a + b i) is (a - b i) / (a²+b²). (That's a division by a _real number,_ or a multiplication by 1/(a²+b²), so no problem there.) To get complex division, you merely have to multiply the inverse of the second term by the first.