Imaginary Numbers Are Real [Part 1: Introduction]
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- Опубліковано 28 вер 2024
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Want to learn more or teach this series? Check out the Imaginary Numbers are Real Workbook: www.welchlabs.c....
Imaginary numbers are not some wild invention, they are the deep and natural result of extending our number system. Imaginary numbers are all about the discovery of numbers existing not in one dimension along the number line, but in full two dimensional space. Accepting this not only gives us more rich and complete mathematics, but also unlocks a ridiculous amount of very real, very tangible problems in science and engineering.
Part 1: Introduction
Part 2: A Little History
Part 3: Cardan's Problem
Part 4: Bombelli's Solution
Part 5: Numbers are Two Dimensional
Part 6: The Complex Plane
Part 7: Complex Multiplication
Part 8: Math Wizardry
Part 9: Closure
Part 10: Complex Functions
Part 11: Wandering in Four Dimensions
Part 12: Riemann's Solution
Part 13: Riemann Surfaces
Want to learn more or teach this series? Check out the Imaginary Numbers are Real Workbook: www.welchlabs.c....
![Imaginary Numbers Are Real [Part 2: A Little History]](http://i.ytimg.com/vi/2HrSG0fdxLY/mqdefault.jpg)
![Imaginary Numbers Are Real [Part 2: A Little History]](/img/tr.png)
![The most beautiful equation in math, explained visually [Euler’s Formula]](/img/n.gif)
thank you Gauss. It sounds much less awkward to say that I have a lateral girlfriend
@Jalfire: Me . . . I just keep it to myself and don't mention it to anyone else at all.
so original
@@QED_ lateral girlfriend= mistress
Lmfao😂😂
still awkward though
By the way, imaginary numbers ARE called “lateral numbers” in China.
It could just because it’s easier to pronounce(less syllables in the Chinese language), but Gauss would be proud
Very cool
Well, imaginary numbers in Chinese still has the ‘imaginary’ meaning. It’s called 虛數 I think
“The Tiananmen Square protests are lateral”
@@masterspark9880 LMAO
@@masterspark9880 legendary
its all fun and games in math class until the graph starts speaking 3d
You will see Fourth Dimension in future, which you will not express or understand in 2d papers like you do 3 dimensional shapes.
be still my heart!
@@Email5507 impossible to understand, impossible to imagine, we can only "speak" about it, i love it!!!!
@@Email5507 stoner 1
@@vladymartinez1232 stoner 2
The presentation of math has never been so fun and interesting like this one here. Kudos to thee. 10/10
Thank you!
@@WelchLabsVideo Keep Making *Great* Videos. And Thank You For Such An *Amazing* Explanation.😀
The easiest way to understand negative numbers is by picturing my bank account.. 😔
And if u don't have any account like me
@@zekzimbappe5311 Watch other peoples poor bank accounts.
@@zekzimbappe5311 then that's lateral bank account
I LOVE YOUUU
And mine imaginary numbers. ;)
I never knew I could have that much fun watching a math video, well done.
Standupmaths mang
More real world applications would've been nice for us noobs. So, thumbs down.
Numberphile has some cool video too
KommentarKanal I knew I was in for a show the minute the video title mentioned imaginary numbers being real. Better Explained already demonstrated how the number line is really a number plane, and how multiplying by /i/ is like rotating rather then scaling or stretching, but seeing it visualized like that made my day.
Mathologer is cool too :)
I show this first video of the series every single semester that I teach Algebra students about "imaginary" numbers for the first time. Really gets through to them!
I just did the same an hour ago.
Math: If I have two apples, and I give you one, I will have one apple left.
Finance: If I have two apples, and I give you one, you will have to repay me the apple in full after a set period of time, plus interest which is to be calculated as a percentage rate of the apple divided by the amount of time it took you to repay me the apple in full.
Very true indeed.
How do I always see see you? On every geography now video I've seen ur comment and now on math? Holy crap man
Politics: If I have one apple, and I give you one, everyone will shout & scream that they didnt get one & band together to try to force me to give them apples.
Economics: I have two apples, I give you one, but few people realize that apples are produced in a farm, and are worried that there isn't enough, and not even Apple farmers seem to know where apples come from (except the Bank of England Apples which plainly stated the truth).
I'm MMT. A Neoclassical Economist would describe things that I think are not true and responsible for the mess economies are in (because they are run on the assumption that the currency issuer should behave like a currency User, & other things that don't apply anymore to modern money):
ua-cam.com/video/TDL4c8fMODk/v-deo.html
Finance is math
this is truly why a lot of people find math difficult to understand. a lot of the names are grotesquely indescriptive. if they had more intuitive names, people would be able to pick things up much quicker, instead of having to first memorize what it means, in addition to learning how it works.
Awesome video! I loved that visualization where you pulled the surface out of the flat paper, that was a big WOW moment! I've worked with imaginary numbers a ton, I studied physics in college, but this video still had an affect on deepening my understanding. Excited to watch the rest!
A picture is worth a thousand words
What I don't understand about that visualization is that after he pulls the surface out, there are an *infinite* number of roots. I thought he just said that there are exactly as many roots as the degree of the polynomial?
frother - There actually only two roots. The "infinite" intersection of the 3d parabola to the imaginary plane is actually just the extension of the whole parabola through 3 dimensions (x, y, i ). Two roots can be seen by taking a different "slice" view point along the new dimension parallel to the coordinate plane (3 units above paper). This will give a new coordinate view of the parabola that does indeed intersect at two points.
That's a good point, but it's only a problem with the visualization. In fact there are only two roots.
The problem is that to really plot the function, we would need 4 dimensions, not just 3, since the input of the function requires 2 dimensions (real and imaginary/lateral) and the output is also a complex number so it would also need 2 dimensions to plot properly. In this visualization they simply didn't plot the imaginary part of the output value of the function, only the real part. And there are indeed infinitely many complex numbers whose square's real part is -1. But for most of them there is a nonzero imaginary part (except for the 2 actual roots, i and -i).
Thanks, I never expected to get such a clear and helpful answer from the youtube comments!
Here again, after a few years. Just wanted to let you know that, watching this was definitely one of the most memorable moments in my math journey. I got a whole lot more interested in Graphs and Complex Numbers, learnt to accept them as a concept that weirdly works.
euler: -1 > ∞
He predicted integer overflow
Can you specify what integer overflow is? I'm sorry I dont know lol.
@@xwqkislayer7117 in computer systems, if a number is too big to be stored, it loops back to a negative number
example: Let's say we have a binary system that can store 8 numbers: 000, 001, 010, 011, 100, 101, 110, 111
If we want to represent negative numbers, it makes sense to put them before the positive ones, so let's say:
000 = -4
001 = -3
010 = -2
011 = -1
100 = 0
101 = 1
110 = 2
111 = 3
so the biggest number we can represent is 3. If we had another digit, we could have:
1000 = 4
But we don't. So if we tried to ”add 1” to our 3, it would be:
111 + 1 = (1)000
so our system would see 000 and think it is -4
This is integer overflow, when we don't have enough digits to represent big numbers which causes a mistake that turns it negative.
@@nuklearboysymbiote Thanks I didnt know that lol
@@xwqkislayer7117 i simplified it a little bit to get the idea across, please keep in mind this is not exactly how computers represent numbers. computers are actually built to represent negative numbers using a thing called two's complement: if you have a positive number, flip all the digits, then add 1, that will be how you represent its negative.
This way, we can actually represent 0 as 000
e.g.: 2 is represented as 010
so to get -2, you do 101 + 001 = 110
this way, you can add the individual digits to get 0 back:
2 + (-2) = 0
010 + 110 = (1)000
The maths is easier this way. That also makes it easier to recognise which numbers are negative, as the first digit will be 1 if it's negative, and 0 if it's positive (-2 = 110, +2 = 010)
@@nuklearboysymbiote ah ok ill keep that in mind. Thanks for the info
Alternately, a better way to think about it is that no mathematical systems are 'real' in that they are necessary to describe physical observations, they're all models we made up, imaginary numbers are just a useful extension to one system of math that allows us to describe a certain system of useful relationships a fairly compact way.
This is one of the best explanations I heard about anything. Incredibly well done and "easy" to understand! I wish they could teach at university or school like this :/
It's an (uncommon) misconception that Euler "didn't know what to do with negatives". Euler was the most productive mathematician to ever have lived. He dealt with complex numbers and complex functions in full generality, it is simply nonsense to say that he didn't know what to do with negative numbers. (It is true that he assigned negative values to some positive (divergent) series, but that was 100 % intentional.)
1:47, it seems the solution to x^2+1=0 is a curve (or two) instead of 2 point (+i and -i). actually x is in 2 dimensional plane, so is f(x). so it requires 4 dimensional to show the function.
never stop doing these videos they are the best out there. thank you so much for taking the time to share them. with us.
"Numbers are lame. Let's invade something!" - LMAO! Subscribed! =D
I "learned" imaginary numbers at some point in school, very briefly. But I never truly understood them. You managed to do a better job in 5 minutes of UA-cam video than 20+ years of education. Finally I truly get it, and it's not even hard. Completely demystified, like a great cloud has been lifted. You are a legend.
Woohoo!!
you should change the text of "0!" to just "0" or "0." since 0!=1
Word.
Welch Labs 0!=0, 1!=1, 2!=2, 3!=6; no?
Leonardo Aielo Tassi nope, 0!=1
One way of seeing it is by thinking that the factorial function tells us how we can order stuff, A&B can be ordered {AB} and {BA} 2!=2
{A}gives just one "{A}" (1!=1)
And the empty set { ø } can be ordered in one way {ø} 0!=1
Dalitas D
WOW! A totally unexpected but revelatory and logical answer.
Graham Lyons
x! = x * (x-1)!
If 0! = 0
Then
1! = 1 * 0! = 1 * 0 = 0
When you "pull" the graph up and make it three dimensional, then yes it crosses the X axis, but it suddenly looks like it crosses it in a lot more places than just 2... and it should only be 2. So I dont think that 3d model was a good representation
yeah, that's what i was thinking
The actual function values would be the outermost edge of the shape, the actual extension of the plotted line, not the interior area. Which would be a _different_ but related function (probably involving calculus). It was filled in only to provide visual context for us viewers.
Wait for the last part. He explains this specific issue.
The exact point is mentioned in the workbook, take a look at it.
Actually, this is a prank video by some jerk, cuz for the eq f(x)=x²+1, we are working with only 2 dimensions. Where the hell did you get the 3rd dimension from ? so for every question, just simply add another dimension if can't solve it?
21st century: "let's call them fake numbers"
22nd century: "flat numbers, because Earth is flat, so is everything"
23rd century: "Numbers are individuals too! Each number should have a name! Isn't that right, Richard?; *-3:* _Yes._ "
Every number has already an own name. So your theoretical statement makes no sense.
25 Century: numbers get to choose their gender.
@@gdash6925 No, my 3 is called Richard. Your 3 is called, I believe, Timothy. Your statement is so numberist.
@@_Killkor my 69 is called..... wait
Microsoft Hites
26 century: Numbers become humans.
I kinda miss when I was a kid and these parts of math seemed absolutely baffling. They're still amazing, and there are still of course very mysterious areas of math, but learning these things for the first time was like learning magic
@5:15 we needed students to know things like negative numbers so they can understand what debt is
AWESOME : Watched the whole serie! THIS was the best and most intuitive explanation of number theory and complex number ever, where also math newbies could follow and get a deep understanding! Thank you soo much was this highly entertaining and educative masterpiece! ❤️👍🏻💡 that was a tremendous effort of work and brain you put into it! 😇
you were really bad at sticking to your guns. lateral numbers lasted like 30 seconds
Where my doom fans at?
Thank you so much! 'Imaginary' numbers were my big stumbling block in A level maths, and my maths teacher was unable to explain them (because he only got the job for being the headmaster's old chum). This video has a made it clear for the first time to me. If only my maths teacher had explained it as another dimension like this, instead of "You don't need to know how it works, just memorise how to use it to pass your exam". I might have passed that A level and become an astrophysicist as I wanted.
No, don't go there. You were a rational human being before when you couldn't understand imaginary numbers. You actually knew intuitively that it was all a load of garbage and just fantasy. Now you've come to accept them as real when they aren't. Go back to the light.
@@tomjscott They're demonstrably real. Physicists have demonstrated that our most fundamental powerful theories of reality only work when using complex numbers. They're as real as any other number system; to assert otherwise is ignorance
so my imaginary friend was real?
Yep, he was just in another dimension
Dude, no maths in the Medieval Europe? What about Fibonacci and Oresme?
Fibonacci was counting breeding rabbits
Or possibly he was breeding counting rabbits... ;-)
You need to read more about the history of the civilizations in this era and see where was the math around the world around that time and before!!!
Muhammad Nour Elmogy and you need to learn better english before you comment on something... the guy asks about european mathematicians in the middle ages...
Bill Killernic What's wrong with my English !!!!, My Answer was more generic for his exclamation, have you read my comment properly?! or you just commented without even bother reading it!! have you even read his comment!! you need to have more wide open sight for other's comments, I am sure you misunderstood my comment or his or both! and BTW I was confirming the phrase and emphasizing it, medieval Europe didn't have much until it's late years and after that a lot have changed, read the history for your own good and you will know (if you were searching and reading in the right places) how far was Europe in this era and before from anything related to something called science!
Welp... negative numbers appear in a lot of formulas in physics, that DO describe the real world :P
the earliest you would see negatives is with charges, we use the negative sign to describe the opposite charge to what we call the positive charge c:
same with complex numbers in quantum mechanics!
Honestly this is the most important video out there explaining imaginary numbers. This has to be archived in museums for generations to come. Thank you very much for the important work!
Glad you enjoyed it!
I could happily be studying for this now!
Gauss's way of thinking is pretty cool
Agreed!
Because of the new video posted by Veritasium, a lot of videos about imaginary number are being recommended to me.
1 minute in, can't wait for the next part!
Nicely Done :D
Wow, when you pulled that 3rd dimension plane up that graphic was incredible! I still don't fully understand **why that plane is the √-1** but this is the first video I've found that put the context of **what √-1 is functioning as** instead of just stating **that i=√-1** with no context
Me: well, math isnt even that bad, atleast it all makes sense.
Teacher: today were gonna learn about imaginary numbers.
Me: 就能5得到的确有
I remember find this series years ago, it made me think of complex and imaginary numbers as completely natural and not strange at all, i want to thank you for being such a great teacher!
Thank you!!
@@WelchLabsVideo Hello, please I think this is an amazing video but I would love if you could include the resources where you found all this info so people can do further reading . Thanks
4:51, Nah, not the anti apple, it simply means *"I was supposed to lose one more apple if I had it"* in the real world.
And well, we live in a 3rd dimension which is a "real dimension", so for us to practically visualise a negative apple would be impossible.. imagine if there were negative and lateral dimensions :)
Love the visuals you created, makes things so clear! Thank you!
If there's one thing that's more incorrectly named than imaginary numbers, it's the fundamental theorem of algebra.
@Mariana Duque It's not really that fundamental to algebra. It probably should be called the fundamental theorem of complex roots.
Thank you sir❤ it was very helpful to visualise the things... the efforts you took to make these videos was really appreciable ❤... I wish u make some more visualisation vedios on math🥰
X^2 + 1 is a polynomial?
What’s a polynomial?
N N Oh boy here we go. Practically anything that contains a series of variables or variable to a certain "degree" or power. Thats the def. in my own words, if you want a more broad explanation do some research.
i think its a type of plastic made of numbers but i could be wrong
4:53 the anti apple haha hahaahahaaa...😅
한글자막도 있어서 너무 잘 보고 있습니다. 이곳에 항상 축복만이 가득하시길.... 감사합니다.
if we call "imaginary" numbers "lateral", what do we call "real" numbers?
also why didn't you stick to your convention to call them "lateral"?
non-lateral numbers? ¯\_(ツ)_/¯
since "lateral" seems derived for "latitude" i was think more a derived term form "longitude" 😂
Fuseteam
Direct + Inverse, I guess.
Gauss didn't say anything other than those.
we don't have to ask Gauss, are we not imaginative ourselves? 🙃
the first that comes to mind when i hear "lateral" is latitude, which is also used on maps in terms of latitude and longitude for example so "longitudinal"? feels clunky but surely there is a word that would show the relation between the two, since that was the intention of gauss calling them "direct, inverse and lateral"
You can call the real numbers as Johnny if you want. I wouldn't mind at all.
2:42 I find it most disagreeable you write G as 6
i never understand why people want to have real things in maths. Everything is abstract, wether it is real numbers ( which is not a proper name), an imaginary number, a function, a polynomial, a ring, a field, nothing is "real", it only exists in our mind
Agree. Math is abstract but it works!!!
True. Imaginary numbers are no more imaginary than rational or irrational numbers.
That's because modern views and traditional views are different. Times changed and so does ideas of people. Back then, math was heavily grounded to reality. Hell there wasn't even algebra back then and solving math problems mostly involved geometry.
Math wasn’t abstract back then. It was used to explain reality and solve real problems as the video outlined. They had no idea of the applications of negative numbers because they were new and didn’t exist.
I would say quite otherwise: in math (or within math, if you will), everything that's not self-contradictory (and not in conflict with other math established so far) is real.
Negative numbers connect very well with the real world. The answer is one simple world, relativity.
x = 50 meters
y = x - 100 meters
Where is y relative to x?
-50 meters
GG EZ
Where is a negative direction in the real world?
Lol, negative numbers in physics mean the opposite direction for direction quantms. Good try tho.
@@DajesOfficial the opposite of a positive direction, if x units in front of you then -x, x units to the left = -x units to the right, x units up = -x units down
y relative to x is -100 meters :)
@@nandakoryaaa exactly... what a fail of him :D
The beauty of the imaginary number is that it proves science and even math are based on the definitions of an observer. i has many applications but so too does quantum mechanics exist as a discipline. It amazes me to see that these two are not often, if ever, discussed in the same sentence. Humans apply so many things in binary terms (one vs another) that the imaginary number seems more an artifact of a fractured hive mind than a prerequisite for understanding the cosmos.
Numbers are lame, let's just invade something. HAHAHAHHHHHHHHHHAHAHAHAHHAHAHAHAHAHAHAHAHAHAHA
May be Poland
Lol what?
ترجم لنا ايش قلت لهم ؟
a benzene molecule but muslim arabs invented algebra
+Poison Cake algebra was even before
I like to call them multi-dimensional numbers :D
This touches on a theory i've had for a while now which is that the Cartesian system limits our ability to solve problems since by design it's linear. What if we were to make the x and y axis non-linear functions? For example, if a coordinate system was based on a parabola, then a plot of a parabola would appear as a straight line.
Yes, I actually had an exam last year with a question that required to do exactly that, there are lots of types of coordinate spaces for different problems.
I am tripping RN, but i cannot be the first to notice how lovely his voice is
Big gay
I never realized how important negative numbers were.
Then I checked my bank balance. :')
Thaks bro .im so tired to research in several month how to understand asyntope.but now 👍,your the best explanator
Medieval Europe whitout Maths? Are you serious? I think it's absurd as so many people try to describe the Medieval age as the age of darkness.This is the work of modernist ideology.
As is the bashing of the name "imaginary". It's pretty good name for those numbers and I don't see how is it confusing. You do need more imagination to perceive them, than say, natural numbers
yeah, we lost the knowledge how to make concrete for nearly half a century, but do tell me about this period of unparalleled scientific development /s
Some things never change. Even hundreds of years ago it was apparently a meme that mathematicians are the worst at naming things, as per Gauss and his, “ill adapted notation,” quote.
At 1:45 how to do this animation? what tools, please? somebody knows?
Negative numbers are still abstract even when the duality of nature is considered as in positive and electric charge. The abstraction is in the recognition of a sign convention applied to measurement.
When you have a string with beads of some size. There is a point beyond which the beads have to bend to get out of line. Imaginary are such strings attached. Why they bend at two is because they occupy parallelism. Parallelism is quite common planar vectors. Even charge shows parellelism.
I LIKE YOU I LIKE THE WAY YOU TEACHING IMAGINARY NUMBERS ...... FIRST INSTANT GIVING THE IDEAS TO UNDERSTAND IMAGINARY NUMBERS ................
@0:05 That is not a nice parabola. It is pointy.
@woowooNeedsFaith: LOL. If you don't have any imagination . . . then this video is obviously not for you.
pointy is good. it scares people.
This is helpful for my brother who's mid school students.
It's a cool concept of a category of numbers that basically have the reverse properties of "normal" numbers. I learned multiplying 2 negative i together the answer is still negative, which is the essential feature for the square root of -1 to be possible
What crazy other categories of numbers will be possible?
Fun fact: Imaginary time is equivalent to temperature.
Well the question is if even Negative Number are real, or if they are just a summary of rules and methods.
And the Complex Numbers (or Imaginary Numbers) just a method to overcome situations where the rules for Negative Numbers do not work.
my favorite imaginary number has always been -0
aeroman5000 the fuck are you talking about? -0 isn't a thing you moron.
This video is about IMAGINARY NUMBERS so I mentioned my favorite IMAGINARY NUMBER you dip shit. Your not the brightest black crayon in the box.
-0 is the same as 0. It lies on the imaginary axis, so it is imaginary. It is real at the same time. Also, try to ask and answer questions without swearing and insults.
You should mention that to aquaified before mentionioning insults to me as if it wasn't for his response mine would have been different. Did you even read their comment? .....😧
aeroman5000 dude, did you watch the video? Imaginary numbers aren't some made up thing, they are specific and useful numbers. -0 means exactly the same thing as 0, there is no negation to 0.
Title: Imaginary Numbers Are Real
My Brain: Stops Working
Videoyu izleyince anladım ki , bizlere liselerde üniversitelerde matematik öğreten hocalarımız sadece ezberlemiş ve ezberlerini anlatmışlar. Anladıkları sanmışlar ve farkında olmadan sadece ezberlemişler ki bize de böyle altı boş bir şekilde anlatmışlar. Yıllardır bildiğimi sandığım şeyler oysaki 1 adet görselle ne kadar da yanlış bildiğimi fark ettim. Thank you for this video.
That animation @01:41 , my mind just blown , maths hasnt been like this.
-1» infinity
Pythogoras And Archimedes wants to know your location.
Sorry I don't have a greater than sign... I have a double greater than sign lol 😂😂😂
>
Good video. At 1:56 please don't put exclamation marks after numbers in math videos. I read that as 0 factorial and was confused for a minute.
How did he graph the two dimensional graph in 3d space when there is no z-component in the 2d function expression?
Infinity is an imaginary number that's impossible to imagine.
Its not a number that u can imagine or not imagine it denotes a limit condition I guess.
So is 0 in a sense (particular mindset)... its not number but a condition that denote simply absense condition....so does negative numbers has some not so clearly define boundaries....for me at least as of now. Till we dont go into power stuff it real its behave just consistently with other normal concepts but just as soon as we step into power domain it becomes imaginary.... Had -2 x -3 been = -6 instead of +6 ..underoot of -4 wud just have been -2 I suppose not a imaginary stuff. But then why is -2 x -3 a plus 6 ..is the question. Why it had to be like that?
I think I need to retake primary school maths classes again. :)
So does that mean my imaginary girlfriend is real?
Jk I don’t have one but I had to make this comment.
Yes. It just means she is a negative debt and lacking a dimension idk.
Traducción de Lo que dijo Gauss para mis hermanos latinos.....yo si deseo tenerlos en cuenta 🤬
"que este tema haya estado rodeado hasta ahora por una misteriosa oscuridad debe atribuirse en gran parte a una notación mal adaptada. Si, por ejemplo, 1, -1 y la raíz cuadrada de -1 se hubieran llamado unidades directas, inversas y laterales, en lugar de positivas, negativas e imaginarias (o incluso imposibles), tal oscuridad habría estado fuera de discusión."
but doesnt that animation prove that imaginary numbers are in the second dimension, when what we've been taught is that complex numbers are two dimensional?
Thank God for this video I was going crazy all this time thinking I was jus “imagining” that imaginary numbers was jus a part of my imagination 😂😂😂😂
Hence it proved there are muti-dimensions in world.
How does it prove that? Because of some arbitrary word choice in a conceptual logic language that people invented?
My 9-year-old son could pretend to be a cowboy before he watched this video, but now he can't wear hats because it made his brain expand too much.
can I use this to teach my kids? Haha kidding, I wouldn't put children in this cruel world :P
first take care of the ones already here :-)
Like we invented a numeral for zero, we can invent a numeral that equals minus one. Call it X. An example is a special base 3 that uses X,0 and 1. We can develop all math functions using these digits. An advantage is that we never need to use the minus sign. If the first digit is X we know it is negative. Of course humans will not want to use this system, but computers might. Is there an advantage to never having to deal with the minus sign?
This video is amazing. Keep it up sir
How did I get through two years of Algebra without being taught the Fundamental Theorem of Algebra? Our school curriculums are bizarre to say the least
4:58, well it's true tho... -ve is indeed greater then infinity..
We all know that, 1+2+3+4+...∞ = -1/12
:)))
Well I think, in imaginary plane, all the numbers make an infinitely big circle, where after infinity, at somewhere transitions to -infinity and keeps moving forward approaching "0" and then enters the +ve numbers again, thus it all has a connection, so his point of view wasn't wrong tho :)))
I maybe wrong, it's just my imagination.. imagining is good ;o
(Btw that 1+2+3+..∞ = -1/12 is true tho..)
This was a really interesting video, thank you😊
WHY ARE THERE ARROWS IN BOTH DIRECTION OF THE AXIS
Great video! But, number 0 was first used by Aryabhatta- Indian mathematician. “Sunya” or zero was found in Hindu scriptures before 1 B.C.
who else stopped understading at 00:00
Brilliantly animated and Interesting
Thank! Now i can visualise complex numbers more effectively
I´m not quite sure, if Euler indeed did not get the negative numbers done, or if it´s rather you who hasn´t fully comprehended the depth of Euler´s Identity yet!?
;-)
Or conveniently applying part of a theorem outside of the appropriate context to prove his point
Are you sure the fundamental theorem of algebra says that? What if f(x)=x^5+1, n would be 5 but we wouldn't have 5 roots, we would have only 1.
These guys will say anything to make a catchy title.
I love imaginary numbers 🔢. Great work
“We’ll be using the term lateral from now on”
*continues to say imaginary*
where are your comments?
It's your imagination, dude.
@@neh1234it's his lateral* now
√anti-apple = banana
@@smallgreen2131 Wut?
Brah, he pulled a rainbow out of his paper.
Drugs
damn
LOL!
*Bruh not brah lol
How did he did that?
JATIN GANDHI editing
You delighted me with the 3D lateral-plane visualization. Well done.
Me too! How did he do it?
After Effects?
+Neithan magic
William Cannon It may look 3D, but it's 4D. It's explained in part 10-13
You´re not the only one!!! XDD!!
I remember a time where I was joking around with my algebra 1B teacher;
"Hey it's kind of wacky that the calculator responds no real numbers does that imply the existence of imaginary numbers?"
"Yes."
I cannot describe the internal panic I had at the idea of seemingly non-existent numbers.
Yo, that's a kick in the discovery, I wish I had thought of that before when I was taught about the set of all real numbers
Wait, calculators don't respond real numbers?
Rip
Something similar happened to me lol. And then the teacher just breezed right by it! It was mid lesson, and she was just like “Oh yeah numbers that don’t exist exist, but that’s high school stuff, anyway…”
@@the_demon149so sad they didn't digress for a minute. minds are open far before H.S....perhaps more so