Imaginary Numbers Are Just Regular Numbers

[@timothystockman7533]

"We're sorry. You have reached an imaginary number. Please hang up, rotate your phone 90 degrees, and dial again,"

[@EtemKaya]

😂😂😂😂

[@2MC]

New life goal. Use the phrase "they darkened the doctrine of equations" whenever discussing complex math.

[@upandatom]

haha let me know how that goes

[@Will-le8yj]

best comment ever ahhahahahhaa

[@yosefmacgruber1920]

So how would you describe the quaternions? Instead of 1 imaginary part, they have 3.

[@Gypsy_Danger_TMC]

I say this all the time when talking to physics students

[@gonza8148]

Now i need to find someone that want to discuss complex math hahahaha

[@SusiBiker]

The rotation method of visualisation for complex numbers was taught in my school back in the mid '70s. I never understood why it seemed to fall out of favour in later years. When I had young electronic engineer trainees working for me in the '90's, complex numbers were always awkward for them - they had learnt by rote, without gaining a deeper understanding. Don't get me started on one poor kid who was freaked out by 'i' and 'j' - "But why were we taught differently..?!" :) Once I explained it the way you did, they all began to understand. What the hell is up with schools today? Oh yeah -> "Targets." Hit the marks specified in the curriculum, but have little or no deeper understanding. It's really quite sad the way things are declining in this respect. Thank you so much for making this video. I am sure you have helped a lot of people grasp what is going on.

[@Lawrence330]

Rote memorization is a terrible way to learn. Especially today, with nearly infinite access to references (both good and *ahem* less-good quality), I believe that (most) school should be about demonstrating the ability to solve problems rather than the ability to memorize tabular data. In my engineering curriculum most testing was open-notes, and even closed testing sometimes allowed a "crib sheet" of your own design. My core physics and math classes, however, expected memorization of relationships and formulas. Memorization comes with repetition, but let me say that repetition should come with use (Ohm's Law, for example), not studying for hours to memorize equations for the heck of it. The equations that you use all of the time WILL start to stick, and the lesser-used relationships will always be there in your reference books when you need them. It seems to me that I'd rather trust a bridge built by someone who checked their references than someone who is "pretty sure" they remember correctly. Everyone I know has had at least one test where they thought they did well and were later surprised by a poor grade...

[@yosefmacgruber1920]

I figured out the rotation thing, from playing with my TI-89 graphing calculator. What happens with (-1)^x when x is not an integer? The math expression that it gave me, showed that it rotates along the unit circle on the complex plane, and that -1 and 1 were just special cases of when x is an integer. Thus *_i_* now makes more sense, as it is just the half-180º rotation of multiplying by -1. As shown in (-1)^(1/2) or √(-1) as it is often stated. Half a factor would result in a 90º rotation to where? Somewhere *not on* the real number line? *_i_* and *_j_* ? Exactly how do those relate to the quaternions, which also has a *_k_* ?

[@aleksandari.7834]

I could give you the outline of the magnitude of the problem . Imagine this: 1. my math teacher from high school would made some clumsy comments about hotter girls in my class, and he was in his 50s. 2. I did not understood any new mathematical field we tackled from his explanations, as they were so damn poor and vague. 3. From a friend who took private lessons with him, I heard that his explanations were excelent. My logical deduction - he knew math, yet hesitated to teach us, because he wanted to look smart, especially to the girls. I spent 3 years with him, and I sensed some things. Might not be, but it does not matter... The point is , that It is not only a problem of curiculums, and such. but that we are dealing with very problematic people. This guy made me think that I am stupid for math, and altered my life course onward in a sense of education, because he was satisfying his patological needs. And I am just one of many with that story. We are dealing with problematic teachers who are in need of therapy.

[@DavideVerde]

I can try to answer for the "i" and "j" ambiguity.. in the 16th century some Italian mathematicians were competing to solve the cubic equations and they all came up with solutions involving imaginary numbers; they were Scipione dal Ferro, Niccolò Tartaglia, Gerolamo Cardano, Rafael Bombelli. The point is that in Italian "i" and "j" were just two different graphical ways to print the "i" letter. "j" was just a nicer way to print "i". You could write Julius Caesar or Iulius Caesar, it was just a graphical variation. Letter-shapes were standardized later, thanks to the huge spread of Gutenberg's press. I would just use "i" as "imaginary" and forget about the j usage

[@amramjose]

Yes, the rotation explanation was excellent. Cheers!

[@AmitSenguptaPlus]

I'm happy that we didn't end up calling sqrt(-1) as undefined.

[@NiHaoMike64]

Don't forget division by zero.

[@notafeesh4138]

Yeah, why can’t 1/0 be defined as j?

[@HeroicJay]

@@notafeesh4138 There are branches of math where the square root of a negative number is useful and has real-life applications. And using i as a number expands math without breaking anything. No one has ever found a real-life application for a number that is the result of dividing a real number by zero (sometimes the LIMIT of such a procedure is useful, though), and such a number breaks math pretty easily were it to exist. (Nearly all "proofs" that 1=2, or whatever obvious nonsense, get there by dividing by zero and pretending they didn't.) 0/0 is a much more complicated case, as in some contexts there's an actual number hidden by that 0/0, but you can't tell what it is just by looking at the 0/0. (After all, the "result" of 0/0 is the x in 0 * x = 0, which could be basically anything.) And sometimes it's just as meaningless as 1/0.

[@Lawrence330]

@@notafeesh4138 It's tempting to thin of n/0 as "dividing zero times," but it's actually dividing into 0 parts. I would be tempted to say n/0 = 0, but I'm not a "math-magician" as my physics prof used to say.

[@jan31416]

Actually it is undefined, if you don't want to modify the power calculation laws - and I don't want to do that. But i is a solution of the equation x² = -1. Why is sqrt(-1) undefined? A small trick can illustrate that: 1 = sqrt(1) = sqrt(-1 * -1) = sqrt(-1) * sqrt(-1) = i * i = -1 One bad solution for the problem is, that sqrt(-1) is sometimes -i, sometime i. But even than it is not well defined.

[@CorneilleBlanc]

As a senior in physics and having worked with complex numbers all the time, I’ve never imagined the rotation like you did it. Wonderful to have new simple ways of viewing a deep topic.

[@MoempfLP]

3:07 R.I.P. Hippasus

[@drenzine]

I mean. His Hip is hella Sus (Yes, it is 2 years ago the meme probably did not exist)

[@MoempfLP]

@@drenzine very sus

[@acbthr3840]

So... i is literally 1 rotated by 90 degrees? Why isnt this explained to people when they learn about this??? Woulda made phasors a lot more understandable in electronics classes!

[@davidsonjoseph8991]

Acb Thr "j" in electronics

[@obinator9065]

@@davidsonjoseph8991 90° rotation is more important.

[@obinator9065]

People always think of the square root of -1, when hearing i, but really, almost every idea that uses it is interested in the algebraic rotation.

[@acbthr3840]

Dvd Ftw Uh... yeah. Thanks. I know. Most software uses i regardless.

[@rmsgrey]

Because that's not how imaginary/complex numbers came about historically. Originally, they came up when trying to find a general way of solving cubic equations - if you try to find a solution algebraically, you get intermediate steps involving taking the square roots of negative numbers, even when the final answers are all real numbers. So mathematicians imagined that those values were meaningful, and worked out how to manipulate them without ever treating them as really being numbers. It was a couple of centuries after that before anyone thought of a geometric interpretation of complex numbers as a complex plane. So, tradition, mostly. There is also the point that there are more natural ways to conceptualise rotation - in general, multiplying by a complex number both rotates and scales, so introducing complex numbers as a way of doing rotations immediately raises the question of why you'd invent some weird two-dimensional numbers to represent rotation when you can just continue to use a 1-dimensional angle. The algebraic approach gives you a scenario where complex numbers are actually necessary, not just useful, even if polar form is more convenient for many purposes.

[@ericpatterson8794]

The term "imaginary numbers" was originally a derogatory name given by a skeptical mathematician when they were first conceived. Somehow it stuck.

[@upandatom]

same for the irrationals

[@marctelfer6159]

And for the Big Bang, if I remember rightly

[@Lucky10279]

@@upandatom Aren't they called irrational because they can't be expressed as a RATIO of integers?

[@ragnkja]

Up and Atom No, in the case of “irrational” it’s the non-mathematical meaning that is unfortunate, since the mathematical one is pretty literal and straightforward.

[@lyrimetacurl0]

Angular numbers could be a better term (well anything could be a better term as most people say!)

[@vikoroller8446]

Jade, It has been 17 years since I got my Ph.D in Theoretical Physics at Universitat de Barcelona, back in 2002, and let me tell you that I have never seen a physics communicator as you! Keep that way!!! Even though I have studied everything you explain in your videos, It’s a pleasure to watch them remembering my undergraduate years....good memories!!! Congrats for your channel!!! 😊🖖🏻

[@pauligrossinoz]

I'm just like you - I earned my Engineering degree (25 years ago) by rote learning the mathematical rules, then one day, long after I graduated, I really thought through the underlying meaning of all that complex number stuff. Only then did it finally make proper sense to me. And I agree with you the key intuition with complex numbers is that *multiplying by **_i_** is a 90-degree rotation.* I wish it was taught your way in school too! Outstanding video. Thanks!

[@mttbernardini]

I'm familiar with complex number since I study engineering, and particularly I got familiar with their use to represent rotations when I studied Signal Processing. However this video still helped me by putting things in order and in the right perspective, starting from that intuition of multiplication by rotating! I wish I could watch this video when I learned complex numbers in high school... Your video deserves a lot, it's so well explained in a smooth way and without the abuse/misuse of the notation i = sqrt(-1). Thanks!

[@MatthewAppleby42]

Did anyone else notice that the music for most of the video was in 9/8 timing? Complex time signatures for the win!

[@RandomCatFromFrance]

Imagine -i/4 timing

[@MatthewAppleby42]

Challenge accepted!! @@RandomCatFromFrance

[@JJ-kl7eq]

I only ever use bowls to eat off of. I’m a non-Platenist.

[@AgentOccam]

Yes. Also, that's Plato for you: not a full philosopher, only a dwarf philosopher.

[@Ratio16180]

Nice

[@Big-The-Dave]

And this totally makes it clearer to me why Quaternions do what they do.

[@faceplants2]

@@AndreaCalaon73 FYI, posting a URL in a comment without any additional info often gets that comment thread suppressed by YT. It might be different since the OP comment got a ❤️ but in general, YT dislikes seemingly random links to external sites.

[@teipeu9033]

I had no idea about the axis rotation, really helped me understand them more. Great video.

[@SONYAdicto]

This was the best explanation of anything I’ve seen in my entire life. It literally turned my thinking around 180º when it comes to imaginary numbers, and I’m an engineer... wow. Thank you!

[@upandatom]

glad you found it useful!

[@pablosartor6715]

You're an engineer and didn't know this? Are you serious?

[@faisaltaher851]

Good pun dude

[@FGj-xj7rd]

Real numbers *Lv1 crook* Complex numbers *Lv35 boss* That's how math works

[@aniofri]

mathia

[@metallsnubben]

Quaternions *Lv100*

[@The_NSeven]

Me too thanks

[@aniofri]

@@The_NSeven haha yes

[@The_NSeven]

@@aniofri include me in the screenshot

[@ascetic3312]

3:04 - A perfectly rational response to irrational numbers.

[@Baekstrom]

So, the imaginary numbers aren't really any more or less imaginary than any other number, and the real number line is mostly made up of irrational numbers that have no representation in the real world, so the real numbers are mostly numbers we can only imagine. No wonder people get confused.

[@NetAndyCz]

Well imaginary numbers is bit weird name, but irrational and transcendental numbers do not have the best names either. And real numbers are not all that real either.

[@upandatom]

this is true, numbers as a rule are badly named. I kind of like how the transcendental numbers sound though :)

[@kappadistributive]

@@upandatom Surreal numbers also deserve their name ;-)

[@jfb-]

They're called irrational numbers because they're not ratios, and transcendental numbers because they transcend the algebraic numbers

[@ragnkja]

jfb-1337 It’s the non-mathematical meaning of “irrational” that’s unfortunate.

[@ThePotaToh]

Surreal numbers seem apt to me

[@atimholt]

Now do quaternions! :D In all seriousness, check out 3-Blue 1-Brown's videos explaining them. The general idea is that you can describe 3D rotation by rotating into the 4th dimension by half the desired angle (around two axes, a four dimensions thing), then rotating back by “the next” half of your desired angle (flipping the sign of the 4th-Dimensional-inducing axis). Just imagine that the xyz axes are all trying to rotate at the same time, and it “pops” everything into hyperspace. You can rotate it all back in while still having a net effect. If this is a bit heavy, 3-Blue 1-Brown's visuals help immensely!

[@axelinedgelord4459]

SCREW QUATERNIONS

[@jamesabber7891]

Negative numbers are just as imaginary. You cannot have less than zero apples. But that means some equations like x-3=2 have no solution. The solution to this problem is to invent negative numbers. Imaginary numbers are just the same. Some equations like x*x=-1 have no solution, but you can invent new numbers to solve that problem. And imaginary numbers are really useful in many areas other than just physics. The difference between these two kinds of imaginary numbers is that the first was invented in the realm of addition while the second was invented in the realm of multiplication. Because addition is easier to understand than multiplication it is not surprising that complex numbers are harder to understand than negative numbers.

[@doougle]

Since it takes imaginary numbers to plot the Mandelbrot Set, I can't imagine life without them.

[@eomoran]

The discovery thing is accurate once you learn about change of bases in linear algebra, it’s literally everything. Points exist relative to some other space. We decide an arbitrary measurement scheme that according to that under a given coordinate system you can locate it. Take a polynomial f(x) such that it’s ax^2 + bx + c, this is basically just hundreds, tens and units. The function f(x) is just map of all x’s to certain y’s and when plotted against one another gives you a graph that can be used to show how one vector changes relative to another. Every nummver from 0-999 can be constructed using any a, b and c for where the x values are fixed

[@thejiminator8816]

It all started with the invention/discovery of the natural numbers i.e. positive integers >0. Acknowledgement of the fact that two oranges and two apples share something in common i.e. their quantity, gave rise to these counting numbers. Therefore, the original numbers that made sense were 1,2,3,4,..... Addition came about i.e. 1 apple plus 1 apple = 2 apples as did its opposite, subtraction. But with subtraction, it became apparent that the numbers were incomplete. What happens when you take 5 from 3? Do we say that this cannot be done and stop there! No, So came the negative numbers. What happens when you take 2 from 2?, so came zero. The set of integers result and are complete from addition/ subtraction. Then comes along multiplication, and division. A new problem arises. dividing 4 by 2 is fine, but it's inverse gives birth to rational numbers. We then get operations like squaring and taking roots. With this comes irrational numbers and imaginary numbers. The square root of 2 is a classic example of an irrational number. Then comes along the square root of -1. We do exactly the same as we have done before. We give birth to the imaginary numbers, which are just as real as all other numbers. They solve the problem of being a quantity, which when squared gives you a negative number. So, just like when we found a problem with subtracting big numbers from smaller ones, thereby creating negative numbers, we create imaginary numbers to solve the problem of taking the roots of negative numbers. Both numbers, are as real as each other.

[@yosefmacgruber1920]

And all this complicated mess just because we couldn't restrict the domain of the function to the original limited domain? Trying to represent 2/4 or 1/2 by counting fingers, is holding a finger out half-way? That violates the digital principle of bi-stable bits. After a while when my finger gets tired, is it now 1/3? Or 2/3? So why is my fancy cheap scientific calculator, unable to calculate 1.5! ? Domain simplistic much? So finally the complex numbers are the complete closed set of numbers that result from all algebraic operations. But then, aren't the quaternions quite cool? But what operation can I do, to a real or complex number, to produce a quaternion, other than simply positing that quaternions ought to exist? And then are the quaternions the ultimate numbers? Can I stop there?

[@JustNow42]

How about : the imaginary i is not a number but a structure constant. There are others like various matrices, the 4 Pouli 3 by 3 matrices , Dirac's 4 by 4 matrices and others .they function to enable additional structures so it is easier or even possible to solve problems.

[@bjarnes.4423]

Since I am studying Biotechnology I will never use complex numbers in my future job. But I will use them in my free time, because math and physics are awesome!

[@MrMctastics]

I would recommend taking calculus and ordinary differential equations as they are used quite a bit in the mathy side of biology.

[@kamranahmad4592]

They come up in the physics of MRI machines... don't "count" them out!

[@stevedoe1630]

Silt Biotechnology without complex numbers? I have 3 words for you...... Mantis shrimp eyesight

[@hypat1aa]

I used to think because I was into art... I'd never need this math. Then I got into computer graphics, ha ha ha ha ha ha

[@tinkerduck1373]

"Since I am studying Biotechnology I will never use complex numbers in my future job." As a biotech engineer I can tell you, I'm always impressed how much from my study I could use again in some way. You'll never know what might turn out to be helpful in the future. After Steve Jobs dropped out of university, he might have thought, this calligraphy course was a waste of time. However, it turned out, that this skill contributed to the outstanding graphics of the Macintosh.

[@theboombody]

I've seen Gauss use imaginary numbers on the complex plane combined with modular arithmetic to show what regular polygons are constructible, and even after all that, I still don't know what the heck imaginary numbers really are.

[@Omnifarious0]

Thanks, this was a very clear explanation of complex numbers. I think one of the niftiest uses of something like complex numbers is quaternions. They're used in computer graphics all the time. As well as in spaceflight. I'd love a good video on them.

[@drdonothing7991]

Imaginary and complex numbers suddenly became real for me with damped LRC circuits. The output waveform for a step input is described by e^x where x is a function of the values of L (inductance), C (capacitance) and R (resistance). Now, in an over-damped circuit where R is large, x is real and you just see an exponential decay or growth, just like the simple charging of a capacitor with time. But in an under-damped circuit where R is small, x becomes imaginary. e^x can now be written using sin and cos (Eulers"s identity). And how does this under-damped circuit behave? You get ringing (ie decaying oscillations). You can literally see imaginary numbers on an oscilloscope.

[@IXSigmaXI]

"I have no idea why they didn't tell us in school!" - Jade, sweetly "I have no idea why they didn't tell us in school!" - Me, Vehemently while throwing books and flipping tables great channel, thanks for the video!

[@mathfullyexplained]

Try my KZbin channel mathfullyexplained.

[@Mothuzad]

2:42 why you gotta call out my bank account like that?

[@brharley0546]

I'm so happy i found your channel you explain these interesting topics with such a simple and understandable way, so thank you! (sorry for my english mistakes btw)

[@owenfulkerson7291]

So happy you made this video! We just started oscillatory motion in my physics class at college today. And I was wondering about this exact thing and getting very confused with the imaginary numbers. Doesn’t help that my prof isn’t that great either. This helps me understand a lot more

[@totlyepic]

Although trivial, given that this is meant to be a very entry-level video for the topic, it might have been worth pointing out that complex numbers are an extension of our normal framework, not an entire replacement; you can express any real number as a complex by simply giving it an imaginary component of 0. I think that probably makes it easier to digest for someone when they realize it's not something completely different but just that we get to ignore this element in our everyday lives.

[@lusher00]

Its been 10 years since I finished my electrical engineering degree and I didn't really understand them until last night. A course on Brilliant sparked my curiosity. I spent a lot of time looking at the unit circle in signals analysis and again in DSP and it wasn't until last night when I thought to my self "Oh!, it's a circle!"

[@QlueDuPlessis]

If this doesn't helo one get better grades, it's only because the education system is broken.

[@upandatom]

it is pretty broken

[@ThePotaToh]

I recommend the 2 videos by Ali Abdaal on evidence-based techniques to study for exams. They're not about how to cheat the system but rather how to learn, understand, and retain new knowledge. It's similar to the way Brilliant works but on your own education!

[@newsgo1876]

00:57 "As imaginary as other numbers, or as real." 2:07 "What does -50 physically represent?" - These comparisons are impressive. 3:27 "What is the square number of a negative number?" - This is exactly how I was introduced to the imaginary number. But I think it is a bit abrupt. *Why do we need the square root of a negative number in the first place?* 3:46 "We just multiplied a quantity less than nothing. It's not too much of a stretch of the imagination that sometimes we'd need to take the square root of a quantity less than nothing." - Unfortunately, this reasoning may not be good enough for some rigorous souls. 4:22 Adding an extra 1 is an refreshing/helpful way to explain your idea. I really like your magic. 7:17 "It's the same kind of complex as in a housing complex, in that one whole can consist of different parts." --- This is a very very helpful clarification. I have some background in computer programming. And I think the numbers are just like objects in Object-Oriented Programming. *What an object/number is doesn't matter. What matters is its behavior* . Each kind of number is *a class of objects with specific behaviors* . Human starts with natural numbers with some natural behaviors. Then our minds generate various *concepts of new behaviors* as a response to real world stimulations. To *embody* these new behaviors *mathematically* , we create new numbers, like 0, negative number, irrational number, and imaginary number. From this perspective, the paradigm of mathematics as I see it is essentially *symbolization of conception/behaviors* . And the conceptual behaviors dictate the various mathematical operations. For the imaginary number "i", the critical concept is to add the "direction manipulation" into the conventional concept of multiplication, which takes the good old multiplication to another level. (BTW, while doing this we keep the notation of square as multiplication 2 times.) This is just one of the many cases of *concept expansion* in mathematics. And why we choose the vertical direction? Because it's independent of the horizontal direction. We can choose other directions but their correlation to the horizontal direction will only make things unnecessarily complex. Did I just say "complex" ? ;) And why no more directions? Because with 2 directions we can fully describe what happens on the plane. I'd like to conclude with one sentence: God only gives human natural numbers. All the others are just fabrications by human minds... Fortunately, some of them are useful.

[@DarkNeutrino_R]

Well you explained it better then my math teacher did.

[@obook9340]

4:41 You can not write 5*1=5*5=25 cause equalities are wrong.

[@pixelapse9613]

Fr

[@Henrikko123]

Thank you for this! Really helps the understanding

[@phillair3813]

Wonderful explaination, Jade. I used complex numbers about 55 years ago as an undergraduate science/math student and eventually became a science teacher. At the time using complex numbers was routine but always left me feeling uneased. Now, having retired, revisiting the meaning of i gives me some math joy. Thanks

[@ericherde1]

From what I heard, the Pythagoreans didn’t drown the guy just because they were upset; they drowned him because they were certain that all numbers must be rational, and he was destroying the beauty of mathematics.

[@upandatom]

drowning someone seems a bit drastic tho

[@ericherde1]

Up and Atom yeah, the Pythagoreans were pretty hardcore, and not in a good way.

[@coffeestainedwreck]

@@ericherde1 To this day, Math department meetings haven't changed...

[@zeeshanmehmood4522]

So basically they were upset

[@sonosofisms]

Here’s what AC Grayling writes about it:” The discovery of irrational numbers was so traumatic for the Pythagorean’s, legend has it, that the man who made the discovery (or, some of the legends say, the man who revealed it after the order’s members had been sworn to secrecy about it), namely Hippasos of Metapontum, was punished by being drowned.” (History of Philosophy, p. 23)

[@jacobscott2473]

Shout out to my Maths C Teacher, who straight up drew that diagram at 05:42 to explain this concept to us. Although he was never able to give much practical applications for them apart from vague references to electrical engineering, physics and such. So learning that they can be used to keep track of rotating systems was really cool!

[@liranpiade4499]

That was amazingly helpful in grasping them!

[@jeffalbertson804]

There are concrete constructions of complex numbers. One is: The set of congruence classes of real polynomials modulo 1+x^2. That is, take two real polynomials to be congruent if they have the same remainder when divided by 1+x^2. You can easily see that the set of these classes has all the desired properties of complex numbers. I can give another concrete construction, but not in a KZbin comment section! So they are NOT just numbers, and yet they are numbers.

[@drwijnen1729]

I like your explanation of how complex numbers work, but you're not quite right on why they're called imaginary. While I agree that it's a confusing name (which AFAIK is why they were renamed to "complex", but not everyone changed), there was a very good reason for it: originally, they were used to describe harmonic oscillations. Those move as a sine function, which is annoying when doing things like differentiating. So instead they imagined that the oscillator was doing a rotation. That way the movement is described by an exponential, which is much easier to work with. The imaginary part of the number was literally made up. So at the end of the computation it was discarded. It still resulted in the correct answer for all calculations. Later complex numbers were found to be useful for many more things, and for many of them the "imaginary" part is no less real than the "real" part. But unfortunately it's still called the imaginary part...

[@mnp3a]

i remember reading that they were called "imaginary" by Euler. Haven't checked it in sources, though

[@MuffinsAPlenty]

What you described here is _not_ historically accurate. I highly recommend Veritasium's video on how imaginary numbers were invented. As a brief summary (this history takes place over the course of 300 years, but I've managed to reduce it to 4 paragraphs), square roots of negative numbers started out as a _necessary_ intermediate step to find (real number) roots of cubic polynomials. Much like there is a quadratic formula, there is also a cubic formula to find the roots of cubic (degree 3) polynomials. However, depending on the coefficients, plugging numbers into the cubic formula would sometimes give square roots of negative numbers. This was a bit perplexing to mathematicians of the 16th century, since, unlike with quadratic polynomials where some of them had no (real number) roots, _every cubic polynomial has a (real number) root._ So in order to find this real number root, sometimes you _had_ to use square roots of negative numbers. The term "imaginary number" was coined by René Descartes. And this was in line with European thinking of numbers. Because mathematics in Europe had a strong Greek tradition which was rooted in geometry, European mathematicians always thought of numbers as representing geometric ideas, such as length, area, and volume. European mathematicians at the time disliked negative numbers, but they could still kinda make sense of them geometrically if thinking about length/position in a certain direction. However, to European mathematicians, lengths, areas, and volumes were _always_ nonnegative. There was no possible length, even considering directions, which gave a square of negative area. As such, there was no _number_ (length) which squared to (resulted in a square of area) -1. So in order to work with such a number (length), you had to _imagine_ it. That is how Descartes first used the name "imaginary". Euler developed his famous formula (e^(it) = cos(t)+i*sin(t)) in the mid-18th century, but even this, alone, did not make the full connection between complex numbers and rotations that we think about today. At this point, Euler still viewed i as a number which had to be imagined, since it did not represent a valid length. It wasn't until Jean-Robert Argand developed the complex plane at the beginning of the 19th century that a geometric interpretation of complex numbers took shape, where, much like negative numbers represented a _signed_ direction/position, so too did imaginary numbers (within a planar configuration, rather than a linear configuration). Euler's formula could then be applied to view complex number arithmetic as movement within a plane (as opposed to lengths of a square yielding certain areas, as numbers had been previously thought of). (Note: Caspar Wessel also had a geometric understanding of complex numbers, and he did so roughly 10 years before Argand, at the end of the 18th century. However, Wessel's publication went unnoticed for roughly 100 years, but Argand's work was noticed fairly quickly, about 7 years after he disseminated it.) But it was Gauss who really cemented complex numbers' place in the "mathematical canon" so to speak since he showed just how necessary and applicable they were within all sorts of mathematics. Gauss also was the person who coined the term "complex number". But you also have a bit of a misconception about the distinction between the terminology "imaginary number" and "complex number". The term "complex number" did not _replace_ the term "imaginary number". An _imaginary number_ was always a number which squared to a negative number. In other words, an imaginary number is one of the form bi where b is a real number and i^2 = -1. A complex number is any number of the form a+bi where a and b are real numbers and i^2 = -1. As such, the imaginary numbers form a subset of the complex numbers (a+bi where a = 0). (To be fair, the real numbers also form a subset of the complex numbers, a+bi where b = 0.) Gauss disliked the terminology "real" and "imaginary", so he proposed a new naming convention: positive real numbers would be called "direct numbers", negative real numbers would be called "inverse numbers", and imaginary numbers would be called "lateral numbers" (since they moved _laterally_ to the direct/inverse directions). Then, a number compromised both of a direct/inverse component and a lateral component was called a "complex number", where "complex" comes from the meaning of being comprised of multiple parts (much like an apartment complex is comprised of multiple apartments). While the name "complex number" stuck, Gauss's preference for direct/inverse and lateral numbers could not overturn the 200 years of momentum the terms "real" and "imaginary" had built up.

[@elijahchang7589]

LOL!!, "In my opinion, it's the worst name that anyone could have possibly come up with in the history of anything ever"

[@philwesom8784]

whoa...a totally new way to look at "imaginary" numbers cool !

[@terryendicott2939]

Another way is to get familiar with 2X2 matrices and one can map a+ib -> a b -b a this way if you are comfortable with matrix arithmetic over the reals then the complex numbers can just be considered as a subset (subring actually).

[@Suman-zp8gs]

Wow Awesome explanation!👌 It's so useful! Thanks👍

[@teancumpusey3406]

Currently a Junior in physics and have been struggling with the incorporation of imaginary numbers in optics. This video cleared up my roadblocks! Please keep posting content.

[@hardikmhatre7075]

Hey jade !! this video really helped me to understand complex number. Great video jade 😀😀!!

[@upandatom]

No worries that's awesome you understand them better!

[@harrybarrow6222]

Excellent! I learned about imaginary and complex numbers in high school (almost 60 years ago). The square root of -1 was introduced essentially as a “let’s pretend it exists” concept, represented algebraically by i. The consequences, I discovered, were amazing and beautiful. All algebraic equations now have solutions, algebra and trigonometry are linked, and, yes, we can represent rotations of vectors. Presenting i as a rotation of the number line right at the start is a much better approach. And it makes more sense this way. For me, even though I have been using complex numbers for 40 years and was completely happy with them, this video really did give me an “Aha!” Moment. Thank you.

[@nektariosmusic]

I really like your explanations. Maybe you can cover quaternions and octonions in a different video? :-)

[@reallyWyrd]

I've seen several explanations of the imaginary numbers recently on youtube, and so far yours is by far the most concise.

[@awertyuiop8711]

Gauss had it just right with “direct”, “inverse” and “lateral” numbers.

[@AgentOccam]

From about 0:46 - such a great explanation/rant. Truthful, accurate, and funny! Love it!

[@adamkendall997]

That's easy -$50 means you're broke.

[@Vistico93]

It also means even if it's free, you can't afford it ;-)

[@kmbb9376]

what she said at 7:29 is sooo true. I feel that you can nail any skill or knowledge if you can have / were shown "a peek at the intuition"

[@alexandrebarret5843]

Next: Quaternions!

[@carultch]

Quaternions = real * (imaginary + joke + kooky)

[@phengkimving]

The imaginary number i wasn't created out of thin air, it wasn't "defined" as sqrt(-1), it was a product of the effort to make every negative number also have a square root, for example sqrt(-9) = sqrt((9)(-1)) =sqrt(9) x sqrt(-1) = 3 sqrt(-1). So if you have sqrt(-1) then you have the square root of every negative number. So efforts were made by mathematicians to construct a set of 2-part numbers based on R^2 with axioms that allowed them to arrive at a number (0, 1) that when squared equals (-1, 0), and the axioms allowed (-1, 0) to be identified with the real number -1. Then they called and denoted (0, 1) by i, and voila, i^2 = -1. Thus, for example, sqrt(-9) = 3i. The imaginary number i isn't "defined", it's the intended consequence of the construction of a number system. If you just "define" i as sqrt(-1), then you must prove that sqrt(-1) is well-defined before you can use i. Now that i is a consequence of a construction based on universally-accepted axioms, it's well-defined. The presenter in the video is right that the "imaginary" numbers aren't any more "imaginary" than negative numbers. Just like negative numbers, "imaginary" numbers have found applications in science, engineering, and even economics.

[@MuffinsAPlenty]

Your first paragraph is historically inaccurate. No mathematician wanted square roots of negative numbers. Instead, in trying to use the cubic formula to find roots of cubic polynomials, sometimes square roots of negative numbers _had_ to be used. Unlike the quadratic formula where square roots of negatives come up if and only if the polynomial has no real-valued roots, every cubic polynomial has at least one real-valued root (even in cases where square roots of negatives popped up in using the cubic formula). So, in the 16th century, mathematicians came to the realization that using square roots of negative numbers were sometimes a necessary step to find real-valued solutions to real-valued problems. The type of mathematical rigor you're talking about (with constructing number systems) didn't start taking place until the 19th century. And the specific construction of the complex numbers you're talking about (ordered pairs of real numbers with specially defined operations) didn't come about until 1831, nearly 300 years after mathematicians started using square roots of negative numbers. The revolution of mathematical rigor that began in the 19th century has greatly shaped how we think about mathematics today. But you have to realize that mathematicians, for centuries prior, were able to do some rather non-rigorous work with these concepts anyway. Descartes and Euler made great use of sqrt(-1) without having any formal or rigorous meaning for the symbol other than "a number which squares to -1".

[@nuclearnyanboi]

Real Axis and Nether Axis. Lovely ...

[@debdip7]

This video made my day. I knew the concept, but the way you approached this difficult concept was amazing!!!! Kudos Jade!!! ❤️

[@GMPStudios]

Fun Fact: If we accept imaginary numbers with open arms in functions, the concept of domain and range will disappear.

[@davidsonjoseph8991]

GMP Studios I was about to argue until it struck me that you are indeed right. 🤔

[@pleaseenteraname4824]

How's that?

[@lerarosalene]

What about noninteger powers of complex numbers?

[@ripno2672]

Where is the 3D axis on the number line?

[@Testequip]

I was never taught the rotation method. Thanks for making this so incredibly clear!!!

[@mathfullyexplained]

Try my KZbin channel mathfullyexplained

[@Stan_144]

Imaginary numbers with real numbers make 2D number space. Now the question is: do we have other types of imaginary numbers that would create 3D number space or maybe even higher dimensions ? To put is differently: if imaginary numbers could be related to negative area, where is the imaginary number equivalent of negative volume?

[@mnp3a]

great video! my take on this: basically: there is no square root of a negative length, BUT if we view real numbers as expansions, contractions, 180° rotations (that is, transformations of the line), then negatives will have square roots. (For me) it is very important to emphasize that you don't just "add sqrt(-1) to the real numbers": you also change the way you interpret real numbers! So: complex numbers are rotations in the plane *when* positive real numbers are contractions/dilations and negative real numbers add a 180° turn to contractions/dilations. when we talk about natural numbers: 1, 2, 3, .... we understand that we are counting stuff: how many? then with real numbers, we are not counting anymore: we are measuring! So, the same symbol : "1", when understood as a natural number, means "one object", and when understood as a real number it means "a length of one", or stuff that is associated with lengths: areas, distances, time intervals and so forth. Then it became usefull to understand minus "-" as direction, so real numbers are basically sizes with a direction. now, the square of any size will be positive. Next step: dynamics. You can *think of* any real number as defining a way to move points around: the number x moves the number y by multiplying. Lets call that "expand-by-x" (or whatever), so: expand-by-2 moves 1 to 2, moves 2 to 4, moves 1/2 to 1 and so on expand-by-4 moves 1 to 4, 2, to 8, 1/2 to 2, and so on expand-by-(-2) moves 1 to -2, -2 to +4, 4 to -8, and so on observe that we can do that twice, thrice or whatever: expand-by-2 followed by expand-by-2 is the same thing as expand-by-4. We can say that (expand-by-2)-squared = expand-by-4 And the thing is: negative real numbers, *when viewed as lengths*, have no square root: no length squared will be negative, and that also makes no sense in the real world. But, negative real numbers, *when viewed as movements of points in the plane* do have a square root exactly as the video explains: rotating stuff on 90° twice moves points on the line in exactly the same way as "expand-by-(-1)"

[@JVMultiProds]

I laughed way too hard at the animation of the random dude drowning in the Mediterranean. I think I may have bigger problems than not understanding imaginary numbers :(

[@rb5519]

3:26 No, that's not how the concept of "i" confronted me when I first saw it. For me, the question was, what number do you multiply by itself to give you 5, for the example of root 5. You could solve for that. What number do you multiply by itself to give you -1? Nothing on the number line gives you that, so they just made up this thing called "i" and said - well whatever it is, let's call it i and say it does. It's a made up number. So instead of imaginary, maybe we can call them the "made-up" numbers?

[@lukiepoole9254]

That is correct. sqrt(-1) does not exist. Imaginary numbers are not numbers and cannot be expressed as a number. Hence, the one TRUE "sqrt(-1)" would be sqrt(Matrix[-1]) sqrt([-1 0] [ 0 -1]) = [ 0 1] [-1 0] (counterclockwise rotation, inductive) Rotation of vector either by 90 degree clockwise or counterclockwise. Truth is simple isn't it? The fact that we live in multisubspacial reality.

[@robfielding8566]

Geometric Algebra has the most absolutely perfect explaination of it. You calculate "i" rather than it being an axiom. Any two unit vectors in space that are length 1 and perpendicular multiply to be the directed area that the parallelogram spans (area 1). And the vectors self-cancel, and anti-commute. Squaring to -1 is a consequence, rather than an axiom. (e_1 e_2)^2 = (e_1 e_2 e_1 e_2) = -(e_1 e_2 e_2 e_1) = -(e_1 e_1) = -1. It doesn't even matter what these vectors are. if they are unit length and perpendicular, then they are an "i" for the plane that they span. There many "i" for each plane.

[@PhilipWalley]

@jade - Thank you. You have a rare and valuable ability to explain things in a simple and understandable way. These videos are what KZbin and the internet are for. Keep up the great work!

[@kiranramarao5627]

Stumbled up on this accidentally. Great explanation. Not sure how many people are driven away from physics/signals by not explaining the basics right. Again, great job.

[@jessstuart7495]

-1 = ±180° rotation i = +90° rotation Assuming for a moment that your numbers are allow to have two orthogonal components, then you can represent complex numbers as matrices. a*[[1,0];[0,1]] is a real number. b*[[0,1];[-1,0];] is an imaginary number. a+i*b = [[a,b];[-b,a];] i is really just the matrix [[cos(90°),sin(90°)]; [sin(90°),cos(90°)]] that rotates some complex number by +90° √i is [[cos(45°),sin(45°)]; [sin(45°),cos(45°)]] = [[1/√2,1/√2];[-1/√2,1/√2]]

[@bjornmu]

And if you express the complex number using the angle (from the positive real line) and the distance from 0, you can multiply two complex numbers by adding the angles and multiplying the distances!

[@KaustavMajumder]

5:10 - Bam! Pure gold.

[@estebanalfaro5069]

Thank you!!! You have no idea how much peace of mind you gave me after decades of not understand imaginary numbers! It is so clear now!!! THANKS!

[@gavincraddock5772]

The discussion of multiplying by -1 being good for systems that flip and multiplying by i being good for systems that rotate was absolutely the missing piece for me - I learned about complex and imaginary numbers in A Level Maths over 25 years ago but they seemed so abstract, so unnecessary, so strange. This is amazing, thanks!

[@hasan7275]

e^ipi = -1 seems like a beautiful equation because of all the common constants contained with it but what it really shows i feel is the nature of numbers. -1 is 180 degrees/pi radians out of phase with 1. i is 90 degrees/pi/2 radians out of phase with 1. real numbers scale when multiplying and the other e^ipi numbers rotate numbers around. it’s nice that this video is able to cover this idea pretty simply. phasors are pretty useful. the connection between exponentials and trigonometric functions is also really helpful and makes differential equations, which describe many, many systems, a lot easier than it could be but also more intuitive i think.

[@braaimanook]

When I think back to the battles I had with j notation, and the completely incomprehensible explanations I was given, it is now apparent that my tutors didn't understand it either. Clear and concise presentation. Thank you.

[@AfdhalAtiffTan]

Could you make a video on quaternions?

[@kri249]

Do you know how many videos and websites I looked at on this subject until I found this? Your visuals and simplistic breakdown just made it click. Thank you and love your work.

[@thoperSought]

two things: *1.* that's great! I'd never heard it explained this way before!! so much easier to understand. *2.* as for why things aren't taught this way, my sister is a H.S. English teacher, and she's a bit frustrated at the moment because she has a big, state mandated, standardized test that she has to prepare the kids for. it leaves her relatively little freedom for creativity about how to teach. I can't say about the U.K., but at least in the U.S., this is one of the contributing causes. also, in my case, most of my H.S. math teachers didn't really know math all that well.

[@SleepyCJCC]

As much as you say this doesn't help me with homework, it actually did! I'm taking a course on Complex variables, but considering I only got a "crash course" on Complex numbers in high school, I just tucked it away. After this video, not only did I gain a better appreciation for imaginary and Complex numbers, but I actually understand it now and can apply the very simple explanation you gave to my assignments. Thank you!

[@upandatom]

that's awesome! glad you found it helpful :)

[@Achrononmaster]

@7:30 Yes, that knowledge will not be all that grade-worthy (a pun if you read further). But for advanced quantum mechanics and robotics I can tell you what will be: *_geometric algebra multivectors._* These Clifford algebras are the natural universal algebra for all of mathematics. The Complex numbers are just the even subalgebra of the Pauli-Clifford Cl(3) algebra for instance, and the Pauli algebra is just a subalgebra of the full Dirac spacetime algebra Cl(3,1). Once you learn Geometric Algebra you never want to think in terms of complex numbers or quaternions ever again, they lose all the geometric insight. In other words, the correct generalization of scalar number systems are graded-algebras, not just complex algebra. In the graded Clifford algebras (geometric algebra) all objects of _all grades_ are rotated by rotors in _exactly the same way,_ whereas in vector algebra if you use matrices you can only operate on vectors. What sold me on Geometric Algebra was realising rotors with timelike bivectors are just the Lorentz boosts, and yet they act on multivectors exactly like 3-sapce rotors, by two-sided multiplication. So for both numerical and computational work they far surpass using quaternions or complex structures. They have all that structure embedded, but with more power and elegance. When I re-wrote Wigner's classification of the Poincaré Group irreps. in Cl(3,1) geometric algebra it was all very basic and simple. Matrix algebra by contrast now seems incredibly ugly and ham-fisted to me, good for torture. For more see the DAMP archives and papers by Lasenby & Doran and David Hestenes. They will show you how complex numbers are baby stuff.

[@Achrononmaster]

Also, once you learn the spacetime algebra you find the Dirac matrices used in relativistic QM are actually just spacetime basis vectors (frame vectors), and the Dirac spinors are just spacetime rotors. So they lose all mystery. Wavefunctions in QM are literally just "instructions to rotate your laboratory frame onto the particles' rest frame". This shows you wavefunctions are not _real_ things, they are information theoretic, not ontological. Then QM becomes a whole lot easier to understand, it is all geometry. With pure spacetime geometry you can get superposition by including non-trivial topology, so closed timelike curves, and then _all_ of quantum mechanics is geometry --- provided you are willing to accept the existence of CTCs, which of course is tricky, because they show up as time-ravel effects, and so cannot be macroscopic, they have to be dynamically unstable and so restricted to around the Planck scale (but most physicists agree that is fine, it's just spacetime foam but in a general relativity context with nothing "weird and quantum" about it). The spacetime metric will always appear to fluctuate if there are CTCs around, but the fluctuations are due to the atemporal aspects of the Planck scale Lorentzian wormholes, not due to anything "quantum mechanical". Or to put it another way, we have QM all right, but QM comes _from_ GR+CTCs. The final nail in this bed of QM is that the Standard Model gauge groups are embedded in the full 16-dimensional spacetime graded Clifford algebra. So there shoudl exist geometric particle models for the entire Standard Model, if someone is just clever enough to look (it's not easy to do the translation from Lie algebras to geometric topology though, few people are good at this, even the Langlands program people focus mostly on algebras, not so much on visualizing geometries with non-trivial topology). The difficulty being that the geometric (Clifford) algebras are describing transformations, not topology. So there is no direct translation available from a given algebra to a topological particle model. There must be one I reckon, but physicists are not looking because they rare too enthralled with String Theory. You cannot get the quantum mechanics in GR without including the CTC topology, so this dictionary from algebra to topology has to be worked on seriously, by people brighter than me. If only I had Grothendieck's brain power, minus his paranoia.

[@Ny0s]

5:40 Wow, mind blown. I had to stop the video to digest that lol. This a such a revelation... Great explanation, thank you very much for your channel.

[@thecrapadventuresofchesimo420]

Thanks Jade! That was super helpful!!! The one time in high school I asked about the square root of a negative number I was told 'don't worry about that'

[@upandatom]

that sucks. i was given a similar response when i asked what triple integrals were doing

[@rohitsohlot]

The intuition of imaginary numbers as rotation of vectors is completely new to me and taught me a new way to look them. Thank you🙂

[@samjebaraj24]

I was thinking why you didn't upload video for a while, and here you are , good to see you 😁!!

[@upandatom]

Yes I've had a very slow start to the year, but it should (hopefully) be smooth sailing from here!

[@engelbertus1406]

for those who like to ponder deeper into this: the “ - “ sign shouldn’t be seen as a calculation, a value indicator or such, but rather as geometrical indicator. It states: i’m moving away from zero in exactly 180 degrees opposite of real numbers. Real numbers are only real, because they are our point of reference numbers. As 0 is the reference point for the numbers, real numbers simply state: we keep moving away on a line from 0 that’s the reference line for measuring at what degree numbers are moving away from 0. Real numbers are 0 degree numbers, negative numbers are 180 degree numbers. If we have a 2D Circle, numbers can move away from zero all along the 360o circle; so, +1 = 1 (0o), -1 = 1(180o). In this way numbers can be expressed, in relation to their degrees on the circle, and no “-“ sign has to be introduced. Therefore, imaginary numbers don’t have to be introduced ( tho we’re dealing with the same geometrical consequences). Now imagine a 0 with a 3D sphere around it, real/positive numbers can run away from zero in any direction, and numbers are now expressed by their respective degrees in the sphere. We can define any number system as such: +1 = 1 (0o, 0o), -1 = 1 (180o, 0o) As such we have a bigger source of calculation operations than we would have if we would only use positive, negative and imaginary numbers!

[@mallninja9805]

I've seen the 2d representation of the imaginary number line intersecting the real number line before, but I'm so used to X & Y axes that I just found it confusing. The rotation example (1, i, -1, -i, back to 1) finally made something click! Thank you!!

[@PaulPaulPaulson]

Thank you for spreading this knowledge. I hope one day the students will be able to start to learn it this way. Simple 2D numbers with rotation and scaling as basic operations. With this in mind, even the famous Riemann hypothesis is getting a lot easier to understand (but still hard to solve). Repeated rotation and scaling, with smaller steps and less rotation in each step, resulting in spirals around certain centers. Riemann found a pattern in where those centers are and nobody could prove or disprove that pattern to this day.

[@rrr00bb1]

there isn't just one "i". every pair of unit vectors that are perpendicular will square to -1. directions in space square to 1. instead of "x" and "y", use "right = -left", and "up = -down"... counterclockwise... (right up) = (up left) = (left down) = (down right), (right up) = -(up right). This anti-commuting multiplication squares to -1. (right up)(right up) = (right (up right) up) = (right (-right up) up) = -(right right)(up up) = -(1 1) = -1. this is basic geometric algebra, that generalizes complex numbers to all dimensions, eliminates handedness issues, and shows "i" to be ambiguous about which plane and orientation the rotation is in.

[@jonthecomposer]

I love that you said i keeps track of systems. When first trying to wrap my head around i, the thing I did was to give it a chromatic series of whole number powers, 1, 2, 3, 4, etc... i^1=i, i^2=-1, i^3=-i, i^4=1, i^5=i, etc... so that it did exactly what you were talking about. Granted, I didn't understand as well as I do now, but I got the idea that in certain equations, if you can't transform a negative number through exponential operations, it either disappears and the whole equation ceases to function, or there's no way to keep track after a certain point. Great job as usual :)

[@LuckyKumar-wb9xm]

That right angle explanation blew my mind!! It was added in such a subtle way... I wasn't even ready😅!! Great Explanation 👌

[@seanloughran6714]

I saw someone visualize the Mandelbrot set and explained i as a rotation. And he showed as you kept multiplying out, it was like dancers circling a dance floor getting further and further out on the floor. It blew my mind, I don't know why we're not taught the rotation aspect earlier in school.

[@efeguleroglu]

In every video, you take concepts in a very different point of view. I sincerely appreciate and am looking forward to see your channel making visible progress.

[@gloweye]

I am fully in agreement with you about the naming of "Imaginary" number. I got a degree in applied physics myself, and I overheard things in the corridor along the lines of "Do you believe in Imaginary numbers?", which demonstrates... well, everything I want to say.

[@pdaphuulz8219]

omg. my mind was blown when she explain 'i'. this is brilliant