Why don't they show this on the first day of Algebra 2 in high school? I might have passed if I understood what they so plainly stated in such a short video. I guess that's what happens when you're math teacher looks more like a walrus than a man...
@dekmaskin I interpret as a mathematical construction rather than a "physical concept". It is just an intermediate quantity (in a mathematical calculation) that makes much of the math easy. However, at the end, we need to get rid of it to get something "physical". For example, the probability density by taking norm of a wave function in QM, or the individual real components of a complex current in EE.
It seems to me that imaginary numbers are not necessary. I say that because a computer can calculate something that "requires" imaginary numbers, and it can do it without using imaginary numbers.
@MrAlexGTV Απ' οτι καταλαβα είσαι Έλληνας και γνωρίζεις τι μαθαινουμε εμεις οι Έλληνες μαθητές σε σχεση με τα παιδια του εξωτερικου.Θα μπορουσα να πω πως θεωρούμαστε προνομιούχοι...... :) :) :) :)
The reason engineers use j is because in electrical engineering i is for current; (as Afroman would say, damn alphabet needs more letters.) It is also used in that discipline for calculating impedance, which combines resistance with a phase shift.
Check the Eigenvalues video and the vibrating rules. Eigenvalues are usually complex (imaginary) numbers. In the case of the rules, how fast the rules stops shaking depends on the real part of the eigenvalue, while the rate at which it shakes depends on the complex part.
I don't like this video, if people are interested in j (or i), they should go to mathematicians not physicists. There is nothing magic or mysterious in the number, it is just an extension to the real numbers. Engineers use j because j is usually associated with oscillations and in that sense you can "measure" it. Things are not that clear in quantum physics, but this is due to the lack of knowledge of physics not mysterious math.
@TriMiro8107 I chuckled when I heard i labeled "immeasurable" and "not an observable physical quantity." I might as well declare "3 doesn't exist because we can't measure it to infinite precision." Such is the perspective of a scientist. Mathematicians have different goals entirely.
I was thinking exactly what that dude said when he said 'J conceals a structure that we have not discovered'. It's like a rule of thumb, for example Newtons laws worked, even though they were not 'true'.
@nvdkooy Well, is J2 is -1 then it is the square root of -1, -*-=+ +*+=+ Therefore J is imaginary because it doesnt exist, as no real number squared can equal a minus so by saying J2 = -1 the square root is still J
Keep in mind that x^2 = 1 has two roots, x = -1 and x =1. x^2 = -1 also has two roots, x = j and x = -j. So you threw out one root at -1 = j^2 and chose the spurious root at sqrt(1) = 1.
Since j allows us to get out of the trigonometric circle, and into the other side of a parabole, I think of j as another " dimension" that links unlinkable things. I intuitively feel one jayish ship could exit a blackhole, for instance (but I am really just making a crazy assumption here).
Maybe j is just a hard to observe part of reality.
@a1a2a3skurr1l from a simple check around wikipedia, turns out the thing about imaginary speed particles scaping blackholes is right, proving my intuition. So, it is possible to have an intuitive feel about imaginary numbers, an if I can, I am sure there are many others that do too.
I think an interesting thing to think about is how irrational numbers are just as weird as imaginary numbers. Natural numbers (0,1,2) are defined in terms of counting collections of objects. Fractions can be defined quite easily in terms of naturals (a third is the thing which if you have 3 of you have 1). But what about root 2 which can't be written as a fraction? What does that number mean? What significance can it ever have? A physical measurement could never give an irrational number either.
@LamaPaj How? We measure stuff with rulers, which consist of a series of unit lengths marked on a stick, and you line up two units with either end of what you're measuring and get a result, which is ALWAYS a rational number because it consists of counting individual units. If you measure the side of the triangle you describe you will never get exactly root 2, no matter how small you make your scale divisions. I'm not even sure a perfect triangle could ever exist in nature because of this?
@LamaPaj When irrational numbers are defined properly in pure maths they are defined in terms of a kind of limit I think. And in the same way you can never measure the instantaneous velocity of an object, even though that does have a theoretical meaning, I don't think you can ever measure an irrational length, you could only take a sort of limit of a series of more precise rational measurements.
@wowsa0 Well, numbers like sqrt(2) and pi are used in a LOT of practical calculations that are not related to purely measuring. Any way my interests are more in the area of pure mathematics then measuring things.
@LamaPaj Oh no don't get me wrong I'm not saying that root(2) or pi are some sort of mystical entities cooked up by deluded mathematicians which have no real meaning... I'm merely pointing out that you don't have to jump straight to imaginary numbers to find numbers which are weird. I don't think imaginary numbers can be said to 'exist' with any more or less confidence than irrational numbers can.
@wowsa0 Well, irrational numbers are weird, but can still be grasped somewhat intuitively. i is on a whole other level, since the rules we use to calculate things usually seems to prohibit there being such a number.
@LamaPaj No it wouldn't, it gives a value of root(2) if you calculate it, but you cannot measure something that has an irrational length. 1.41cm is NOT root(2), no matter how accurately you measure it you will not end up with root(2)
@LamaPaj Do you always behave like a fuckwit when someone tells you you are wrong, and why?
Grow the hell up. If it's SO obvious you wouldn't have made such a stupid statement in the first place would you. Geez! Wowsa was right, a physical measurement cannot give an irrational number. So you're just plain wrong
@salerio61 Well, if you would have read my reply i was really not talking about measuring was I?
If I measure up two distances of 1 arbitrary unit each from a point A, where one of these distances would be the distance from point A to a point B and the other would be the distance from point A to a point C, and the angle between AB and AC would be 90 degrees then the distance B to C would be sqrt(2) au. Then if you can measure up 1 au you know this to be true and therefore of significance.
@LamaPaj "Well, if you would have read my reply i was really not talking about measuring was I?" Let's see "well, a triangle with a ninety degree angle and two sides of 1 would give you sqrt(2) as the hypotenuse if you measured it." Oh, yes you did! FFS don't post arguments when you're drunk or stoned "the distance B to C would be sqrt(2) au" Yes it would "Then if you can measure up 1 au you know this to be true" No it wouldn't because when you measure you get a rational number.
@salerio61 I am not talking about measuring... If you would have read the old argument you would have seen that i realized my mistake, and went on making a point about the numbers still having very real and practical uses. Jeez...
Also, if you can measure the sides AB and AC to exactly 1 au you know that the hypotenuse BC is sqrt(2).
You cant measure it, but that doesnt mean its not true.
Fucking incredible! You are totally delusional. I refer you to your previous posts ...
" would give you sqrt(2) as the hypotenuse if you measured it."
Sure sounds like measuring
"Then if you can measure up 1 au you know this to be true and therefore of significance"
Yet again measuring. Look it's ok to admit you're wrong, but you don't have to make yourself look like an arse defending the indefensible. Give up already!
Fucking incredible! You are totally delusional. I refer you to your previous posts ...
" would give you sqrt(2) as the hypotenuse if you measured it."
Sure sounds like measuring
"Then if you can measure up 1 au you know this to be true and therefore of significance"
Yet again measuring. Look it's ok to admit you're wrong, but you don't have to make yourself look like an arse defending the indefensible. Give up already!
can you guys by any chance just look up the curriculum of all my courses and just do videos on them (yr 11 and 12 queensland maths c, maths b, physics, chemistry) cheers
And then of course, there are quaternions, which have i, j, and k, *all* of which are the square-root of -1. But they aren't the *same* square-root of -1.
I always thought there was supposed to be some deeper meaning to j, so it is good to know that it is just a tool to be used and that if there is a deeper physical meaning, these guys don't understand it either.
@TheRightNeutrino Uhhhh... dude... it was a joke? I know i=root -1, and how you can do i^4 =1, so all powers of i cancel in 4's etc, it's simple at school level... the joke was 'god=i', 'god=your ass', thus 'i=your ass'. Anyway, if we're going to get immature... BAD grammar, that should be "No, didn't you watch the video or *go* to high school", cor, didn't you go to school?
@AscendingAshTree a. not funny, b. there is no grammar in youtube comments, I can write however I want as long as normal people can understand what I mean, I'm certain that if you were to read a comment from Shakespeare's time you wouldn't understand what it meant, so language is flexible, it has one single purpose and that is to allow people to communicate so your stupid grammar corrections are useless. c. I wasn't born or raised in an english speaking country like yourself.
@TheRightNeutrino There's still a few problems with this, a. Whether or not it's funny is entirely subjective, so either way you don't get to be an arse. b. If there're no grammar in youtube, and there's also no humour either apparently, why ARE there obvious maths corrections? c. In which case, sorry for the constructive criticism, and finally d. This has spammed up the video's comment box.
@AscendingAshTree a. ? b. I didn't think it was an obvious correction in your case, and why should the three things you mention be somehow related? the obvious mathematical correction was sarcasam on top of your "joke", d. i'm sure they don't care. e. 2.7183, f. (fi) 1.618, g. 9.8m s^-2, h. 6.610^-34 J s, i. sqrt(-1), and we're back where we started, it's like we passed through a wormhole, amazing. next sixty-symbols video - wormholes.
@TheRightNeutrino Ok then... uhh... R. 8.31 JK^-1mol^-1 ? Gone brain dead, can only thing of e, earth's,gravity, golden ratio, planck's constant, molar gas constant and pi... Those aren't even alphabetical! ...Holidays are too long...
They should add to this that although "imaginary numbers" do seem counter-intuitive and abstract at first, real numbers are just as much a mathematical construct.
What is a real number? It's not a thing, you can never find one in the real world, the reals are a useful mathematical structure and so are the complex numbers. The imaginary numbers are no less "pure" or "unreal" than the reals, which is why I hate the term "imaginary" number. All numbers, in some sense, are just abstract constructs.
@jamma246 Yes, but I believe the problem lies in the fact that we can relate "real" numbers to amounts. In contrast, an "imaginary" number such as i cannot be applied in the same sense. That fact makes it very hard to conceive what it is exactly without a concept of what it does or how it works. I do, however, agree with you.
We should make it universally "i". It makes more sense, it's "i" for imaginary, it's what the mathematicians call it (and we invented it!) and when you go to quaternions it's nice having a base of 1,i,j,k.
Maybe the engineers should start referring to their things which already have an i with an iota instead.
In most math classes they use i to represent the sqr-1 because it stands for imaginary, most physics/engineering classes use j to distinguish it from the i that represents a variable current. J is also used for current density, but normally the current density isn't a variable its a constant and is therefore represented with a capital letter J.
I don't understand why they find j or i to be a difficult concept. I can see how one can be amazed at how incredibly useful they are in solving 'real' problems, and I share this feeling, but as for the definition, existence etc... I see no problem whatsoever.
@sorrysonofa Well, maybe you can relate to struggling with some abstract concept some time ago. Maybe there was something you wanted to learn, once upon a time, but nobody told you or you couldn't figure out why you really needed to learn it, or you couldn't see a real world usage for the concept, and you gave up on it. That's how it is for a lot of people with imaginary numbers. Ages ago it was the same thing with negatives - people couldn't imagine that there could be "less than nothing".
@sorrysonofa Well I believe they are talking from an engineering aspect more so in that they are using j in a real system. Not just a math problem, and that is where it can be confusing. I am an electrical engineer and one of the times it came up for me is when you put a capacitor in the system with a sin signal into it. you have a real voltage and an imaginary voltage. When you combine them together you get the actually measurable voltage. See a bit confusing.
@sorrysonofa Then i guess that you haven't quite understood how fucking weird it is to have a number that has the property that when you square it it equals minus one.
@LamaPaj No, I think you haven't understood why that is nothing out of the ordinary : many number systems besides the complex numbers have minus one as a square.
@sorrysonofa quaternions and octonions. Also please do remember that complex numbers don't have -1 as their square roots as a rule. Also, consider this: there is no real answer to this question "Which number is bigger, 1 or i"
@LamaPaj quaternions and octonions don't even scratch the surface : take the integers modulo a prime that is 1+4k, take matrixes of even order over any commutative ring, take any commutative ring that has no square root of minus one and *add* such a root via a quotient space of polynomials....
C is not an ordered field, so what? Minus one is still commonly a square, and there's nothing special about that.
It is probably because you are a non-physicist/mathematician. If you do not have knowledge in theoretical physics but read general physics books, you may find quantum mechanics understandable. But a physicist would find it really awkward. Certain things just require an intense level of study to finally realize that they are just ridiculously abstract. The same is true for imaginary numbers. It just doesn't make sense as to why a nonexistent value has such great existential value.
It is probably because you are a non-physicist/mathematician. If you do not have knowledge in theoretical physics but read general physics books, you may find quantum mechanics understandable. But a physicist would find it really awkward. Certain things just require an intense level of study to finally realize that they are just ridiculously abstract. The same is true for imaginary numbers. It just doesn't make sense as to why a nonexistent value has such great existential value.
I think it's the fact that it's called an "imaginary number" that causes confusion. It aint really a number. A vector isn't a number, but people don't think vectors are weird. You can't have "i" apples, but equally you can't have "3 miles north north east" apples, because neither of these things make sense. "i" usually means something like "an oscillation with an amplitude 1 and a phase of pi/2". Nothing weird about it when it's in the right context.
@Chronosaur yes, if you deal with i in a purely applied context its relatively easy, and helps to make things more simple. however ive found the more you learn about i, the less comfortable you feel making the analogy with R^2. things like Cauchy’s integral theorem can be proved or explained using vector calculus (green's theorem), but im sure in doing this you miss something more fundamental. personally i hate fake numbers and am sure they conceal some deeper structure.
@abrelosojo Well I wasn't really stressing the analogy between complex numbers and 2d vectors. I'm just saying they're both examples of mathematical widgets we've invented, which make sense in context, but maybe their name makes people think of them in the wrong context. Maybe the perception of them as fake comes from their name as well - I don't think they're any more fake than vectors, tensors, or even real numbers. I think they're all just tools we invent to describe various bits of reality
"i" is a mathematical tool, simple as that. You need it because you want your mathematical frame of reference have X and y (and perhaps Z or more) be around the 0 mark, and use graphs to display something of value around the 0 point. This means you get a discrepancy from "reality" only because you're framing the data in a mathematical frame which is limited, and "i" helps turn certain things the proper way.
Negative numbers can't be 'seen' in the conventional sense either. You can't have -5 apples. Yet you can use negative numbers and still end up with a real result, for example having 4 apples and removing 3, leaving 1.
In the same way, you can't have sqrt(-1) grams of sugar, but you can still get a real output from an equation using imaginary numbers.
I was used to i but the new book I bought (Linear Systems and Signals, from B.P. LATHI) represents it with j, although every software I know also uses i.
@alexanderhulse It's not an observable quantity yet. Mathematics has a neat way of predicting real phenomena, which says something about how these relatively abstract structures may relate to reality better than our limited observations realize.
It's unfortunate that this dimension has a bit of a misnomer. It's not merely some "imaginary" thing, nor would I say it's merely a convenience. As it turns out, the real numbers are a subset of the complex numbers.
@Lyrelia so what your saing s thatfrom our 3 dimentonal human perspective we have no ability to understand such things. ANd so would an imaginary number only beable to quantify (i dont even know if i made up that word or no but you get the gist) imaginary quantity such as vertual particles which seem to be some limited in this world that they would need such advance level of mathmatics to make sense of. otherwise im just making silly link according to myown very limite knowlage of reality.
@alexanderhulse From the three spatial and one temporal dimension we grew up in, we may not be able to witness these things directly, but we can still understand them. We can use technology to supplement out senses, and failing that, we can use math. It's our eyes and ears in abstract places.
Imaginary numbers are well-defined quantities in math, but does it have a physical analogue? Besides fluid dynamics, can it quantify a counterpart to real objects? I don't know. Interesting thought.
@oryxfreeride O.K., explain to me just why imaginary numbers play a central role in quantum mechanics. Why is that we need to use complex notation to express probability amplitudes? Feynman and Penrose, amongst many others, claim that in QM complex numbers *aren't* just a mathematical shortcut, they're an essential element of the physics. Why is that the time-dependent Schrodinger equation requires "i" ? These are far from trivial questions.
@Moriarty2112 Complex numbers are just an automatic consequence of defining inverse powers real numbers, it doesn't matter that there's no solution to the square root of a negative number, if you're going to try and create a number system with an exponent operation then its only natural to have to define the imaginary unit. Define them as having a real and imaginary part so that the reals are a subset, and the rest pretty much fall into place, except Euler equation isn't quite so obvious.
What I'm alluding to at the start of the video is that a wavefunction is a complex-valued mathematical function. As an undergraduate I found the complex nature of the wavefunction (and the link between observables and real-valued quantities) a fascinating and difficult-to-fathom aspect of QM. As a professor of physics, I still do. Perhaps you don't? If not, *please* explain it to me!!
More on this in the Sixty Symbols "wavefunction" video...
@Moriarty2112 Pretty glad you wanted to chat some more Philip ! I'm a student in mechanical engineering here in the Polytechnique of Montréal.
I see imaginary numbers as a change of variable that gives a more accurate model of the situation (ex. Alternate Current, negative discriminant)
Also, you mention about wavefunctions, it's not much about wavefunction themselves, but their periodicity. Again if I refer to Euler, the circle goes round and round periodically !
Hi, Philippe. Thanks for responding. I mean wavefunction in the quantum mechanics sense. This is much, much more than a question of a periodic function and the use of Euler's identity. (I taught a module on Fourier analysis for five years and am therefore reasonably well-versed in the use of complex functions in describing periodic functions).
In QM, complex numbers are not *just* a mathematical "trick" (as in AC circuit theory/resonant systems etc...).
The phase of a complex number has a central signficance in quantum mechanics as it underpins the phenomenon of wave interference. But in QM imaginary numbers are not just a "change of variable", as you put it. The imaginary component is essential for the *physics*.
I am very much aware of how complex numbers are used in engineering applications but their use in QM goes well beyond this. The imaginary part has a "reality" of its own...
Given that you already have a good mathematical grounding in complex numbers and complex functions, I would recommend the Feynman Lectures in Physics as a good place to learn about the use of complex numbers in QM.
Rafael Bombelli is generally credited with the discovery/invention (in itself, an interesting distinction in mathematics!) of complex numbers in the 16th century.
@Moriarty2112 I'm sure you already knew that though. I didn't mean to offend you, and I don't know how else to say it but they just make sense to me, especially given how rigorous the rest of mathematics is. As far as their use in Physics and Engineering I can't say much since i'm a Mathematics major. When they are used in an equation I just recognize it as a number which sort of doesn't represent anything 'yet', but it might 'soon' when you multiply it by some other complex numbers.
@Moriarty2112 It's kind of like defining irrational numbers in a number system when you're only concerned with integers, it might not make sense to use the square root of 5, but if you sqaure that number it makes sense, just as if it was the square root of a negative number in a number system where were we want to assign each number a 'measurable' value, such as length or velocity. Again, i'm sure you know this, but it seems perfectly natural to me.
You didn't offend me in the slightest. It's good to see complex numbers discussed on YouTube (rather than the pros and cons of yet another X Factor or "dancing cat" video...!)
See my responses to @ContempoREX below re. complex numbers in quantum mechanics (QM)
In QM you can't just say that the modulus (or modulus squared) of the complex number is all that matters. If we take the wavefunction then, yes, the modulus squared represents a real (observable) quantity.
In QM the imaginary part (or, more accurately, the phase) of the complex number plays an essential role - you can't just say that the complex number itself doesn't represent anything "yet". The phase is essential because it embodies the interference phenomena at the heart of QM.
This is why I said to ContempoREX below that complex numbers in QM are not used in quite the same sense as they're used in engineering/Fourier analysis/resonance problems.
You're absolutely correct that in the majority of physics and engineering problems one need only really worry about the use of complex numbers as a mathematical trick (...and having taught Fourier analysis for 5 years I entirely appreciate this usage).
But when I say at the start of the video that complex numbers "do my head in", I'm specifically referring to their usage in QM (although admittedly, that's not clear from the video!) which is more subtle.
For those who still does not get it, in an imaginary system it's (i,j) instead of the usual (x,y) in cartesian. Where i is the reel part and j the imaginary part.
Go check out Euler's Formula in wikipedia for better understanding of them imaginary ''not so weird'' numbers lol...
@SolidSnyder Euler's identity is e to the power of i * pi and as he wrote it was e to the power of i * theta. I thought the same till I rewound it though.
i can't imagine how much dept i'm in but then it shows me how to imagine it is possible to borrow energy from the future to be matter now. Then I realise nothing really matters and feel much better.
If I gave you -4 apples you would have 2 apples, however if I gave you i apples you would have 6+i. It is the impossibility of converting this into real values which makes it 'imaginary'.
@orochimarujes In fact there was a time when negative numbers seemed just as awkward. Maybe because people used to think of numbers strictly as devices to count material things. So in this scenario is perfectly understandable to reject negative numbers as well..
@orochimarujes In fact there was a time when negative numbers seemed just as awkward. Maybe because people used to think of numbers strictly as devices to count material things. So in this scenario is perfectly understandable to reject negative numbers as well..
@orochimarujes It depends if you take zero for nothing or the relative point. For instance, if you take sea level as 0 height, below water level is negative height. Now, that distance
The clip is referring to the role of the imaginary unit in physics!
For example, why does it appear in the Schrödinger equation?
LeconsdAnalyse 1 day ago
It's i, not j. !@@!!@
IamBread18 2 days ago
Why don't they show this on the first day of Algebra 2 in high school? I might have passed if I understood what they so plainly stated in such a short video. I guess that's what happens when you're math teacher looks more like a walrus than a man...
mynameismatt2010 4 days ago
Dark side of the engineers ... I LOLed at that!
ashwinnarayanVlog 5 days ago
e^(i*pi) + 1 = 0
I shat a brick when I realised that :P everything comes together!
xTheGerbilx 3 weeks ago 2
we started using " i " in year 8 at school.
engelteir 3 weeks ago 3
I love imaginary numbers baby: r*exp(j*theta) = r*[cos(theta) + j*sin(theta)]
quidproquo2004 3 weeks ago
The concept of imaginary numbers is very disturbing and unnatural to me :( Only good thing about it is that it's really easy to use :D
dekmaskin 3 weeks ago
@dekmaskin I interpret as a mathematical construction rather than a "physical concept". It is just an intermediate quantity (in a mathematical calculation) that makes much of the math easy. However, at the end, we need to get rid of it to get something "physical". For example, the probability density by taking norm of a wave function in QM, or the individual real components of a complex current in EE.
subh1 20 hours ago in playlist From Sixty Symbols
It seems to me that imaginary numbers are not necessary. I say that because a computer can calculate something that "requires" imaginary numbers, and it can do it without using imaginary numbers.
Ye, I don't like imaginary numbers.
antiHUMANDesigns 4 weeks ago
Just so everyone knows, all students in Greece learn how to use Complex numbers in the third class of Senior Highschool.....!!!!!!
vassilis15m 1 month ago 2
@vassilis15m I know, it's sad also that we neglect evolution wich is in the last pages of biology, never to be done. :(
MrAlexGTV 1 month ago
@MrAlexGTV Απ' οτι καταλαβα είσαι Έλληνας και γνωρίζεις τι μαθαινουμε εμεις οι Έλληνες μαθητές σε σχεση με τα παιδια του εξωτερικου.Θα μπορουσα να πω πως θεωρούμαστε προνομιούχοι...... :) :) :) :)
vassilis15m 1 month ago
@vassilis15m Those the same numbers used when your country calculated its bank balance?...Sorry I really could not resist.
Oafing 1 month ago 2
the guy in the thumbnail looks kinda like Chris Pontius. :3
PoliMeim 2 months ago in playlist More videos from sixtysymbols
The reason engineers use j is because in electrical engineering i is for current; (as Afroman would say, damn alphabet needs more letters.) It is also used in that discipline for calculating impedance, which combines resistance with a phase shift.
Desmaad 2 months ago
"sqrt(-1) love you!" - Markus Persson.
derick1259 2 months ago 13
@derick1259 j love you?
TheMrNoxia 2 months ago
@TheMrNoxia No, I love you :D
derick1259 2 months ago
Check the Eigenvalues video and the vibrating rules. Eigenvalues are usually complex (imaginary) numbers. In the case of the rules, how fast the rules stops shaking depends on the real part of the eigenvalue, while the rate at which it shakes depends on the complex part.
TriMiro8107 2 months ago
I don't like this video, if people are interested in j (or i), they should go to mathematicians not physicists. There is nothing magic or mysterious in the number, it is just an extension to the real numbers. Engineers use j because j is usually associated with oscillations and in that sense you can "measure" it. Things are not that clear in quantum physics, but this is due to the lack of knowledge of physics not mysterious math.
TriMiro8107 2 months ago
@TriMiro8107 I chuckled when I heard i labeled "immeasurable" and "not an observable physical quantity." I might as well declare "3 doesn't exist because we can't measure it to infinite precision." Such is the perspective of a scientist. Mathematicians have different goals entirely.
karlkid333 3 days ago
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LeconsdAnalyse 2 months ago
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LeconsdAnalyse 2 months ago
I was thinking exactly what that dude said when he said 'J conceals a structure that we have not discovered'. It's like a rule of thumb, for example Newtons laws worked, even though they were not 'true'.
TheBiddleMan 2 months ago
They're right: j^2 = -1
However, j is NOT the square root of -1. Let me show you:
-1 = j^2 = j*j = sqrt(-1) * sqrt(-1) = sqrt(-1*-1) = sqrt(1) = 1
In other words, if you say the square root of -1 is j, you have proven that -1 = 1 (Which isn't true , obviously)
nvdkooy 2 months ago
@nvdkooy Well, is J2 is -1 then it is the square root of -1, -*-=+ +*+=+ Therefore J is imaginary because it doesnt exist, as no real number squared can equal a minus so by saying J2 = -1 the square root is still J
8BitMusicStudios 2 months ago
@nvdkooy
By definition, when j^2 =-1 then j has to be √-1. You can't multiply imaginary numbers with each other as you have done.
From wiki answers:
To multiply two imaginary numbers aj and bj [lol], start by pretending that j is a variable (like x).
So aj x bj = abj^2. But since j is √-1, j^2=-1. So abj^2=-ab.
Examples:
6j x 7j =-42.
(-5j) x 2j =-(-10)= 10.
smaakjeks 2 months ago
@nvdkooy
Keep in mind that x^2 = 1 has two roots, x = -1 and x =1. x^2 = -1 also has two roots, x = j and x = -j. So you threw out one root at -1 = j^2 and chose the spurious root at sqrt(1) = 1.
dhmorgret 2 months ago
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LeconsdAnalyse 2 months ago
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LeconsdAnalyse 2 months ago
@nvdkooy Actually, that's utter nonsense. Be definition of j, sqrt(a)*sqrt(b) in general is not sqrt(ab), as you've shown. This only works for a,b>0.
safibn12 2 months ago
Well, I have an intuition about "i" or " j":
Since j allows us to get out of the trigonometric circle, and into the other side of a parabole, I think of j as another " dimension" that links unlinkable things. I intuitively feel one jayish ship could exit a blackhole, for instance (but I am really just making a crazy assumption here).
Maybe j is just a hard to observe part of reality.
a1a2a3skurr1l 2 months ago
@a1a2a3skurr1l from a simple check around wikipedia, turns out the thing about imaginary speed particles scaping blackholes is right, proving my intuition. So, it is possible to have an intuitive feel about imaginary numbers, an if I can, I am sure there are many others that do too.
a1a2a3skurr1l 2 months ago
To acquire MRI images using the signals received from body tissue, the process utilizes imaginary numbers. A practical use for them
ScottOrTheApocalypse 3 months ago
I'm an accountant. I don't like the idea of something like this. At all.
Versudan 4 months ago 35
@Versudan lol as an accountant, i think you'd be lookin at some trouble from the SEC or something if you used it in financial reporting.
UltraProle21 3 months ago
@UltraProle21 lol, oh yes I would. That being said, I'm tempted to bring this symbol in to show my workmates and see if any of them recognize it.
Versudan 3 months ago
@Versudan Ironically, negative numbers where devised for debt. Now tell me you don't want to have negative square roots!
MrAlexGTV 1 month ago
Listen to 2:37 of this video and then listen to the first minute or so of /watch?v=q9w26JXXpWU
Same person?
GrumbleJones 4 months ago
Heck, we were taught this in high school trigonometry and precalculus.
OOZ662 4 months ago
I am an engineer, and yet I use i, not j...perhaps j is just a UK thing...
1RadicalOne 5 months ago
@1RadicalOne j is a german thing, too. In electronics noone uses i, everyone uses j.
zurechtweiser 5 months ago
I think an interesting thing to think about is how irrational numbers are just as weird as imaginary numbers. Natural numbers (0,1,2) are defined in terms of counting collections of objects. Fractions can be defined quite easily in terms of naturals (a third is the thing which if you have 3 of you have 1). But what about root 2 which can't be written as a fraction? What does that number mean? What significance can it ever have? A physical measurement could never give an irrational number either.
wowsa0 5 months ago
@wowsa0 well, a triangle with a ninety degree angle and two sides of 1 would give you sqrt(2) as the hypotenuse if you measured it.
LamaPaj 4 months ago
@LamaPaj How? We measure stuff with rulers, which consist of a series of unit lengths marked on a stick, and you line up two units with either end of what you're measuring and get a result, which is ALWAYS a rational number because it consists of counting individual units. If you measure the side of the triangle you describe you will never get exactly root 2, no matter how small you make your scale divisions. I'm not even sure a perfect triangle could ever exist in nature because of this?
wowsa0 4 months ago
@LamaPaj When irrational numbers are defined properly in pure maths they are defined in terms of a kind of limit I think. And in the same way you can never measure the instantaneous velocity of an object, even though that does have a theoretical meaning, I don't think you can ever measure an irrational length, you could only take a sort of limit of a series of more precise rational measurements.
wowsa0 4 months ago
@wowsa0 Well, numbers like sqrt(2) and pi are used in a LOT of practical calculations that are not related to purely measuring. Any way my interests are more in the area of pure mathematics then measuring things.
LamaPaj 4 months ago
@LamaPaj Oh no don't get me wrong I'm not saying that root(2) or pi are some sort of mystical entities cooked up by deluded mathematicians which have no real meaning... I'm merely pointing out that you don't have to jump straight to imaginary numbers to find numbers which are weird. I don't think imaginary numbers can be said to 'exist' with any more or less confidence than irrational numbers can.
wowsa0 4 months ago
@wowsa0 Well, irrational numbers are weird, but can still be grasped somewhat intuitively. i is on a whole other level, since the rules we use to calculate things usually seems to prohibit there being such a number.
LamaPaj 4 months ago
@LamaPaj No it wouldn't, it gives a value of root(2) if you calculate it, but you cannot measure something that has an irrational length. 1.41cm is NOT root(2), no matter how accurately you measure it you will not end up with root(2)
salerio61 4 months ago
@salerio61 Well, to begin with, thank you for jumping into a discussion that happened over two weeks ago.
Also, thank you for pointing out the bloody obvious, wankhead. I answered to his question "what significance can it ever have?"
LamaPaj 4 months ago
@LamaPaj Do you always behave like a fuckwit when someone tells you you are wrong, and why?
Grow the hell up. If it's SO obvious you wouldn't have made such a stupid statement in the first place would you. Geez! Wowsa was right, a physical measurement cannot give an irrational number. So you're just plain wrong
salerio61 4 months ago
@salerio61 Well, if you would have read my reply i was really not talking about measuring was I?
If I measure up two distances of 1 arbitrary unit each from a point A, where one of these distances would be the distance from point A to a point B and the other would be the distance from point A to a point C, and the angle between AB and AC would be 90 degrees then the distance B to C would be sqrt(2) au. Then if you can measure up 1 au you know this to be true and therefore of significance.
LamaPaj 4 months ago
salerio61 4 months ago
@salerio61 I am not talking about measuring... If you would have read the old argument you would have seen that i realized my mistake, and went on making a point about the numbers still having very real and practical uses. Jeez...
Also, if you can measure the sides AB and AC to exactly 1 au you know that the hypotenuse BC is sqrt(2).
You cant measure it, but that doesnt mean its not true.
LamaPaj 4 months ago
@LamaPaj " I am not talking about measuring..."
Fucking incredible! You are totally delusional. I refer you to your previous posts ...
" would give you sqrt(2) as the hypotenuse if you measured it."
Sure sounds like measuring
"Then if you can measure up 1 au you know this to be true and therefore of significance"
Yet again measuring. Look it's ok to admit you're wrong, but you don't have to make yourself look like an arse defending the indefensible. Give up already!
salerio61 4 months ago
@salerio61 Well, since you dont fucking read what i write this conversation is over.
LamaPaj 4 months ago
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@LamaPaj " would give you sqrt(2) as the hypotenuse if you measured it."
Yep, time to give up on your fuckwittery.
salerio61 4 months ago
This has been flagged as spam show
@LamaPaj " I am not talking about measuring..."
Fucking incredible! You are totally delusional. I refer you to your previous posts ...
" would give you sqrt(2) as the hypotenuse if you measured it."
Sure sounds like measuring
"Then if you can measure up 1 au you know this to be true and therefore of significance"
Yet again measuring. Look it's ok to admit you're wrong, but you don't have to make yourself look like an arse defending the indefensible. Give up already!
salerio61 4 months ago
I like the video, but do prefer the way Calvin and Hobbes explain imaginary numbers.
MrStaggerLeeee 5 months ago
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nmuller789 5 months ago
can you guys by any chance just look up the curriculum of all my courses and just do videos on them (yr 11 and 12 queensland maths c, maths b, physics, chemistry) cheers
nmuller789 5 months ago
@nmuller789 .... Really, don't try to cheat on your curriculum. It will get you nowhere.
itsmanofpopsicle 5 months ago
@itsmanofpopsicle huh is it cheating if Im being taught?
nmuller789 5 months ago
And then of course, there are quaternions, which have i, j, and k, *all* of which are the square-root of -1. But they aren't the *same* square-root of -1.
Because *that* makes sense.
GeneralBolas 5 months ago
The guy with the glasses and goatee so looks like chris pontius :P
J4K34RM0UR 6 months ago
I always thought there was supposed to be some deeper meaning to j, so it is good to know that it is just a tool to be used and that if there is a deeper physical meaning, these guys don't understand it either.
MisterMANada 6 months ago
sqrt-1 LOVE THIS VIDEO :D
(spoiler) it meant i LOVE THIS VIDEO
hall0fgamez 6 months ago
@hall0fgamez i thought you loved it so much you started squirting
assassinbbx 6 months ago
God = i?
wennafied 6 months ago
@wennafied god=my ass
TheRightNeutrino 5 months ago
@TheRightNeutrino wait..... so i=your ass ??? D=
AscendingAshTree 5 months ago
@AscendingAshTree no, didn't you watch the video? or went to high school? i=sqrt(-1)
TheRightNeutrino 5 months ago
@TheRightNeutrino Uhhhh... dude... it was a joke? I know i=root -1, and how you can do i^4 =1, so all powers of i cancel in 4's etc, it's simple at school level... the joke was 'god=i', 'god=your ass', thus 'i=your ass'. Anyway, if we're going to get immature... BAD grammar, that should be "No, didn't you watch the video or *go* to high school", cor, didn't you go to school?
AscendingAshTree 5 months ago
@AscendingAshTree a. not funny, b. there is no grammar in youtube comments, I can write however I want as long as normal people can understand what I mean, I'm certain that if you were to read a comment from Shakespeare's time you wouldn't understand what it meant, so language is flexible, it has one single purpose and that is to allow people to communicate so your stupid grammar corrections are useless. c. I wasn't born or raised in an english speaking country like yourself.
TheRightNeutrino 5 months ago
@TheRightNeutrino There's still a few problems with this, a. Whether or not it's funny is entirely subjective, so either way you don't get to be an arse. b. If there're no grammar in youtube, and there's also no humour either apparently, why ARE there obvious maths corrections? c. In which case, sorry for the constructive criticism, and finally d. This has spammed up the video's comment box.
AscendingAshTree 5 months ago
@AscendingAshTree a. ? b. I didn't think it was an obvious correction in your case, and why should the three things you mention be somehow related? the obvious mathematical correction was sarcasam on top of your "joke", d. i'm sure they don't care. e. 2.7183, f. (fi) 1.618, g. 9.8m s^-2, h. 6.610^-34 J s, i. sqrt(-1), and we're back where we started, it's like we passed through a wormhole, amazing. next sixty-symbols video - wormholes.
TheRightNeutrino 5 months ago
@TheRightNeutrino Ok then... uhh... R. 8.31 JK^-1mol^-1 ? Gone brain dead, can only thing of e, earth's,gravity, golden ratio, planck's constant, molar gas constant and pi... Those aren't even alphabetical! ...Holidays are too long...
AscendingAshTree 5 months ago
theres a phone constant in the background noise.
bone656 6 months ago
They should add to this that although "imaginary numbers" do seem counter-intuitive and abstract at first, real numbers are just as much a mathematical construct.
What is a real number? It's not a thing, you can never find one in the real world, the reals are a useful mathematical structure and so are the complex numbers. The imaginary numbers are no less "pure" or "unreal" than the reals, which is why I hate the term "imaginary" number. All numbers, in some sense, are just abstract constructs.
jamma246 6 months ago
@jamma246 Yes! Thank you for saying it.
wkrepelin 6 months ago
@jamma246 As a saving grace, at least the term 'complex number' is very good.
lethalsub 6 months ago
@jamma246 Yes, but I believe the problem lies in the fact that we can relate "real" numbers to amounts. In contrast, an "imaginary" number such as i cannot be applied in the same sense. That fact makes it very hard to conceive what it is exactly without a concept of what it does or how it works. I do, however, agree with you.
CrazyMonkey124 6 months ago
We should make it universally "i". It makes more sense, it's "i" for imaginary, it's what the mathematicians call it (and we invented it!) and when you go to quaternions it's nice having a base of 1,i,j,k.
Maybe the engineers should start referring to their things which already have an i with an iota instead.
jamma246 6 months ago
In most math classes they use i to represent the sqr-1 because it stands for imaginary, most physics/engineering classes use j to distinguish it from the i that represents a variable current. J is also used for current density, but normally the current density isn't a variable its a constant and is therefore represented with a capital letter J.
theviolator23 6 months ago
I saw the imagenary numbers in 4th grade in hi-school, i thought everybody whom started graduate knew it. Aparently not.
wybo2 7 months ago
Loving the Gordon Freeman beard going on there.
gcndavidmn 7 months ago
@gcndavidmn He is a physicist :)
zorbakpants2 7 months ago
@zorbakpants2 it's very appropriate.
gcndavidmn 7 months ago
can't you use vectors and a coordinate system in most cases where i/j is used?
nybotheveg 7 months ago
I don't understand why they find j or i to be a difficult concept. I can see how one can be amazed at how incredibly useful they are in solving 'real' problems, and I share this feeling, but as for the definition, existence etc... I see no problem whatsoever.
sorrysonofa 7 months ago 2
@sorrysonofa Well, maybe you can relate to struggling with some abstract concept some time ago. Maybe there was something you wanted to learn, once upon a time, but nobody told you or you couldn't figure out why you really needed to learn it, or you couldn't see a real world usage for the concept, and you gave up on it. That's how it is for a lot of people with imaginary numbers. Ages ago it was the same thing with negatives - people couldn't imagine that there could be "less than nothing".
agentvirgo 6 months ago
@sorrysonofa Well I believe they are talking from an engineering aspect more so in that they are using j in a real system. Not just a math problem, and that is where it can be confusing. I am an electrical engineer and one of the times it came up for me is when you put a capacitor in the system with a sin signal into it. you have a real voltage and an imaginary voltage. When you combine them together you get the actually measurable voltage. See a bit confusing.
gimlic 6 months ago
@sorrysonofa Then i guess that you haven't quite understood how fucking weird it is to have a number that has the property that when you square it it equals minus one.
LamaPaj 4 months ago
@LamaPaj No, I think you haven't understood why that is nothing out of the ordinary : many number systems besides the complex numbers have minus one as a square.
sorrysonofa 4 months ago
@sorrysonofa quaternions and octonions. Also please do remember that complex numbers don't have -1 as their square roots as a rule. Also, consider this: there is no real answer to this question "Which number is bigger, 1 or i"
LamaPaj 4 months ago
@LamaPaj quaternions and octonions don't even scratch the surface : take the integers modulo a prime that is 1+4k, take matrixes of even order over any commutative ring, take any commutative ring that has no square root of minus one and *add* such a root via a quotient space of polynomials....
C is not an ordered field, so what? Minus one is still commonly a square, and there's nothing special about that.
sorrysonofa 4 months ago
@sorrysonofa
It is probably because you are a non-physicist/mathematician. If you do not have knowledge in theoretical physics but read general physics books, you may find quantum mechanics understandable. But a physicist would find it really awkward. Certain things just require an intense level of study to finally realize that they are just ridiculously abstract. The same is true for imaginary numbers. It just doesn't make sense as to why a nonexistent value has such great existential value.
finalfantasiest1234 4 months ago
This has been flagged as spam show
@sorrysonofa
It is probably because you are a non-physicist/mathematician. If you do not have knowledge in theoretical physics but read general physics books, you may find quantum mechanics understandable. But a physicist would find it really awkward. Certain things just require an intense level of study to finally realize that they are just ridiculously abstract. The same is true for imaginary numbers. It just doesn't make sense as to why a nonexistent value has such great existential value.
finalfantasiest1234 4 months ago
engineers in belgium also use "i" :)
ArnestyInternational 7 months ago
1/0 = Undefined do a video on that PLEASEEE
69MrUsername69 7 months ago
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LeconsdAnalyse 7 months ago
@69MrUsername69 Its simply not a valid operation.
LamaPaj 4 months ago
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LeconsdAnalyse 8 months ago
The product of i and circular displacement is velocity.
webmastertool 8 months ago
i love the thumbnail of the guy looking pensive as if he's trying to imagine the number i.
dannyb21892 8 months ago
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LeconsdAnalyse 8 months ago
Engineering is not the dark side. I am disappointed.
Moratorium 8 months ago
If you notate "j" for "i" how do you notate quaternions!
orangegold1 8 months ago
I think it's the fact that it's called an "imaginary number" that causes confusion. It aint really a number. A vector isn't a number, but people don't think vectors are weird. You can't have "i" apples, but equally you can't have "3 miles north north east" apples, because neither of these things make sense. "i" usually means something like "an oscillation with an amplitude 1 and a phase of pi/2". Nothing weird about it when it's in the right context.
Chronosaur 9 months ago
@Chronosaur yes, if you deal with i in a purely applied context its relatively easy, and helps to make things more simple. however ive found the more you learn about i, the less comfortable you feel making the analogy with R^2. things like Cauchy’s integral theorem can be proved or explained using vector calculus (green's theorem), but im sure in doing this you miss something more fundamental. personally i hate fake numbers and am sure they conceal some deeper structure.
abrelosojo 8 months ago
@abrelosojo Well I wasn't really stressing the analogy between complex numbers and 2d vectors. I'm just saying they're both examples of mathematical widgets we've invented, which make sense in context, but maybe their name makes people think of them in the wrong context. Maybe the perception of them as fake comes from their name as well - I don't think they're any more fake than vectors, tensors, or even real numbers. I think they're all just tools we invent to describe various bits of reality
Chronosaur 8 months ago
I learned to use i in stead of j.
Muscleduck 9 months ago
Yes, the darkside, Engineering
iburnaga 9 months ago
Physicists generally use i
i = current for electrical engineers so they use j instead
Richy15251 9 months ago
"i" is a mathematical tool, simple as that. You need it because you want your mathematical frame of reference have X and y (and perhaps Z or more) be around the 0 mark, and use graphs to display something of value around the 0 point. This means you get a discrepancy from "reality" only because you're framing the data in a mathematical frame which is limited, and "i" helps turn certain things the proper way.
kashgarinn 9 months ago
i said to pi "be rational", pi replied "be real"
Bissmarkpl 10 months ago
@Bissmarkpl maths jokes - if you get them you probably have no life
MANGEYHUNTER1 9 months ago
@MANGEYHUNTER1 Coming from the guy who just watched 4 minutes of a physicist talking about imaginary numbers.
rockerlkj 9 months ago
@rockerlkj its a fucking joke get over it
MANGEYHUNTER1 9 months ago
@MANGEYHUNTER1 Sorry. It's hard to tell sometimes on youtube.
rockerlkj 9 months ago
well..of course you gotta use j
dreasim 10 months ago
Think of it this way:
Negative numbers can't be 'seen' in the conventional sense either. You can't have -5 apples. Yet you can use negative numbers and still end up with a real result, for example having 4 apples and removing 3, leaving 1.
In the same way, you can't have sqrt(-1) grams of sugar, but you can still get a real output from an equation using imaginary numbers.
Freechips1 10 months ago
I agree on talking about "e", i think that would make an excellent video.
Forrester 10 months ago
@Forrester - log means base 10, ln means base e
njimko23 10 months ago
@njimko23 I understand the application of e.
Forrester 10 months ago
On a similar note to using i and j, at university do you use log or ln? (log base(e))
PianoKwanMan 10 months ago
I was used to i but the new book I bought (Linear Systems and Signals, from B.P. LATHI) represents it with j, although every software I know also uses i.
leovailati 11 months ago
I thought it was i...
mangoismycat 11 months ago
ok can you give me an example yes i can understand how something can be used to help gain an estimate but is not an observable quanitiy.
alexanderhulse 1 year ago
@alexanderhulse It's not an observable quantity yet. Mathematics has a neat way of predicting real phenomena, which says something about how these relatively abstract structures may relate to reality better than our limited observations realize.
It's unfortunate that this dimension has a bit of a misnomer. It's not merely some "imaginary" thing, nor would I say it's merely a convenience. As it turns out, the real numbers are a subset of the complex numbers.
Lyrelia 9 months ago
@Lyrelia so what your saing s thatfrom our 3 dimentonal human perspective we have no ability to understand such things. ANd so would an imaginary number only beable to quantify (i dont even know if i made up that word or no but you get the gist) imaginary quantity such as vertual particles which seem to be some limited in this world that they would need such advance level of mathmatics to make sense of. otherwise im just making silly link according to myown very limite knowlage of reality.
alexanderhulse 9 months ago
@alexanderhulse From the three spatial and one temporal dimension we grew up in, we may not be able to witness these things directly, but we can still understand them. We can use technology to supplement out senses, and failing that, we can use math. It's our eyes and ears in abstract places.
Imaginary numbers are well-defined quantities in math, but does it have a physical analogue? Besides fluid dynamics, can it quantify a counterpart to real objects? I don't know. Interesting thought.
Lyrelia 9 months ago
ooh there's tension between these two
alexanderhulse 1 year ago
OKAY whose dumb idea was it to put the alphabet in mathmatics
possesedbunnystudios 1 year ago
I have an imaginary friend. (-1)^1/2
rbandgh3 1 year ago
This is one of my favourites. Can we please have a video on all of the ludicrously unbelievable properties of e?
hooloovoo1st 1 year ago 55
@hooloovoo1st Yes I am with this guy! why e?!
I've been wondering about this so much, why is e everywhy just why?!
CrazyPerson03832 8 months ago
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@hooloovoo1st Yes I am with this guy! why e?!
I've been wondering about this so much, why is e everywhere just why?!
CrazyPerson03832 8 months ago
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LeconsdAnalyse 7 months ago
Oh I remember these from highschool, little bastards > u <
Though we used an i^2, even the textbooks.
MarkArandjus 1 year ago
no offense, but its not that hard to fathom, and you still dont understand by the 3rd or 4th year of an undergrad, well.. I don't know what to say.
oryxfreeride 1 year ago
@oryxfreeride O.K., explain to me just why imaginary numbers play a central role in quantum mechanics. Why is that we need to use complex notation to express probability amplitudes? Feynman and Penrose, amongst many others, claim that in QM complex numbers *aren't* just a mathematical shortcut, they're an essential element of the physics. Why is that the time-dependent Schrodinger equation requires "i" ? These are far from trivial questions.
Philip (first person in video)
Moriarty2112 1 year ago
@Moriarty2112 Complex numbers are just an automatic consequence of defining inverse powers real numbers, it doesn't matter that there's no solution to the square root of a negative number, if you're going to try and create a number system with an exponent operation then its only natural to have to define the imaginary unit. Define them as having a real and imaginary part so that the reals are a subset, and the rest pretty much fall into place, except Euler equation isn't quite so obvious.
oryxfreeride 1 year ago
@oryxfreeride
What I'm alluding to at the start of the video is that a wavefunction is a complex-valued mathematical function. As an undergraduate I found the complex nature of the wavefunction (and the link between observables and real-valued quantities) a fascinating and difficult-to-fathom aspect of QM. As a professor of physics, I still do. Perhaps you don't? If not, *please* explain it to me!!
More on this in the Sixty Symbols "wavefunction" video...
Philip (first person in video)
Moriarty2112 1 year ago
@Moriarty2112 Pretty glad you wanted to chat some more Philip ! I'm a student in mechanical engineering here in the Polytechnique of Montréal.
I see imaginary numbers as a change of variable that gives a more accurate model of the situation (ex. Alternate Current, negative discriminant)
Also, you mention about wavefunctions, it's not much about wavefunction themselves, but their periodicity. Again if I refer to Euler, the circle goes round and round periodically !
Philippe Léveillé
ContempoREX 1 year ago
@ContempoREX
Hi, Philippe. Thanks for responding. I mean wavefunction in the quantum mechanics sense. This is much, much more than a question of a periodic function and the use of Euler's identity. (I taught a module on Fourier analysis for five years and am therefore reasonably well-versed in the use of complex functions in describing periodic functions).
In QM, complex numbers are not *just* a mathematical "trick" (as in AC circuit theory/resonant systems etc...).
..contd...
Moriarty2112 1 year ago
@ContempoREX
..contd...
The phase of a complex number has a central signficance in quantum mechanics as it underpins the phenomenon of wave interference. But in QM imaginary numbers are not just a "change of variable", as you put it. The imaginary component is essential for the *physics*.
I am very much aware of how complex numbers are used in engineering applications but their use in QM goes well beyond this. The imaginary part has a "reality" of its own...
Philip
Moriarty2112 1 year ago
@Moriarty2112 I don't know much about quantum mechanics unfortunately.
At this point, I'd like to know how the imaginary numbers were first introduce.
ContempoREX 1 year ago
@ContempoREX
Hi again Philippe.
Given that you already have a good mathematical grounding in complex numbers and complex functions, I would recommend the Feynman Lectures in Physics as a good place to learn about the use of complex numbers in QM.
Rafael Bombelli is generally credited with the discovery/invention (in itself, an interesting distinction in mathematics!) of complex numbers in the 16th century.
Best wishes,
Philip
Moriarty2112 1 year ago
@Moriarty2112 Keep the good work with the vids, I like them a lot =P
ContempoREX 1 year ago
@Moriarty2112 I'm sure you already knew that though. I didn't mean to offend you, and I don't know how else to say it but they just make sense to me, especially given how rigorous the rest of mathematics is. As far as their use in Physics and Engineering I can't say much since i'm a Mathematics major. When they are used in an equation I just recognize it as a number which sort of doesn't represent anything 'yet', but it might 'soon' when you multiply it by some other complex numbers.
oryxfreeride 1 year ago
@Moriarty2112 It's kind of like defining irrational numbers in a number system when you're only concerned with integers, it might not make sense to use the square root of 5, but if you sqaure that number it makes sense, just as if it was the square root of a negative number in a number system where were we want to assign each number a 'measurable' value, such as length or velocity. Again, i'm sure you know this, but it seems perfectly natural to me.
oryxfreeride 1 year ago
@oryxfreeride
You didn't offend me in the slightest. It's good to see complex numbers discussed on YouTube (rather than the pros and cons of yet another X Factor or "dancing cat" video...!)
See my responses to @ContempoREX below re. complex numbers in quantum mechanics (QM)
In QM you can't just say that the modulus (or modulus squared) of the complex number is all that matters. If we take the wavefunction then, yes, the modulus squared represents a real (observable) quantity.
...contd...
Moriarty2112 1 year ago
@oryxfreeride
In QM the imaginary part (or, more accurately, the phase) of the complex number plays an essential role - you can't just say that the complex number itself doesn't represent anything "yet". The phase is essential because it embodies the interference phenomena at the heart of QM.
This is why I said to ContempoREX below that complex numbers in QM are not used in quite the same sense as they're used in engineering/Fourier analysis/resonance problems.
..contd...
Moriarty2112 1 year ago
@oryxfreeride
..contd..
You're absolutely correct that in the majority of physics and engineering problems one need only really worry about the use of complex numbers as a mathematical trick (...and having taught Fourier analysis for 5 years I entirely appreciate this usage).
But when I say at the start of the video that complex numbers "do my head in", I'm specifically referring to their usage in QM (although admittedly, that's not clear from the video!) which is more subtle.
Philip
Moriarty2112 1 year ago
j*j = -1 ?..... jesus * jesus = -1 ? .... jesus is imaginary???? what are you trying to say here sir?!?!! :P did I just bring religion into this?
VCSandARM 1 year ago
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runnybabbit12 1 year ago
For those who still does not get it, in an imaginary system it's (i,j) instead of the usual (x,y) in cartesian. Where i is the reel part and j the imaginary part.
Go check out Euler's Formula in wikipedia for better understanding of them imaginary ''not so weird'' numbers lol...
ContempoREX 1 year ago
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Leudast1 1 year ago
I spy with my little eye...Euler's formula!
SolidSnyder 1 year ago
@SolidSnyder Euler's identity is e to the power of i * pi and as he wrote it was e to the power of i * theta. I thought the same till I rewound it though.
Furiouslyfappin 1 year ago
i can't imagine how much dept i'm in but then it shows me how to imagine it is possible to borrow energy from the future to be matter now. Then I realise nothing really matters and feel much better.
lthp001 1 year ago
The way I see it, imaginary numbers are no more imaginary that negative numbers.
orochimarujes 1 year ago 33
@orochimarujes
Say for instance that you had 6 apples.
If I gave you -4 apples you would have 2 apples, however if I gave you i apples you would have 6+i. It is the impossibility of converting this into real values which makes it 'imaginary'.
RazorN6 1 year ago
@orochimarujes:As far as math goes, your right - except - there's twice as many of 'em.
puncheex 1 year ago
@orochimarujes In fact there was a time when negative numbers seemed just as awkward. Maybe because people used to think of numbers strictly as devices to count material things. So in this scenario is perfectly understandable to reject negative numbers as well..
bilthon 9 months ago
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@orochimarujes In fact there was a time when negative numbers seemed just as awkward. Maybe because people used to think of numbers strictly as devices to count material things. So in this scenario is perfectly understandable to reject negative numbers as well..
bilthon 9 months ago
@orochimarujes There is an old joke about mathematics:
"Mathematics means that if 3 people are in a room, and 5 leave, 2 have to enter it again to make it empty."
ThatGuyFromAustria 9 months ago
@ThatGuyFromAustria i dont get the joke...can you try it again in the style of say, groucho marx or robin williams?
MasterOfSuprise 9 months ago
@MasterOfSuprise, unfortunately, no. If you don't get it, there is no help for you. Sorry.
ThatGuyFromAustria 9 months ago
@MasterOfSuprise unfortunately, no. If you don't get it, there is no help for you.
ThatGuyFromAustria 9 months ago
@orochimarujes It depends if you take zero for nothing or the relative point. For instance, if you take sea level as 0 height, below water level is negative height. Now, that distance