How Imaginary Numbers Make Real Physics Easier to Understand
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- Опубліковано 26 лип 2024
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#imaginarynumber #complexnumbers #physics
In this video, we'll look at the basics of complex and imaginary numbers, and how they are used in physics!
To begin with, we define the "imaginary number", i, as being the square root of -1. We're often told that negative numbers cannot have a square root, but imaginary numbers are based on the idea that they can. Engineers often use j to represent the imaginary number but we'll stick with i.
An imaginary number can be added to a "real" number (one which does not have a factor of i) in order to create a "complex" number. We look at how two real numbers can be added together, as well as multiplied together.
Imaginary numbers do not fall on the (real) number line, but we instead are found on a perpendicular axis to the number line. That way, we have a real axis and an imaginary axis creating an abstract space. This graph/space is known as an argand diagram, and can be used to represent any complex number. The way to do this is to start at the origin, move as many units in the real direction as the real component, and then as many units in the perpendicular, imaginary direction as the imaginary component. The point we end up at represents our complex number.
The complex number can also be represented with a vector from the origin to the corresponding point on the argand diagram, so its horizontal component is the real part, and its vertical component is the imaginary part. Using this knowledge, as well as basic trigonometry, we can define two new quantities known as the absolute value, or modulus (length) of the vector, and the argument (angle from the real axis). These two pieces of information are equally as good at defining a complex number as knowing its real and imaginary parts.
We can take this information to write a complex number in terms of its absolute value, and the sines and cosines of its argument. However this last part can be converted to a much simpler complex exponential using Euler's identity (en.wikipedia.org/wiki/Euler's.... We cover the basics of the exponential function as well as how much easier it is to deal with complex exponentials than sines and cosines (as exponentials are easier to multiply).
We then look at two scenarios in physics where we need to represent systems by using sines and cosines. The first is a mechanical harmonic oscillator, such as a mass oscillating on a spring. Instead of dealing with the sine (or cosine) representing the motion of the mass, we can represent it using a complex number evolving over time, do any calculation necessary, and then simply take the real part of the complex number. Taking the real part involves just reading the real part and ignoring the imaginary part. This works because the two components are separate from each other (or perpendicular on the argand diagram). The same logic can be used to represent electric circuits with a sinusoidal input potential difference. This is useful when we have capacitors, inductors, or resistors in our circuit as the voltage and current are not always in phase.
Finally, we look at how quantum wave functions are complex. Although the square (modulus) of a wave function relates to real, measurable probabilities, and the square modulus is not complex, the complex nature of the wave function can be measured in more subtle and indirect ways in effects such as the Aharonov-Bohm effect. Check out the links below for more info, as I've made a full video discussing it.
Videos linked in Cards:
• Why Ohm's Law is NOT V...
• Wave Functions in Quan...
• We DON'T Understand Ma...
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Timestamps:
0:00 - What are Imaginary, Real, and Complex Numbers, and How Do We Add Them?
3:18 - Representing Complex Numbers on an Argand Diagram
5:08 - The Modulus and Argument of a Complex Number
6:10 - Trigonometric Identities and Exponential Functions
7:59 - Euler's Identity (and Why We Bother With It)
9:28 - Oscillating Mass on a Spring and Complex Numbers
10:23 - Alternating Current Power Sources
12:19 - Quantum Complex-ness
14:31 - Big thanks to Squarespace for Sponsoring!
15:27 - Outro
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As always, let me know what other topics to cover in future videos :)
I feel it to be a shame that you left out complex power in the presentation of complex numbers in electrical circuits.
Yes, certainly interested in a lecture about voltage and current, thanks.
+1
Plus 1 more
Your wish is my command. Math Challenge: Pythagoras, i, AC Circuits ua-cam.com/video/AYFdhlqzH5c/v-deo.html Leave a comment to let me know how many seconds into it, it went off the rails.
Capacitors, inductors, then passive filters, active filters with op amps, digital filtering, FFT, diodes, transistors, circuit analysis, all this is cool.
yes yes yes, I would love such series.
I read once that Gauss wanted to call them “lateral” numbers which, given the complex plane, makes a lot of sense.
Understanding i as a 90° counterclockwise rotation gives all the intuition for what imaginary numbers are.
@@NoActuallyGo-KCUF-Yourself: Yep, especially when you see multiplication by -1 as a 180° rotation that takes one from the positive part of the real number axis to the negative part (and vice versa). _Half_ of such a rotation (90°) necessarily amounts to sqrt(-1).
@@TheWyrdSmythe Yes, the -1 jumping from right to left and back when multiplying really explains in a way why multiplying two negative numbers results in a positive number. At school I just learned to apply the rule, without really getting it.
@@paulbloemen7256 Yep. Multiplication is rotation!
Thank you parth I'm really glad you made this video because I was always intrigued on how imaginary numbers would be used I've heard of them in my math classes but never used them to solve a physics problem so it's nice that you gave me your window onto how that would work
Thanks for watching! I'll try to include even more concrete examples in future videos :)
Another weird way you can use them is to integrate particularly tricky integrals that arise in physics using tricks like residue theorem
Consensus: Imaginary numbers
Gauss: Lateral numbers
Parth: JEFF!
Electrical circuits series all the way! Great video as always
I really enjoy your content. I initially saw some of your videos when I was getting into physics a while back. I really couldn’t understand the math but was able to pick up on your enthusiasm for the subject as well as gain insight with a layer of abstraction. I think personalities like yours are crucial to spreading these physical/mathematical ideas effectively. Upon coming across your channel now after having brushed up on some math, I’m left much more fulfilled and informed with the concision of your explanations, as well as your taste in content.
I wish I had math and physics teachers like you during my school days... You make learning science more fun and beautiful! ❤
Thank you parth the structure of the video is really nice ❤️
Man !!! You just saved my whole Classical Mechanics
Yes. do the electric circuits referred to at time stamp 11:40
“Fundamentals of Electric Circuits” 5E is a good read, rather an easy to follow textbook on things that concern phasors, circuits and complex numbers too. Mathew Sadiku is an excellent writer on the topic of Electromagnetics too. These helped me a lot during my college days..and Parth does well in providing a lot of insight to young students these days..👍🏻👍🏻
Awesome video! Would love to see more on this topic. I’m currently in quantum 2 and still don’t fully understand the interpretation of imaginary numbers in a system lol
Very clear explanation: thank you! I would like to see all the videos you mentioned, me having the feeling I might actually get them.
This was well presented my guy! Well done!
Would love to see a video from you on electric circuits
Good Morning!
The correct is ì² = -1
And
V-1 = { - i ; i }
The principal Square of -1 in The set complex numbers is i
We write V-1 = i ; so you must to indicate That this is a principal Square of -1 and that exist a second Square that is this - i.
I wish you a Good Day!
Make as many videos on as many concepts as you want, will watch them all.
I would really really love to see the video about electric circuits in 5 difficulties you mentioned!
Thank you Parth. VERY much interested in your proposed electric circuits video - particularly the beginner level!
Absolutely would love a video on circuits. Can you talk about the connection between resistance and impedance.
Yes , i´m also interested in advanced circuit analysis & thank you very much for your videos on Physics
I was just waiting for Parth to upload another interesting topic.
Thanks, I understood how to put them in polar form but not what it actually represented
Yes, please do the other videos you talked about on circuits.
yeah it would be pretty nice, if you could do a video about eletric circuits, thanks for the vid btw
Thanks for this wonderful video. Clear as always
Can you go thru the related concepts of j^2=1, but j 1 and epsilon^2=0, but epsilon 0 ?
Thank you! Very helpful!
1:30 Actually, +/- 2i.
5:20 Yes, the square of the imaginary number is the product of that number and its complex comjugate.
I think you mean “the square of the length of the imaginary number”?
@@TheBasikShow I know what I meant. Do you?
Yeah the square of the modulus is the number times its conjugate
8:37 yes! yes! yes!....... Pls make a video on euler's identity
A set of axiomatic operations could be created whose operation was similar to that of complex numbers. The fact is, complex numbers being useful in physics doesn't mean they're fundamental to mathematics. In fact, the rule that multiplying two negative numbers returns a positive number, while useful in many real-life mathematical calculations, fails in many calculation jobs and that is why we need the concept of the absolute value of a number.
your way of explaning is sooo damn good
I'm suspicious of the 13:30 "you can't have a 3i% chance of finding a particle in space" thing. Maths doesn't lie.
Actually at the cutting edge of QM there is some speculation about negative probability but not imaginary as far as I know.
Hey, G your Physics videos are awesome. Can you also make math videos.
Can you make a more in depth video for simple harmonic motion and waves relationship to complex numbers like how we use the properties and all of that stuff.
Ah yes would love to do that!
Thanks
Yes, a video on Euler's Identity will be helpful. Thanks by the way.
I, for one, would like to leave a standing election on any physics videos Parth would like to make: an unqualified YES!
PLEASE DO make a circuit analysis video!
Was thinking of doing my PhD thesis in Jeff analysis
Would love to see those 5 videos!
I would definitely like to see a video on circuits and on electric currents in general
Thank u that was very helpful.
Well I learned complx nbers before but now I understand them more, however I have a little suggestion. Until now I've watched all your videos . I suggest that u add a little bit of soft bass music , it will kinda help with the explanation , don't know why lol. And I am gladly interested in circuit videos that would be super helpful.
thank you so much for such a nice explaination, my main intrest was in understanding quantum physics relation to complex number.
Hell to the yeah would I love to see the Electronic discussion in 5 levels.!!!!
Actually, i is defined such that i^2 = -1. This means that ‘solving for i’ gives two values ( sqrt(-1) and -1*sqrt(-1) ), so this is something you shouldn’t do. This is also the reason why sqrt(-4) is actually undefined, there is no positive square root for complex numbers because i can be considered neither positive nor negative.
sqrt(-4) = sqrt(-1 × 4) = sqrt(-1) × sqrt(4). Both of those factors are well-defined.
Square roots aren't restricted to positive values, only _non-negative._
i = sqrt(-1) _is_ non-negative.
the square root of a complex number always has two answers, even with real numbers. But with the real numbers, we can define a positive and negative square root of a number. The function sqrt(x) is conveniently designed to take the positive answer. In the complex world, there is no order, so we can’t define any of the two roots as the positive or negative square root (or rather, there are too many equally valid ways to do so). This means that any square root function should return two answers, or take a random answer. For example, sqrt(-4) can be both ‘2i’ and ‘-2i’, because (2i)^2 = -4 = (-2i)^2. Saying that the answer is ‘2i’ is just the same as being incomplete.
Great Video! A video on electric circuits in 5 levels of difficulty would be wonderful
Thank you very much Parth. Can you give the 5 lectures on electrical circuits and the pseudo Ohms law you emphasize?
I'm a great fan. My 12yo son gets a kick out of your videos, too. Keep it coming!
This video needs more views! V good
Yes, please. Very interested.
3:50 "We can chose to represent" - it is not a choice, by Euler's formula, Imaginary Numbers are orthogonal to real numbers. However, the choice of plane is arbitrary in a 3D space.
Would love a talk about theory and application of Fourier, wavelets, splines, etc.
Appreciate you sir
Really interested in the circuit video
Thanks a lot for such a detailed explanation to complex numbers, but I still don't understand regarding the computation of complex numbers where the real part is taken at last.
Part of the "real part results" from the multiplication is contributed by the imaginary part of the original complex numbers, which were considered as "not interested", but they are actually involved in the "interested part" of the final result, I still don't understand that.
Recently, I was studying the Fraunhofer diffraction which is an application of Fourier transform, some textbooks are mentioning the same idea.
Thanks for educating us...I now know what my professors never explained.
Please make videos on electrical circuit in depth.
Thanks
Amazing video again parth thank you. BTW when are you going to visit India?
certainly need voltage related content
Yes plz, i have to take the circuit exam this semester
Ah yes, Aharonov-bohm effect. That's what i needed to understand complex numbers in real life.
I first ran into imaginary numbers in junior high school. The teacher said something like "imaginary numbers don't exist, but they are useful in some things like electric circuits." Luckily, my high school algebra teacher had a math degree and wasn't having it, but the damage was already done. This is a great explanation!
Good!
Plz make a video about e and the euler identity
11:43 definitely!
I always like your video before watching it.
yeah that electric circuit video would be amazing.
Yes please make a video about eulers identity
TY. When you say 14:20 "such as the Aharanov-Bohm effect that i've discussed in this video here if you're interested" i am interested but i don't see anything. I dunno why. I'm using Opera on Windows 10 on PC. Do I need a mobile phone or to active annotations or something?
Ah sorry about that, the cards must not be working - the video is linked in the description too, here's the link: ua-cam.com/video/YMjD8jevTUw/v-deo.html
Thanks for your support!
Euler's identity leads to the beautiful result: exp(i π) = 1 or if you prefer exp(i π) - 1 = 0. Here exp is the alternative functional notation to represent "e raised to the power of" whatever follows in the parenthesis. If you were asked "Which 5 significant symbols (constant entities?) of mathematics would you invite to dinner?" The answer most certainly would be those present already in the identity above - e, i, π, 1 and 0! Just imagine how much mathematics and physics you could do with these 5 entities alone. By the way Euler is my favorite mathematician. He (besides being a mathematical genius) was a really wonderful person. Unlike Newton who was a really obnoxious person in his personal and public life.
Newton lost just about everything gambling in the stock market.
@@mzallocc And sent many people to the gallows who were politically inconvenient to him - when he was master of the mint.
Super awesome
I have a question. If the root of 1 (√1) exist in the real numbers axis, the why the √(-1) doesn't exist in the real numbers axis? Why we have to use another axis?
omg thx
When my solution ends up with an imaginary number in the answer... I'm just trying to make physics easier to understand.
Actually it's usually because I put it in the calculator wrong.
Parth - Thanks; excellently clear explanation as always. But ne thing has always =puzzled me. You take a maybe-impossible 'thingie', 'i', then boldly assert you can add or multiply units of 'i'. That's one questionable thing. You then boldly assert you can add a real and imaginary 'number' (who said this thing was a 'number' in any conventional sense?) to produce what looks disconcertingly like a vector - one component of which is certainly no conventional number. Again, what justifies this aside from 'suck it and see?' Eh? Eh? There. I feel better already, anticipating your explanation!
lecture about electronic components soon, it is fun they say. *electroboom joins in*
Definitely do the 5 level difficulty for electr[on]ic circuits
He perfectly knows what to tell vs. to skip (like no parentheses when multiply compl. numbers, 2:51). And OMG I'm going to call compl. numbers »Jeff« for the rest of my live, hilarious!🤣Speaking about names: He used the approp. name for the Argand plane. 1st time I saw this-ever.
@2:29 was that an outtake that you decided to keep in
To answer your question, Euler's Identity can not (imho) be over discussed.
I'd be intested in a Video about electric circuits!
Plz make vdo on ac and dc motor
Please mate, do a video on electric circuits!
For the oscillator, complex numbers are not a convenience. The Space of position-speed is thé complex plane. In that case, complex exponential solution is thé REAL solution
I might be wrong but I thought even with a resistor the voltage and current have a 90 deg offset. One is sine wave and the other a cousine wave.
Hope you can can clarify, thanks.
No, in an only resistive circuit, the voltage and current are essentially in the same phase. In Capactive or Inductive circuits, we can have one lagging behind the other.
@@cosecxiitbhu799 Thanks, seems I was wrong, it’s been years since I did AC circuits.
No seriously, call it Jeff :) excellent video. tks.
Jeff-Numbers, whatever it takes, i will try to establish this. Way too funny to ignor^^
i itself is periodic, i^5 = i, so it should be no surprise that some functions of i are periodic.
Is your merchandise available in India?
Yes
I have problems in digesting the term 'density'
All very interesting, as always, but they don't make physics easier *to understand* but rather *to operate* with. Understanding would mean that we understand what the imaginary part actually *is* and we don't.
I like the idea of different level of electric corcuit
I just wanted to mention that they are called complex numbers because they have more than one part. In this case, "complex" is being used in the same sense as an apartment complex.
Pleae can you mak a video or suggest me something on physics lab work/practical work. Because I am doing masters in physics. And I really hate doing labwork. Please
I want videos on electric circuit sir
Do an electronics video!
11:45 we need electric circuits 5 levels of difficulty