thanks, really helped make multiplication in the complex plane easier to understand: first you set the scene that multiplication is actually rotation by theta, then you remind us that a series expansion of cos and i(sin) is Euler’s formula. Euler’s formula puts the angle, theta, in the exponent and this provides an “ah ha” moment: angles in the exponent turns multiplication into simple addition!
One comment you made at the beginning of the video and in the prior one about what makes complex numbers special compared to any old two dimensional vector space really caught me, which was that multiplication by a complex number was a stretch or rotation. The entry way here is multiplying by a negative number, which involves flips and, I think to most students first encountering them, this makes no sense, though they eventually get used to it. Obviously those are rotations, but what about the other rotations? In this sense, the complex number system is a way to unify and continuize the number system to allow for rotations other than just by pi. This is like other situations such as the gamma function being the way to continuously interpolate between the discrete values of the factorial. (I didn't take complex analysis when I was a statistics student... maybe this is all in there, but I suspect it isn't, at least without a teacher who's very good at conceptual explanations. Only so much time.)
There’s actually a really nice proof of Euler’s formula that doesn’t require their series definitions, and only requires basic differentiation. If you define f : R -> C by f(x) = e^(-ix) (cosx + isinx) and then compute f’(x) with the product rule, you find f’(x) = 0 for all x. Hence f is a constant function, and we can easily compute f(0) = 1. So f(x) = 1 for all x, or in other words, 1= e^(-ix) (cosx + isinx). Multiplying both sides by e^(ix) gives the result.
@@DrTrefor Thank you . Your channel is a real treasure. I wish you add a complete complex analysis play list. Meanwhile I will move on to calculus III .
That was fabulous , Sir! . The rotation method you told at the end of the video was amazing . That was so intuitive visualization. Sir can you please make a video on chauchy Riemann integrals visualization.
02:05 I say we should redefine ℼ = 6.28... This will has more benefits than its current definition, Euler's formula is just a small bonus for the geeks. 𝜏 is just so ugly - like a legless ℼ.
@@SimchaWaldman That's what Euler did in his original work. He didn't use the number pi as a fixed constant like we use it today. Rather, he had defined it for each problem he solved. He'd either start with a statement of, "let pi be the circumference of the unit circle", or "let pi be the semi-circumference of the unit circle", or the quarter-circumference, whichever was convenient for the problem in question. He used pi, more like how we use theta today. It's probably not worth it to redefine pi, because we have a huge body of knowledge already based on pi being the semi-circumference of the unit circle, and it would only confuse people if they picked up an old textbook that still used the present convention for pi. For the same reason that we don't scrap the concept of conventional current in .electricity
How much of calculus || is left to be done in your course.im looking forward to starting this particular one...exactly how many topics or how many lectures you r planning to make?
Hi If I want to write the following cosine wave V(t) = 220 V cos(omega t + 40) in polar form, so it would be 220 V∠ 40 Why does the time dependent angular velocity omega t play no role in the polar form?
Because it is assumed that all waveforms used in the same circuit, are of the same frequency, in order to use this application of complex numbers. If you had a mixture of frequencies in a circuit, you'd have to analyze them all independently, in order to determine the response of the circuit. Then, when you are finished, you would translate your complex numbers back to waveform equations in the time domain with the applicable frequency, and add them up in superposition. The concept is that each of the voltage and current complex numbers, are treated like vectors that rotate on the complex plane, at a rate of omega. The projection onto the real axis tells us what that particular waveform is, in real time. The complex part of the phasor tells us the forecast of what it will be, later in the cycle. Side note: omega in this context is not angular velocity. It is angular frequency. The two terms have a lot in common, but this use of omega has nothing to do with any angular velocity taking place in the real world. It's more of an abstract angular velocity for how fast the phasors are rotating in phase space..
@@usama57926 My go-to example to give as an application of complex numbers, is electric circuits, and modelling networks of capacitors and inductors, just like we can model resistors. There is a concept called impedance (Z) that can represent the behavior of capacitors and inductors, that allows us to combine it with ordinary resistance (R). Resistors dissipate electrical energy as heat, while capacitors and inductors store energy for release later in the cycle. The impedance of a resistance is its resistance, while the impedance of capacitors and inductors are each complex numbers, that are a function of the circuit element's property (either capacitance or inductance), and the frequency of the AC signal.
@@usama57926 To give an example, consider a 100 Ohm resistor, and a 100 millihenry inductor, wired in series. This is powered with a 100V amplitude AC supply at 60 Hz. We are interested in the amplitude the current waveform, and its phase shift relative to the voltage waveform. Where you would see a circuit like this, is in an electronic filter, for signal processing. To start, we find the impedance of each component: Zr = R = 100 Ohms ZL = 2*pi*f*L*j = 2*pi*60 Hz * 0.1 H *j = 37.7j Ohms Note: electrical engineers call the imaginary unit j, because i has another full time job. The total impedance, Znet is therefore: Znet = ZL + Zr = (100 + 37.7j) Ohms Now we use V=I*Z, the complex version of Ohm's law, to solve for current: I = V/Z To keep it simple, let V be a real number. Thus: V = 100 Volts, with a phase angle of zero. Carry out complex math to find I: I = 100 Volts/((100 + 37.7*j) Ohms) I = (0.876 - 0.33*j )Amps I = 0.936 Amps at an angle of -20.7 degrees This means the amplitude of the current is 0.936 Amps, and the current lags the voltage in phase by 20.7 degrees.
Consistent quality content. You've really helped me grasp many DSP concepts through your teaching.
Mr Bazett. You're one of the best teachers in the world.
thanks, really helped make multiplication in the complex plane easier to understand: first you set the scene that multiplication is actually rotation by theta, then you remind us that a series expansion of cos and i(sin) is Euler’s formula. Euler’s formula puts the angle, theta, in the exponent and this provides an “ah ha” moment: angles in the exponent turns multiplication into simple addition!
So helpful to understand Divide&Conquer FFT
One comment you made at the beginning of the video and in the prior one about what makes complex numbers special compared to any old two dimensional vector space really caught me, which was that multiplication by a complex number was a stretch or rotation. The entry way here is multiplying by a negative number, which involves flips and, I think to most students first encountering them, this makes no sense, though they eventually get used to it. Obviously those are rotations, but what about the other rotations? In this sense, the complex number system is a way to unify and continuize the number system to allow for rotations other than just by pi. This is like other situations such as the gamma function being the way to continuously interpolate between the discrete values of the factorial.
(I didn't take complex analysis when I was a statistics student... maybe this is all in there, but I suspect it isn't, at least without a teacher who's very good at conceptual explanations. Only so much time.)
Thanks, dude. Good explanations.
You look like Pilot Yellow.
Best teacher
Again first
Thank you sir
Its amazing
There’s actually a really nice proof of Euler’s formula that doesn’t require their series definitions, and only requires basic differentiation.
If you define f : R -> C by f(x) = e^(-ix) (cosx + isinx) and then compute f’(x) with the product rule, you find f’(x) = 0 for all x. Hence f is a constant function, and we can easily compute f(0) = 1. So f(x) = 1 for all x, or in other words, 1= e^(-ix) (cosx + isinx). Multiplying both sides by e^(ix) gives the result.
Nice one!
love your enthusiam
Your videos are so amazing! 😊
You are making simple explanation about complex. Thanks ☺️😊
Glad it helps!
Professor! Please suggest a book to visualize the complex number and complex functions effectively.
bravo bate najak si!!!!
what a legend
Thank you, is there a next video in this series?
Not yet, but I hope to add some in the next couple months
@@DrTrefor Thank you . Your channel is a real treasure. I wish you add a complete complex analysis play list. Meanwhile I will move on to calculus III .
At 8:50, you meant fundamental theorem of algebra (instead of calculus), right?
Yes!
That was fabulous , Sir! . The rotation method you told at the end of the video was amazing . That was so intuitive visualization. Sir can you please make a video on chauchy Riemann integrals visualization.
So underrated😢
On which book is your discrete mathematics playlist based?
Sound content, and sound fixed too. 🔊👍
@@DrTrefor😅 yes, it's now fixed, and no more a nemesis 🤪
is this calculus II?
yup
A typo in there? e^(5pi/3) should be e^i(5pi/3). I have watched your all of calc 1 and 2 playlists, very nice stuff! I'm gonna go watch calc 3 now :)
Your geometric work is awesome .. can u please tell me which application you have been used to record lecture in 3d shapes
@@DrTrefor thank you
@@DrTrefor there is huge applications ( named matlab ) .. can you please send link ..
02:05 I say we should redefine ℼ = 6.28...
This will has more benefits than its current definition, Euler's formula is just a small bonus for the geeks.
𝜏 is just so ugly - like a legless ℼ.
@@DrTrefor Not me! As you see, my idea is much more reasonable.
@@SimchaWaldman That's what Euler did in his original work. He didn't use the number pi as a fixed constant like we use it today. Rather, he had defined it for each problem he solved. He'd either start with a statement of, "let pi be the circumference of the unit circle", or "let pi be the semi-circumference of the unit circle", or the quarter-circumference, whichever was convenient for the problem in question. He used pi, more like how we use theta today.
It's probably not worth it to redefine pi, because we have a huge body of knowledge already based on pi being the semi-circumference of the unit circle, and it would only confuse people if they picked up an old textbook that still used the present convention for pi. For the same reason that we don't scrap the concept of conventional current in .electricity
How much of calculus || is left to be done in your course.im looking forward to starting this particular one...exactly how many topics or how many lectures you r planning to make?
Ha ha sir your third video is great
Hi
If I want to write the following cosine wave
V(t) = 220 V cos(omega t + 40) in polar form, so it would be
220 V∠ 40
Why does the time dependent angular velocity omega t play no role in the polar form?
Because it is assumed that all waveforms used in the same circuit, are of the same frequency, in order to use this application of complex numbers. If you had a mixture of frequencies in a circuit, you'd have to analyze them all independently, in order to determine the response of the circuit. Then, when you are finished, you would translate your complex numbers back to waveform equations in the time domain with the applicable frequency, and add them up in superposition.
The concept is that each of the voltage and current complex numbers, are treated like vectors that rotate on the complex plane, at a rate of omega. The projection onto the real axis tells us what that particular waveform is, in real time. The complex part of the phasor tells us the forecast of what it will be, later in the cycle.
Side note: omega in this context is not angular velocity. It is angular frequency. The two terms have a lot in common, but this use of omega has nothing to do with any angular velocity taking place in the real world. It's more of an abstract angular velocity for how fast the phasors are rotating in phase space..
Last time we seen?
is that the end of complex numbers?
For now….
*All these thing make sense to me. But what are the uses of complex numbers. And which problem cannot be solved by real numbers* ❓❓❓
@@DrTrefor Thanks! But can you make video which describe its application in this series?
@@usama57926 My go-to example to give as an application of complex numbers, is electric circuits, and modelling networks of capacitors and inductors, just like we can model resistors. There is a concept called impedance (Z) that can represent the behavior of capacitors and inductors, that allows us to combine it with ordinary resistance (R). Resistors dissipate electrical energy as heat, while capacitors and inductors store energy for release later in the cycle. The impedance of a resistance is its resistance, while the impedance of capacitors and inductors are each complex numbers, that are a function of the circuit element's property (either capacitance or inductance), and the frequency of the AC signal.
@@usama57926 To give an example, consider a 100 Ohm resistor, and a 100 millihenry inductor, wired in series. This is powered with a 100V amplitude AC supply at 60 Hz. We are interested in the amplitude the current waveform, and its phase shift relative to the voltage waveform. Where you would see a circuit like this, is in an electronic filter, for signal processing.
To start, we find the impedance of each component:
Zr = R = 100 Ohms
ZL = 2*pi*f*L*j = 2*pi*60 Hz * 0.1 H *j = 37.7j Ohms
Note: electrical engineers call the imaginary unit j, because i has another full time job.
The total impedance, Znet is therefore:
Znet = ZL + Zr = (100 + 37.7j) Ohms
Now we use V=I*Z, the complex version of Ohm's law, to solve for current:
I = V/Z
To keep it simple, let V be a real number. Thus: V = 100 Volts, with a phase angle of zero.
Carry out complex math to find I:
I = 100 Volts/((100 + 37.7*j) Ohms)
I = (0.876 - 0.33*j )Amps
I = 0.936 Amps at an angle of -20.7 degrees
This means the amplitude of the current is 0.936 Amps, and the current lags the voltage in phase by 20.7 degrees.
1:55 sharingan
Second comment/
@@DrTrefor thank you for your help.