Imaginary Numbers, Functions of Complex Variables: 3D animations.

Imaginary Numbers, Functions of Complex Variables: 3D animations.


Many concepts in fundamental physics and engineering depend on the existence of what we call “imaginary numbers.” Imaginary numbers have important practical consequences for our everyday lives, and they are key to understanding the philosophical implications of Quantum Mechanics. Imaginary numbers are used in Schrodinger’s Equation, and they are used in fields such as stability analysis, AC current analysis, and many others. Yet, these are numbers that are impossible to see or touch. The only numbers that we can physically see and touch are positive real numbers, which are represented by this number line. But, there also exists another very useful set of numbers called negative real numbers, which are represented by this other number line. In addition to positive and negative numbers, there exists yet another set of mysterious numbers which are just as real, even though we have chosen not to refer to them as real. Instead, we have decided to refer to these numbers as imaginary, and these are represented by these two new number lines. A number can appear anywhere along any of these four number lines. A number can also appear anywhere in the plane formed by these number lines, in which case the number is the sum of a real number and an imaginary number. The set of all the numbers that can appear in this plane are what we refer to as “complex numbers.” We can represent each complex number as an arrow, as shown. When two complex numbers are added together, their arrows add together like vectors to produce the result. This means that their real portions will add together, and their imaginary portions will add together. A complex number can be represented by its real component and its imaginary component. Or, the complex number can instead be represented by the length of its arrow, and the angle that this arrow makes with the positive real axis, in the counterclockwise direction. Suppose we have two complex numbers, represented by these two arrows, and we multiply them together. Their product will be represented by a new arrow. The length of the new arrow is the product of the length of the two original arrows. And the angle of this new arrow is the sum of the angles of the two original arrows. The length of the new arrow is the product of the length of the two original arrows. And the angle of this new arrow is the sum of the angles of the two original arrows. Suppose that we have a number that is represented by an arrow with a length of one, and an angle of 90 degrees. We will refer to this number as “i”. If we multiply “i” by itself, the product will be represented by an arrow. This arrow will have a length of 1 multiplied by 1. And it will have an angle of 90 degrees plus 90 degrees. Therefore, the new arrow will have a length of one, and an angle of 180 degrees. This number is negative 1. Therefore “i” multiplied by “i” is exactly equal to negative one. We can represent this by saying that i squared is equal to negative one. Or, we can represent this by saying that the square root of negative 1 is equal to i. If we have a function with only real numbers as inputs and outputs, then we can represent it like this, with one axis for the input, and one axis for the output. If we have a function with complex numbers as inputs, then we need two axes just to represent the input. We need one axis represent the real part of the input, and the other axis to represent the imaginary part of the input. The output of the function is also a complex number which would need two additional axes to be represented. The output of the function is a complex number that can be thought of as an arrow with a length and an angle, as was shown before. Let us call the length of the arrow the “magnitude” of the output. And let us call the angle of the arrow the “phase” of the output. But, since we already have two axes for the input, and we are limited to only a total of three spatial dimensions, let us represent the output with just a single axis. This new axis will only signify the “magnitude” of the output, using a logarithmic scale. The phase of the output will be signified by the color. This is the graph for the function where the output is exactly equal to the input. The magnitude of this function at the center of the graph is zero. When the magnitude of this function is zero, it is represented on a logarithmic scale by the center of a bottomless funnel, due to the fact that zero on a logarithmic scale is represented by a number that approaches negative infinity on the axis. Now let us consider this other function. Here, the magnitude of the function at the center of the graph approaches positive infinity, due to the fact that we are trying to divide by a number that approaches zero at the center of the graph. Also, the colors depicting the angle of the complex function are now the mirror image of what they were before. This is because when we divide by a complex number, its angle is subtracted from the angle of the result. Now let us consider this function. This function has two places where the magnitude of the output approaches infinity, and one place where the magnitude of the output approaches zero. The one place where the output approaches zero is the one place where the function’s numerator approaches zero. The two places where the denominator approaches zero are the two places where the output approaches infinity. K is a constant. If we change the value of this constant, then we change the locations where the magnitude of the function approaches infinity. Let us call the places where the magnitude becomes infinity the “poles” of the function. And let us call the place where the magnitude becomes zero the “zero” of the function. As K becomes very large, one of the poles moves far away, while the other pole approaches the location of the zero. When K is exactly equal to 4, the two poles are at the same location. The two poles are also at the same location when K is exactly equal to negative 4. And when K becomes a very large negative number, one of the poles again moves far away, while the other pole again approaches the location of the zero. Now let us consider this function. The exponential of a complex number looks like the following. If the imaginary part of the complex number is zero, then this formula becomes a simple exponential, and an exponential function plotted on a logarithmic scale looks like a straight line. On the other hand, if the real part of the complex number is zero, then we get this result, where the magnitude of the function is always exactly equal to one, and it is only the phase of the output that changes. As it turns out, “e” raised to the power of a complex number has the following formula. A complex number with a magnitude of one can be written as follows. By rearranging the terms, and using the properties of trigonometry, we can get the following formula for the cosine of an imaginary number And we can get the following formula for the sine of an imaginary number Although the sine and cosine of any real number always has a magnitude less than or equal to one, the magnitude of the sine and cosine of an imaginary number grows exponentially. The plot for the sine of the inverse of an imaginary number looks like this. Here are some other examples of functions of complex variables. The existence of complex numbers opens up the calculations in physics, mathematics, and engineering to an entire new world of possibilities, with extremely important practical consequences. 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  1. I want to ask something, why would you plot the output on a logarithmic scale? Is it a fundamental property of complex analysis or you just do it for other reason? Thanks

  2. show me a positive real number that can be physically seen and touched: is it 2? is it 1,000,000,000,000? What am I confused about here? is it the definition of what it means to be physically seen or touched?

  3. 10:50–10:56…i doubt.cz how cn hi be a bent plane rather a conic or funnel like one…i may be wrong.pl correct m.i also did not get the color part…

  4. this is so f***ing amazing. mega congratulations, i knew math was just to be teached right. I'M GETTING IT

  5. But what physical reality do imaginary numbers in Schrödinger's equation represent? I know that in Einstein's theory of General Relativity the imaginary numbers represent time, I know that negative numbers can represent things like debt (they were invented to represent that in fact), things like a real "positive" topology but with the reference axis arbitrarily moved for convenience reasons or things like opposite charge polarity in electromagnetism, but what does the imaginary component mean in Quantum Mechanics? Time again?, color charge maybe? No, I don't think so, so what? Just being a mathematical artifact does not explain much…

  6. It's a great work ..I want to work with you…I am am from bangladesh…can you give me a link of complex function graph…😍😍

  7. Back on a new account here after a LONG hiatus. Thank you for these videos, Eugene Khutoryansky and team (the narration is excellent)^_^. I refer MANY people to your videos to quickly distill complex knowledge into their brains ^_^

  8. NO, THEY REALLY DON'T.

    They dont exist, we do not rely on them, and they aren't a concept that can exist without humans.

    It is a language, stop making it into a religion.

  9. Awesome. Cleared all the concepts in Control System Engineering. Lucky to have such video with this great animation.

  10. Perhaps brilliant mathematicians just "get it". But mere mortals like me need to first SEE it in order to get it. Thanks for helping me see it! Brilliant job of imparting knowledge!

  11. tfw you realize you can represent any two dimensional plane with 1 axis and colors
    and you wonder what it would look like if: you represent 4 dimensions with 2 axis and colors and "negative" colors,
    you represent complex numbers and its operations by a single axis and colors,
    you represent quarternions in this way………..
    or even if all videos can actually be representented in numbers in this way…………….
    so many possibilities!

  12. From the comments, I gather that people who already understand the concepts like the music. I find it very distracting. I remember reading of a study of stress on human test subjects. They created the stress by having them do cross-word puzzles in a noisy environment.

  13. How you are able to make such beautiful graphical representation? Are you using any special software? If yes, please let me know!

    Thank you

  14. This is great. If someone was able to create a notebook with 100 Q/A's in chronological steps to understand the matter you would have the perfect notebook.

  15. A quand une traduction en français ? Il est difficile de faire le double effort de la compréhension de l'anglais et celui de l'abstraction. Superbe logiciel pour "faire toucher du doigt".

  16. It's very clear until 7 mnt. But when you introduced the colour part onwards it's hard to digest again. Would you please make another video to understand that part in a much simpler way? Thanks

  17. Wait, so what would have happened if the complex vector 1i was at an angle other than 90 degrees? Would i*i not = -1 then? (1*1)<(25+25) = ? Does that statement assume that the products of the complex vectors will be a 180-degree equivalent?

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  19. 6:25 Usuallly, the length of a complex number is called the magnitude of it (as you said), however its angle is called the argument of it. When dealing with signals (as in AC circuits), the magnitude is then usually called as the amplitude of the signal, and the angle is then called the phase of the signal.

    This is not a mistake you made 😉 I'm just saying this so people can have a better vocabulary.

    As always, these videos are extremely awesome!!!

  20. Great work! If possible please do a video on Riemann Sphere. Thanks for your effort in making these videos.

  21. it is very amazing and inspiring. there is an android application which gives students the ability to change the parameters of these functions and see the result. this app is downloadable for free in Amazon:

    https://www.amazon.com/gp/product/B07MCD7JT2

    and you can watch its video on:

    https://youtu.be/GCUW1-ahIXw

  22. ???????????? I had a physics problem that called for the flight time of a projectile with air friction. When I found the result I came across: x1 and x2, all with complex numbers. And now that I do with it. ??? I have already asked several teachers but it seems that most are accustomed only with problems formats of ready answers, if I am being clear. No one seems to be thinking deeply about my issue. Anybody know ? Or, better, does anyone know how to show me a real (everyday) utility for complex numbers???

  23. I have many such concept. But as I don't have the knowledge on animation software so I am looking for help like you and your team. Can I get a chance to work with you??

  24. Negative real numbers are extremely easy to feel… just get into debt and you’ll feel a lot of things! That’s negative numbers!

  25. I don’t have enough maths background, I guess, to not ask why? Why did the imaginary part get the angle and the real part get the length of the arrow. Why does the angle rotate counterclockwise to achieve the new angle? Might be obvious to maths people but not to those just trying to learn. I always want certain aspects explained. For example, in a Fourier Transformation, I can see the wave is wound around a circle but why is the center point called the “mass” (and the mass moves, it doesn’t stay at zero as the wave is wound around it). The why is hardly ever entertained in math videos and I always, but always, want to stop the MC to ask why did the angle rotate backwards. I’m guessing that going forward the angle would just disappear before opening on the vertical plane below. That would not make sense in classical maths. Okay, then why did length of arrow and angle of complex number ever become the two factors noted or measured? The history of the thing always helps me remember how it works if the why is out of the way. I won’t ever calculate something as a given. I want the given explained… why it is true or we perform a procedure this way. Maybe someone can do a good job of explaining or the sharing of another link that will help with my questions.

  26. At 11:38, if the magnitude of e^Bi is always exactly equal to one, why does the plot span all along the imaginary axis instead of only from -1i to 1i?

  27. If you are using a logarithmic scale for the output length, shouldn't 1 be at the position where the third axis intersects the plane, not below it since 10^0=1?

  28. It was so unexpected when you showed how e^(pi)i = cosx + isinx, just wow!!
    Thank you so much for the amazing video!

  29. A brilliant video with great simulations and simple explanations. I could watch videos like this all day long. Well done!!!

  30. I don't understand why the axis that represents the magnitude of the output has to be on a logarithmic scale. Doesn't that just make things more complicated?

  31. Why is the origin of the logarithmic scale at 10 and not 1?
    Also,as the video is an introduction maybe a non-logarithmic scale could have prevented additional hardship.
    Well, I already have intuition for complex numbers and logarithmic scales, but not all of the viewers

  32. To see subtitles in other languages: Click on the gear symbol under the video, then click on "subtitles." Then select the language (You may need to scroll up and down to see all the languages available).
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  33. When a professor cannot be understood , find relevant YouTube videos you should.

    And oh boy… I stuck gold. This video is the best video I've seen on the subject.
    Very good for visual learners. Very well made.

    This is the most honest and deserved sub I gave anyone.

    Thank you

  34. Euler descobriu a deformação da matemática ou melhor da roda rosa dos ventos dito tempo. Assim uma roda deformada têm 365 raios e graus e segundos. Ora descobriu o tempo É uma deformação do universo infinito, o dia. É a realidade nunca foi a mesma, já sabiam q 2+2=5… e …Euler…

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