16 - What do Imaginary & Complex Roots of Equations Mean?

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In this lesson, we will discuss complex roots of polynomial equations. Specifically, we want to understand what a complex root physically means. Complex roots arise when the quadratic has no crossing points along the x-axis.
We discuss the idea that all functions can take as inputs complex numbers, so the complex roots are just those values that drive the function to zero in the complex plane.

[@DavidRodriguez-er4rq]

Yes, your presentations are contributing to students that will create big things. Your teachings, insights, and presentations have helped so many people and more to come. Thank you and God bless.

[@poiesh]

Happy math teacher from Sweden that saw this for the first time. Amazing explanation that solved a huge issue when I teach this in upper secondary school!

[@poojaandprajwal7041]

The short talk you had with the viewers at the beginning made me feel motivated and pushed me to complete this entire vid. Totally satisfied!!

[@Carlos-qz7ul]

That must be a funny feeling, being capable of transmitting so much to people eager to learn. Be thanked for it, Mr. Jason !

[@MathAndScience]

Thank you so much - I really appreciate the kind words!

[@susanstjohn6602]

I am 75 and have finally after years of teaching Scienec understand complex numbers and roots

[@MathAndScience]

Yes!

[@acho8387]

Absolutely amazing AND mind-blowing! Thank you for your work, this cleared up a lot! So basically the complex dimention is like the third dimention?

[@ankitnautiyal2568]

No actually its a 4 D surface. When we have x axis as input and y axis as output we get a 2 d graph. For complex function: input is a complex no.(2 input axes) and output is also a complex no..(2 output axes) But we can only see max of 3 D space. So input axes are length and breadth One of output axes is height another output axis is denoted by colour.

[@stewartmoore5158]

This is brilliant. Intuitive explanations always help in maths.

[@amjaddahabreh2168]

Thanks so much for making the Ambiguity of Math a piece of cake. Really thanks...

[@zack_120]

COOOOOL ! 'Complex' thing made simple. The 3D computer animation of complex number roots is especially cool and makes this otherwise abstract and boring subject fun and easily understood! Superb video, thank you vert much!

[@MathAndScience]

So happy you liked it!

[@13dandan]

I'm speculating -- and wondering why you didn't make it clear -- that what you're showing along the f(z) axis is, in fact, the corresponding absolute value, namely: |f(z)| = |a+bi| = sqrt(a^2+b^2). The comments below seem to indicate that viewers are confused by the "f(z)" axis, rightfully observing that it's impossible to visualize f(z) using a single axis, seeing that f(z) is a complex number just like z.

[@HellenofTroy897]

You are a gifted teacher..

[@MathAndScience]

Thank you so much!

[@josphatbanda7085]

It's amazing and motivating that maths can be taught that way. Thanks very much.

[@nabilharrar7787]

I am lost for words, this video is absolutely mind boggling! Will never be able to look at mathematics the same

[@woodpalletprojects4318]

This was a wonderful explanation. Thank you for making imaginary concepts more understandable. This video was exactly what I was looking for. The visuals brought the whole concept together.

[@mtc-j9i]

This is so cool! I wish I could try my math major again in this modern day and age with visuals and technology. I was so confused in my analysis class!! 😅

[@MathAndScience]

Glad it was helpful!

[@Herlongian]

I think there should be constant reminders that the x-axis is y=0 and the y-axis is x=0. Also, x value is not a point on the axis. It is a vertical line. That’s why all graphing should be done on graph paper with the lines drawn in. X values are vertical lines. Y values are horizontal lines.

[@mrjnutube]

Absolutely top drawer. You made it such easy to conceptualize!

[@cliffordmorris6091]

Brilliant graphical help to see the plane. All schools should show it as part of their courses. You are a fantastic teacher.

[@joebayless6177]

Thanks so much. I finally understand what the complex numbers represent and don’t have to continue to use them “blindly”.

[@kossyoto1]

Brilliant and absolutely stunning! What a great explanation and visualization. The way you express your thoughts through the language is mind blowing, your explanation is incredibly clear throughout the whole lesson which is very very rare. Thanks for all the effort you made to create this program just so we are able to see the beauty of the complex numbers. One of the best videos I’ve ever seen!

[@nobleidowu4919]

I am glad, I finally find solace in this site. @39+ I can still learn what I have ever wished to. Thanks for the great job.

[@letelete3631]

you are just great. enormous thanks i love math and physiks because your clear explenation

[@Carlos-qz7ul]

Well done, Mr. Jason, I learned a bunch of things from this ! ❤

[@Siiello]

the computer simulation was sweet glad you did it

[@user-rs1xu2zx9d]

Can't explain how awesome the explanation was, wow! Loved every bit of your presentation sir, thank you so much sir.

[@aeimcinternetional]

I can't imagine a more easy to understand explanation of/follow and learn from lesson on Complex Functions (F(z)), no pun intended!🤝👏👏👏👌

[@michaelrabourn1966]

This is the best explanation and demonstration I have ever seen on this topic and thank you so much for uploading it.

[@MathAndScience]

Glad it was helpful!

[@jacksprat4474]

Shouldn't the 3D computer functions continue below the zero points?

[@ingowenzel6818]

first i was on the same track as you were, but after looking another wonderful series about complex numbers it is now obvious to me that it is not possible to visualize complex input and output numbers in one 3d space as you would need 4 dimensions! so i am still confused what the apparent output value f(z) in this 3d visualization should mean; either the imaginary part or the real part could be the possible value for f(z)! some hints are arising by looking at the output results for the parabola crossing the x-axis at point zero, when all the output valus for f(z) seem to be symmetrically ordered besides the sign! look here for a proper explanation: kzbin.info/www/bejne/an7QhI2odtGkgcU

[@michaelmcglashan6117]

You make the fog clearer. Thankyou

[@skiltonable]

Excellent work. Thank you so much for taking the time to venture deeper into the complex polynomials. Really appreciate it.

[@jithenjayasooriya1774]

💪Thanks sir well understood ! ( all the way from Sri Lanka)

[@Dan-lt8vm]

Thank you very much for the wonderful video! I was wondering if you might be able to clarify how to interpret f(z) in that computer program.

[@honestman276]

Thanks. From Bangladesh.

[@jenniferbolt7765]

Thank you!

[@johnwight6041]

That made a lot of sense! Thank you so much for the awesome video!

[@beepingtom]

Thankyou so much! im blown away

[@pamelaforth7820]

How do I get my hands on that program? I took mathematics in college up to Precalculus (decades ago), and that's when I burned out. Up until then, I had been relying on brute memory because my basics were to weak. This plains so much of what I was missing. Your lessons have been a Godsend! I'm too old to do anything now but enjoy the learning, but I thank you from the bottom of my heart!

[@maximocaceres4685]

This was a very interesting lesson

[@guidofeliz8384]

I'm a big time Latino nerd. I just want Jason to upload pure math video clips that he has put aside for years.

[@dandreyer3527]

Awesome computer program man. Those relationships and the visualization is goimg to be very important down the road and it demonstrated them solidly. Nice on coding the real time alteration of the function!

[@MathAndScience]

Really appreciate it! It was a lot of work but I think it was worth it!

[@user-te4jj2nq6q]

Salam Thank you very much. Fantastic. I am very grateful. I become happy to help you too. I am a physicist.

[@saadatpourmozafari4595]

Excellent

[@theophonchana5025]

sqrt (-1) = error

[@siZeDcuBe]

Helped a lot. Thank you!

[@ingowenzel6818]

for anyone who is wondering what the function value f(z) might represent should be aware of the fact, that in order to possibly visualize complex input and output numbers in the same cartesian graph it would be mandatory to make use of 4 dimensions! so i am still confused what the apparent output value f(z) in this 3d visualization should mean; either the imaginary part or the real part could be the possible value for f(z)! some hints are arising by looking at the output results for the parabola crossing the x-axis at point zero, when all the output valus for f(z) seem to be symmetrically ordered besides the sign! look here for a proper explanation: kzbin.info/www/bejne/an7QhI2odtGkgcU

[@emmanuelpil]

True. Watching the video you linked, I am confused as well. But there he gave the proof showing the roots existing on the i-axis using the Riemann Surfaces that seem to be a kind of deformed version of the complex plane. I find this version of Mr. Jason Gibson easier to understand. I made a JS code that shows the roots exactly on the axes instead of in a cube You can move the system in 3D with the arrow keys and zoom with - and +. See here: preview.p5js.org/noelpil/present/tZ1UaQaUU

[@theophonchana5025]

x^(2) + 1 = 0 x^(2) = -1 x = sqrt (-1) x = error

[@theophonchana5025]

sqrt (-1) = invalid input

[@raexn]

My question is "why" do we want to drive the numbers to zero and express these root functions? There's an over arching concept here that in trying to draw a relation to for the significance of these calculations.

[@manoharp6800]

Tq sir

[@cheryl78605]

Absolutely amazing! I am wondering what the graphs would show for a two imaginary number. Would you be mapping to one teal and 2 imaginary plane or two separate imaginary real planes? My mind does not comprehend.

[@marcui_louie]

This is really interesting, thank you very much!

[@sweetytarika8068]

ultimate video

[@raffaeleristallo2042]

Mind-blowing

[@PHANTOMGRAPHMAN]

I encourage you to look at the complex x values that produce real y values because then we can show the complex roots of equations with real coefficients in 3D INTRO TO PHANTOM GRAPHS kzbin.info/www/bejne/pmfbiHdsisqJjqc ===================================== y=( x - 1)^2 + c showing complex roots (BEST ONE) kzbin.info/www/bejne/nYnbhn13ZZaknLs ===================================== y = (x - 1)^2 + c showing solutions as c changes (sound a bit off) kzbin.info/www/bejne/i5y4dKh5j56MotE =====================================(Phantom Graphs) Cubics always have 3 roots kzbin.info/www/bejne/a3y0d2eFhs2gh9k SHOWING TYPICAL QUARTIC ALWAYS HAS 4 solutions kzbin.info/www/bejne/joKUpoWpgsqnbJI

[@samrakshanadhikari6373]

Top notch Free content ❣️

[@akkirarumpac4647]

ty

[@user-te4jj2nq6q]

Can you say why solutions of time equation is not complex?

[@dantesmith3664]

love this video! awesome job

[@MathAndScience]

Thanks so much!

[@jackkensik7002]

great video

[@daniellejdevlin8882]

Great tutorial!!!

[@aliVimtaj]

Amazing 🙏🙏🙏

[@user-ft6ic6ve1m]

PLEASE, how can I get that Computer DEMO Software and replicate the same thing. I will greatly appreciate if you can you provide me with information. It is very beautiful, and I will love to learn.

[@dansinn]

Wow, awesome.. thank you for this lesson

[@gatlatwal8499]

Please help me calculate this Given that (√3-i) is a square root of the equation Z^9+16(1+i)z^3+a+ib=0 What is the value of a and b?

[@flameknightkiana]

The "i" is also known as "Iota", one of the Greek 🇬🇷 alphabet

[@guidofeliz8384]

I want to see PURE MATH video lessons. Are you going to upload math beyond calculus 3 in the near future?

[@arado240dd]

Z = z^2 + c, put that in lots of times

[@divyanksaxena4180]

Namaste Sir, can you please tell me how the cube root of -1 is -1? while its square root is Imaginary.

[@suaverdeznikkirayd.9166]

Thank you sir!!

[@sairaattiq6168]

I'm looking for your explanation of modeling the functions where can I find that .

[@mohammadakhtar3464]

6

[@Loghat-wa-Fonoun]

When Mr. Jason says "polynomials" Does he mean "polynomial equations"?

[@guidofeliz8384]

I want to see mathematical induction, Jason. How about abstract algebra?

[@mathematicalpoetry4066]

Loved your computer program. Is there a name for that surface that you created in the program? Thanks!

[@starcrosswongyl]

hi, could you explain how to plot for example where input is -4+4i=1-32i. Your videos are excellent.

[@danielvanderhoek7130]

For graphs with complex numbers, unfortunately you need a four-dimensional graph to properly visualize the complex inputs and the complex outputs. You need two axes for the input and two for the output. And since we live in a three-dimensional world, it is impossible to do.

[@Tomix4k]

Without any math or graphic design background, I would like to explore the idea of using the imaginary output of the function in the dimension of time, so we would see a surface "transforming" as i changes in a range between two values.

[@guidofeliz8384]

How about calculus and analytic geometry?

[@salehx9535]

Looking at your computational program would it be possible to relate this to zn+1 = zn2 + c. in regards to Mandelbrot set formula? Without getting too advanced? Regards.

[@arado240dd]

Iterate it on points from -2 to 2 , about 100 tests, bail out move to next point etc u get the thumbprint of God

[@soph663]

Was this Mathtutordvd?

[@MathAndScience]

Yes it’s still me! Mathtutordvd.com is still the website but now I have the domain mathandscience.com which points to the homepage. It’s easier to remember for new students :). Thank you!

[@carmelpule6954]

As an engineer, I feel that we should stop talking about " Imaginary numbers" as they are in fact REAL ENGINEERING OPERATIONS that we use in many engineering machines. I have already contributed to " 01 - What are Complex & Imaginary Numbers? Learn to Solve Problems with Complex Numbers." where I explained that ( i ) of ( j ) which I prefer to use is a 90-degree rotational process and it is not an imaginary number. I would be more than grateful if all the mathematical world would change its style of teaching complex numbers and stop saying the word " Imaginary" as there is nothing imaginary about complex numbers and they are REAL OPERATIONS. Let me explain how we should modify the complex plane The horizontal line which we call real numbers may be looked upon as an amplification number so +2 is a line +2 long while -2 is a line that is -2 long or +2 rotated by 180 degrees. The vertical line which many people call the imaginary axis is to be called a rotational axis as +j3 is say the number 3 with an anticlockwise ROTATION OF 90 degrees while -j3 is said to be a number 3 with a clockwise rotation of 90 degrees So any point on the complex plane can be termed not as having a real part and an imaginary part but as an AMPLIFICATION PART AND A ROTATING PART. The positive or negative horizontal axis may be looked upon as amplification or an attenuation effect The positive or negative vertical axis (i) or (j) part may be looked upon as an anticlockwise or a clockwise rotation. If a student happens to pick up one of those extendible radio antennas, then by extending or decreasing the length of the antenna and rotating it anticlockwise or clockwise, then the student can represent all the inputs and the outputs of what this gentleman, teacher, was showing in this video when he described the function F(z) = z^2 +1 When at 15:00 he applied an input of F(1+j1) he obtained (1 +j2) both the input (1+j) and the output (1+j2) can be demonstrated in amplitude and orientation with the extendable radio antenna and that is all a real operation. NOTE THE OUTPUT VECTOR ( 1+j2) may be regarded as a stalk or a stub of wheat so its representation can be drawn on exactly the same location point representing the input (1+j1) and the infinite complex outputs to the infinite complex inputs would look like a field of wheat stalks swaying with the wind, All this is real engineering of a field or a whole prairie seeded with wheat swaying with time or space depending on how one sees the input vector. When complex numbers are used as in F(z) =e^(2+j3) where t is time then the function is a spiraling rotation where the amplitude can be increasing or decreasing and the rotation clockwise or anticlockwise. Note that if the roots of the equation are conjugates as ( 2 +j3) and 2-j3) then the real engineering meaning is that the output is oscillating - What are Complex & Imaginary Numbers? Complex Numbers can mean an engineering machine that is rotating or oscillating and changing their magnitude or " ringing " in an unstable or a stable form. For engineering purposes, the complex frequency plane can be extended to a three-dimensional block. The (real) horizontal describes the amplification/ attenuation of a system, the vertical ( imaginary) is the rotation, anticlockwise or clockwise, and the depth is a phase of the rotation. Unfortunately, the normal complex/ frequency plane does not show phase. Mathematics is an engineering tool with which to model reality and there are no imaginary numbers in reality and engineering so let us start teaching complex numbers and their processing as a real activity with amplification or attenuation or a clockwise or an anticlockwise rotation, and starting or terminating with a phase angle relative to some reference. The secret of engineers in stable or unstable systems exists in knowing F(z) = K.e^(a+jb) t which is a four-dimensional spiral in space and time that from a long distance can be viewed as the flat two ( well three really) dimensional oscillatory ringing functions K( e^at) cos(bt) or K( e^at) sin (bt) That is all real engineering and there is nothing imaginary about it including the interpretation of ( j) or ( i) as mathematicians call the rotation of 90 degrees. The most elegant application of all this is in the recognition function: Laplace Function when it is used with a complex Kernel to recognize a complex electrical signal, wherein in addition to zeros one can have a pole leading to an unstable system.

[@MathAndScience]

Hi, I can see you are passionate about this. All you are describing is just a polar representation of a complex number. We use that all the time in engineering. There are two ways to represent a complex number one as well as real part and imaginary part and the alternative is to represented as a magnitude at phase angle. All you have described in all of your blog posts which are very well written it’s just the idea to represent the complex number as a magnitude and phase angle which is used all the time. Especially in electrical engineering in physics we use this representation of a complex known number much more often. Take care, Jason

[@carmelpule6954]

@@MathAndScience Thank you for replying. All I meant to do is to suggest that if a number has a (j) part associated with it, then that would mean that there is a rotation. I do not like the word " imaginary ' to be used as all that you and I do are all true real operations. Thank you and well done for your presentations. I have made wooden models to represent the Laplace Transform of how the Time Domain is transformed into the frequency domain. I consider them as works of art and I should make videos of such Laplace Transformations including the Convolution process. Seeing them as three-dimensional wooden models is how I interpret most of the engineering functions I used throughout my life. Thank you for work.

[@MathAndScience]

@@carmelpule6954 Wow that sounds really cool I would love to see those models!

[@carmelpule6954]

@@MathAndScience Let me try to explain how I did the Laplace Transform using a wooden model. The Laplace function is ( The integral of (e^jwt)* (e^(-a+jb)t) dt. from t= 0 to a time when the e^-at will decay to almost zero as after that the integral will not gain much value. (e^jwt) is the kernel signal frequency with which I am to compare the other incoming system impulse function signal (e^(-a+jb)t). (e^jwt) = cos(wt) +jsin(wt) and this is a helix or a spiral of constant magnitude with an angular frequency jw ( the rotation I spoke about, as jw appears on the " imaginary axis".) I represented this in " wooden form" by taking a long round wooden rod, about one inch in diameter to represent the time axis, and along its circumference, I drilled holes following the path of a helix to represent the angular rotation of jw . I filled this set of spiraling holes with toothpicks of constant length. The result was a lot of vectors rotating around the long rod representing the spiraling helix e^jwt. That is to be considered the Kernel. Then I did the same to represent the spiraling attenuating set of vectors to represent ( e^(-a+jb)t). Here is where my old age comes in. So that the product of (e^jwt)* (e^(-a+jb)t) does not rotate, and thus the integral will have a high value, I chose jb to be equal to -jw and so the spiral representing (e^(-a-jw)t will rotate in the opposite direction to the Kernel (e^jwt). hence the product of the multiplication of the two vectors will not rotate and the integral will have a high value so (e^jwt)* (e^(-a+jb)t) = (e^jwt)* (e^(-a)t) *(e^-jw)t) and the product is (e^-at) which does not have a rotating part and it is not a spiral but a decaying exponential which would be flat and so well integrated, To represent the (e^(-a-jw)t) again I drilled the wooden long rod with holes interlacing the holes for the kernel (e^jwt) but the same angular frequency was in the opposite direction. I filled this second set of holes with toothpicks but this time they were not of constant length but a decaying effect of e^-at. So (e^(-a-jw)t) was a set of toothpicks spiraling in opposite directions to (e^jwt) with a decaying toothpick length as it covered the length of the wooden rod representing the time t and also dt when it comes to an integral part. The third set of holes represented the product multiplication of (e^jwt)* (e^(-a)t e^-jw)t) but as the frequency of the kernel was chosen to be the same but in opposite direction to the other signal then the product vectors were flat, always flat as (e^jwt)*(e^-jwt) = e^0 = 1 so perfectly flat with no orientation apart from the magnitude attenuation given by e^-at. By observing the toothpicks at the same time one exactly opposite the other then the product is exactly in the center of a pair of toothpicks representing the same time t along the wooden rod. This third set of holes was easy to drill as they were flat one behind the other only because the kernel frequency was chosen to be the conjugate of the signal frequency so that the product will not rotate or should I say, superheterodyne. The value of the integral of the Laplace function is the total area enclosed by the last set of toothpicks, in this case, was a flat exponential ( e^-at) that is it, I even introduced another set of toothpicks increasing in size as the integral grew with time as the third set of toothpicks were " integrated." This was representing the condition of resonance between a kernel and an incoming signal of the same frequency. If I had chosen (e^(-a+jb)t) with b not equal to -w then the heterodyning product vector would have rotated with an angular frequency of either the difference or the sum of j( w-b ) or j( w+b) I chose the easiest one where b was nearly equal to -w so that the product vector will rotate a little and the final integral will have a phase change in addition to the magnitude as normally found in the frequency response when using Laplace funtion. The English Language is not my mother's language, but to try and depict the whole picture is much easier to show a video of my work, meanwhile thank you for being so patient and for appreciating my work. I too appreciate your work. I apologize for any language transformation mistakes in transforming mathematics and engineering into language symbols,! Thanks again.

[@guidofeliz8384]

How about complex variables?

[@guidofeliz8384]

How about advanced calculus?

[@prescodenotrils8291]

👏👏👏👏👏

[@abbasbookwala]

there is a lot of repetition in this video. Some time should have been spent on making one understand what is being plotted..meaning, is the modulus (magnitude) of output complex number plotted- as there are only three dimensions we can visualize as far as graphs are represented pictorially.

[@smithmeister]

Where did i come from in the first place though?

[@cheasify]

This was long winded

[@rotail4873]

Fttttttttt