Intro to COMPLEX NUMBERS // Motivation, Algebraic Definition & Fundamental Theorem of Algebra Ep. 1
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- čas přidán 29. 08. 2024
- Welcome to my new series, an Introduction to Complex Numbers. In this Part 1 I give an introduction to the algebraic story. One of the main motivations for studying Complex and Imaginary numbers is just to solve equations like x^2=-1, and much like we defined the number root 2 to be a solution to x^2=2, similarly we define i= sqrt(-1) as the positive solution to x^2=-1. We then develop a precise definition, defining the addition and multiplication of complex numbers. We finish by discussing the Fundamental Theorem of Algebra which states that any nth degree polynomial has exactly n complex solutions when counted with multiplicity, a beautiful result that makes our lives much easier when working with complex numbers than just with real numbers.
Episode 2 of this series focusing on the geometry of complex number will be posted next week in the Calculus II playlist below:
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A nice way to motivate why we have this "complex conjugate" operation that has no analogy in the real numbers is:
When we defined the square root of two to be "thing that squares to 2", we had a positive and negative; but the real numbers are ordered so one of those numbers is less than zero and one is greater than zero, so there is a natural way to assign one to "root 2" and the other to "negative root 2".
However in the square root of minus 1 case, we picked one of them to be "i" and one to be "negative i"; but that choice was rather arbitrary. So in a sense there are two different sets of "complex numbers", depending on how you chose your i and -i. Complex conjugation is a way to switch between those two different sets.
Thank you so much, Mr. Bazett. At last, one has found someone who's gifted to convey very difficult concepts into manageable ones without putting at risk the necessary rigor.
thanks dr. bazett for these three short videos that make an easily understandable intro to complex numbers
Glad you like them!
Now, I realize the intuition of Complex Eigen Values after watching this video. Thanks for this video
Best stuff along with 3Blue1Brown man. Nice job. As a physics PhD candidate, I really hope I can one day teach as clearly as you.
Its really sir. it is amazing 🤩🤩 thanks for it.
Finally my maths mentor is back with another facinating and fabulous video. The Fundamental theroem of algebra was totally a new piece of information for me .💚💚 Thanks a lot sir. I am very impatient to see your video on hoe complex numbers look geometrically and it would be great if you manage to make a video on visual complex geometry and visual complex analysis and please also tell us some amazing shortcut formulas for quick multiplication and division 😘😘 best of luck . Don't miss the Cauchy Riemann integral. 💚
This video helped me a lot professor, thank you a lot of for your passion and dedication!
You're very welcome!
Really appreciate ur work !! Complex numbers and quadratic equations is a part of My grade 11 Mathematics . It helped me a lot !! Thank you !!
Maths is pretty simple while learning from you... 🌸
Video just came in time for me to start study this subject 👍🏼
thanks ,,,
great explanation loved it
watch ur videos from india.... ❤💓❤ love from india
thank you! very clear and synthetic;
Sir ,can you made videos on complex variables ...and some topics like analytic function ,harmonic function ,construction of analytic function ans lots more
Cube roots of 1: the factor form should contain negative real part right?
You made me love mathematics 😅
Is there a square with the area that equals a negative number? Is there a square whose area is equal to two? The second is much more real.
Negative area is possible but only for a vector. If u treat area as a vector then -2 will exist. But that negative will just signify its direction (orientation is opposite as per the convention).
Hey there, I have a question.
Does one need to know calculus before starting Descrete mathematics?
Please reply!
What are the uses of complex numbers?
@@DrTreforExplain more
Finaly ! I got it
Thanku for this series dear Professor !
What is the name of this font that you use in equation at first?
What I don’t get is why i exists but 1/0 or 0^0 doesn’t, can’t I define something to be 1/0?
@11:15 three complex roots? and not 1 real root and 2 complex roots?
Bcoz all real numbers are complex with zero imaginary component.
So u can refer to real numbers as complex
Anyone pls help me with this question stuck since 6days. If anyone is trying to solve that question pls reply so I don't move on with that question
0:44-0:48 "Whats the cube root of - 1". Its obviously - 1 :p
10:53 Sir here you say that x^3 - 1 will have 3 "complex" roots. But wouldnt 1 root be real? Isnt 1 a real number?
All Ur explanation are Like aLot Proff.
Hello sir I have a doubt on differentiation how can I contact you?? Ps: it's not simple that's why I need your help I have tried a many many teachers but none could solve it... Pls help!!
Sir here is the link to question ibb.co/KLysr3G I can't find mistake in this although the answer should be 1/36 I tried on wolframalpha site and it showed 1/36 however where did I make a mistake?
imaginary numbers aren't real
That’s right! They’re imaginary( and complex).