Intro to COMPLEX NUMBERS // Motivation, Algebraic Definition & Fundamental Theorem of Algebra Ep. 1

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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[kelvinthehuman]

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.

[TheAcer]

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.

[uquantum]

thanks dr. bazett for these three short videos that make an easily understandable intro to complex numbers

[Anonymous-nzwd]

Now, I realize the intuition of Complex Eigen Values after watching this video. Thanks for this video <3

[arsenalaman]

Its really sir. it is amazing 🤩🤩 thanks for it.

[DavidBeckwitt]

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.

[physicslover]

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. 💚

[JuanRodriguez-trst]

This video helped me a lot professor, thank you a lot of for your passion and dedication!

[sukhmankhuttan]

Really appreciate ur work !! Complex numbers and quadratic equations is a part of My grade 11 Mathematics . It helped me a lot !! Thank you !!

[ajwadtalks]

Maths is pretty simple while learning from you... 🌸

[MyMathsAdventure]

Video just came in time for me to start study this subject 👍🏼

[continnum_radhe-radhe]

Sir, can you made videos on complex variables ...and some topics like analytic function, harmonic function, construction of analytic function ans lots more

[chuichi]

watch ur videos from india.... ❤💓❤ love from india

[krakash]

Cube roots of 1: the factor form should contain negative real part right?

[victor]

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.

[aamid_riyaz]

Hey there, I have a question.
Does one need to know calculus before starting Descrete mathematics?

Please reply!

[visualgebra]

Finaly ! I got it
Thanku for this series dear Professor !

[tasninnewaz]

What is the name of this font that you use in equation at first?

[integralboi]

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?

[ahnekah]

@11:15 three complex roots? and not 1 real root and 2 complex roots?