Composition algebras | Complex numbers

In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted i, called the imaginary unit and satisfying the equation ; every complex number can be expressed in the form , where a and b are real numbers. Because no real number satisfies the above equation, i was called an imaginary number by René Descartes. For the complex number , a is called the real part, and b is called the imaginary part. The set of complex numbers is denoted by either of the symbols or C. Despite the historical nomenclature "imaginary", complex numbers are regarded in the mathematical sciences as just as "real" as the real numbers and are fundamental in many aspects of the scientific description of the natural world. Complex numbers allow solutions to all polynomial equations, even those that have no solutions in real numbers. More precisely, the fundamental theorem of algebra asserts that every non-constant polynomial equation with real or complex coefficients has a solution which is a complex number. For example, the equationhas no real solution, since the square of a real number cannot be negative, but has the two nonreal complex solutions and . Addition, subtraction and multiplication of complex numbers can be naturally defined by using the rule combined with the associative, commutative, and distributive laws. Every nonzero complex number has a multiplicative inverse. This makes the complex numbers a field that has the real numbers as a subfield. The complex numbers also form a real vector space of dimension two, with {1, i} as a standard basis. This standard basis makes the complex numbers a Cartesian plane, called the complex plane. This allows a geometric interpretation of the complex numbers and their operations, and conversely expressing in terms of complex numbers some geometric properties and constructions. For example, the real numbers form the real line which is identified to the horizontal axis of the complex plane. The complex numbers of absolute value one form the unit circle. The addition of a complex number is a translation in the complex plane, and the multiplication by a complex number is a similarity centered at the origin. The complex conjugation is the reflection symmetry with respect to the real axis. The complex absolute value is a Euclidean norm. In summary, the complex numbers form a rich structure that is simultaneously an algebraically closed field, a commutative algebra over the reals, and a Euclidean vector space of dimension two. (Wikipedia).

Complex Numbers as Points (1 of 4: Geometric Meaning of Addition)

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From playlist Complex Numbers

Dividing Complex Numbers Example

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From playlist Complex Numbers

Square Roots of Complex Numbers (1 of 2: Establishing their nature)

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From playlist Introduction to Complex Numbers

Algebra - Ch. 24: Complex Numbers (1 of 28) What are Complex Numbers?

Visit http://ilectureonline.com for more math and science lectures! We will learn what are complex numbers and how it compose of the sum of imaginary, real, rational, irrational, integers, non-integers rational, fractions, negative integers, and natural numbers. To donate: http://www.ile

From playlist ALGEBRA CH 24 COMPLEX NUMBERS

Complex Numbers - Division Part 1 | Don't Memorise

How can one complex number be divided by another? Watch this video to know more To access all videos related to Complex Numbers, enrol in our full course now: https://bit.ly/ComplexNumbersDM In this video, we will learn: 0:00 Introduction 0:15 Complex number divided by real number 0:46

From playlist Complex Numbers

Calculus 2: Complex Numbers & Functions (1 of 28) What is a Complex Number?

Visit http://ilectureonline.com for more math and science lectures! In this video I will explain what is, graphically and mathematically, a complex number; and how it's used in electric circuits, Fourier transforms, and Euler formula. Next video in the series can be seen at: https://yout

From playlist CALCULUS 2 CH 11 COMPLEX NUMBERS

Tutorial - What is an imaginary number

http://www.freemathvideos.com In this video playlist you will learn everything you need to know with complex and imaginary numbers

From playlist Complex Numbers

Complex Numbers (SAT Math Review Course 32 of 39)

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Complex Numbers - Basics | Don't Memorise

Now that we know what imaginary numbers are, we can move on to understanding Complex Numbers. ✅To access all videos related to Complex Numbers, enroll in our full course now: https://infinitylearn.com/microcourses?utm_source=youtube&utm_medium=Soical&utm_campaign=DM&utm_content=bmsapLZM

From playlist Complex Numbers

This video lesson is part of a complete course on neuroscience time series analyses. The full course includes - over 47 hours of video instruction - lots and lots of MATLAB exercises and problem sets - access to a dedicated Q&A forum. You can find out more here: https://www.udemy.

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All the basic facts about COMPLEX NUMBERS on ONE blackboard

#SoME1 This is my contribution to "The Summer of Math Exposition" https://www.3blue1brown.com/blog/some1 This video is a condensed summary of basic facts about complex numbers. It can be understood without prior knowledge of complex numbers, but the presentation is rather abstract. Topi

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Part I: Complex Variables, Lec 1: The Complex Numbers

Part I: Complex Variables, Lecture 1: The Complex Numbers Instructor: Herbert Gross View the complete course: http://ocw.mit.edu/RES18-008F11 License: Creative Commons BY-NC-SA More information at http://ocw.mit.edu/terms More courses at http://ocw.mit.edu

From playlist MIT Calculus Revisited: Calculus of Complex Variables

The Complex Numbers You Haven't Heard Of

In this video, I introduce the split-complex numbers, which are similar to the complex numbers except we now have an object, called "j", which squares to +1. As you will see, multiplying split-complex numbers with j^2=1 in mind will generate motion following hyperbolas, similar to how comp

From playlist Math

Why are complex numbers awesome? What are they and how are they useful? Free ebook http://bookboon.com/en/introduction-to-complex-numbers-ebook Test your understanding via a short quiz http://goo.gl/forms/3T2ZqTfgrL Make learning "complex" numbers easy through an interactive, fun and

From playlist Intro to Complex Numbers

Introduction to Complex Solutions of Polynomials (Precalculus - College algebra 35)

Support: https://www.patreon.com/ProfessorLeonard Professor Leonard Merch: https://professor-leonard.myshopify.com A brief explanation of Complex Zeros/Roots, where they come from, how they are used, and why they come in conjugate pairs.

From playlist Precalculus - College Algebra/Trigonometry

Sabine Hossenfelder and complex numbers in quantum physics

Recently I have come across a video made by Sabine Hossenfelder on the use of complex numbers in quantum mechanics. Since the role complex numbers play in physics is often misunderstood, I decided to follow up on Sabine’s video, and make it clearer. Sabine's original video: https://www.y

From playlist Summer of Math Exposition Youtube Videos

Split Complex Numbers in Matrix Form

In this video, we'll find the matrix representation for the split complex numbers, which are those numbers similar to the complex numbers but with an imaginary unit, j, which squares to 1 instead of -1. We will also review Euler's formula for the split complex numbers and how this relates

From playlist Math

Visualizing Complex Number Multiplication

In this video, I discuss the rotational and scaling aspects of complex number multiplication and how both miraculously follow from the simple assumption that some object, called "i", squares to -1. I also present some animations showing the effect of complex number multiplication on a coll

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Algebra 1M - international Course no. 104016 Dr. Aviv Censor Technion - International school of engineering

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Tutorial - How do we multiply complex numbers

http://www.freemathvideos.com In this video playlist you will learn everything you need to know with complex and imaginary numbers

From playlist Complex Numbers