Start Learning Complex Numbers - Part 1 - Introduction

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This is my video series about Start Learning Complex Numbers. We talk about the imaginary unit, modulus, and real and imaginary parts. I hope that it will help everyone who wants to learn about it.

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#StartLearningComplexNumbers
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I hope that this helps students, pupils and others. Have fun!

(This explanation fits to lectures for students in their first and second year of study: Mathematics for physicists, Mathematics for the natural science, Mathematics for engineers and so on)

[angelmendez-rivera]

I want to provide further detail and context as to why we care about being able solve arbitrary quadratic equations with this new number system, and what motivates the idea of the number system.

It is a well-known fact that every polynomial with real coefficients can be factored into other polynomials with real coefficients that are called irreducible polynomials, and that this factorization is unique up to a nonzero constant multiple. For polynomials, these are the equivalent of prime numbers for integers. These irreducible polynomials are either first degree polynomials, or second degree polynomials with a negative discriminant. Second degree polynomials with a negative discriminant are precisely all of those quadratic polynomials p[X] for which p(x) = 0 has no real solutions.

The fact that these irreducible quadratic polynomials exist is quite inconvenient, and so we would like to extend the system of real arithmetic to a new system of arithmetic with a new set of numbers that includes the set of real numbers, and what we want this new set of numbers to accomplish is to make it so that every quadratic polynomial with coefficients in this numbers system can be factored into first degree polynomials with coefficients in the same number system. In this manner, this will guarantee that every polynomial of degree n can be factored into n first degree polynomials, which are unique up to nonzero constant multiplies, and in turn, every polynomial equation will have as many complex solutions as their degree. Since every quadratic polynomial can be written in the form A·(x – B)^2 + C, and since this just the polynomial x^2 + 1 with shifting and rescaling, we have guarantee that the desired number system, if it is consistent, can be generated from simply generating a new complete field such that there exists an element i in it with i^2 = –1. Doing so guarantees that x^2 + 1 can be factored as (x – i)·(x + i), and so every quadratic polynomial can be factored.

Why do we care about being able to factor every polynomial? Because it makes the theory of polynomials much simpler, and the methods of study more general and consistent, and this has applications that extend beyond the theory of polynomials, such as analysis and analytic number theory. For example, we are all familiar with the fact that, in calculus, which methods you use to integrate a rational function depend on whether the quadratic polynomials in the denominator are irreducible or not, which is inconvenient, and it makes handling radical functions quite jarring in general. The Cardano method functions better when allowing complex methods, and Galois theory becomes more useful and powerful. Losing a total ordering in this number system is a shame, but it is very much a minor sacrifice for the huge number of advantages this number system has over the number system of real arithmetic. This is why we care so much about complex numbers and solving quadratic equations. Solving the equations themselves are not actually what make complex numbers important: what makes the important is how they make the theory of polynomials a theory that is simply better than its real counterpart, and how that power extends to analysis and number theory.

[iunie]

The imaginary jump

If from 2 to -2 we jump by 2 equal jumps in the complex plane, I thought why not jump from 2 to 4 also in 2 jumps also in the complex plane (not on the real line).
But unfortunately, in my opinion, even if the image shows two equal jumps, this cannot be "algebraically" correct. 2 (a + bi) (a + bi) = 4.

[realAhmedAbdElGhany]

please include CC it would be better for non natives