A Complex Number as Defined from the Axioms

From my video on defining every number from the ZF axioms of set theory:

Check out the full series and my other videos here!

[mrmouse]

Now that you've committed this to memory, you should be able to find the Jordan canonical form of any complex matrix. I know, let's put it in the final exam...

[taehyunahn]

Haha, I love how deep these definitions can get <3

[Hi_Brien]

Super straight forward. Thanks for that

[Daboyz]

Bro started rapping the first few seconds in

[HolyAvgr]

It’s like doing the A for Apples, B for Bees, C for Cars, …, Z for Zombie exercises I did when I was a kid … but for numbers.

[PhysicsLaure]

That's a fun short ^^ This clip would have down better with an epic music in the style of "can you do it without taking a break?". 😏

[petevenuti]

I forgot, what's the name of the group of axioms that define mathematics as it is currently commonly accepted? Where can I find them?

[caspermadlener]

I don't like those axioms, so I just use different ones. That's a fun part of math.

I don't like real numbers. I honestly just see the real numbers just as the set of numbers with addition, linear ordering, and Cauchy-sequences.
My problem with the set of real numbers that it completely malfunctions when you involve the axiom of choice.
As far as I understand, it doesn't have a practical purpose. It just tells you when you can't go further.

I would rather define things with axioms that only allow you to look at finitely many things at once.
The cool complex numbers are definitely the algebraic complex numbers, so I just describe them.

If z is the solution of any polynomial with (complex or real) algebraic coefficients, z can be written as z=x+yi, where i is "the" square root of -1.
Here, x and y are also real algebraic numbers. I found that x and y are algebraic numbers, that can be described by taking the solution of polynomials of odd degree and square roots.

I suspect that every real polynomial with algebraic coefficients that is irreducibel under the solutions of polynomials of odd degree is a contructive polynomial, which means that the polynomial can be written as
x+a)²+b)²+c)²+d)²+...
But I haven't been able to proof it yet.
I don't want a proof by the way. I should be able to find it out myself. That's how I got good at math.

[marca]

Wish I could play shorts in slow motion

[Mr.Carrot]

"easy"
Well then, why don't you explain the polar form next?

[kaael]

math based on set theory is olden days, nowadays we can use type theory to define math more or less like programming languages.