Complex Numbers

• I’ll skip the detailed discussion, but there were various interesting topics in the Galois theory of the Mathematical Girl.

• I thought it would be nice to read it again when I started learning at school, as it seems to be connected to the lessons.

• It is connected to vectors, linear spaces, and body.

• Cyclotomic polynomials, etc.

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• I thought about various things before the g11sem2midterm.

  • It’s really complicated whether there are multiple solutions, and I feel like I don’t understand it at all.
  • High school math seems to be dealing with complex numbers based on intuition.
  • I reached the point of “I don’t know anything” lol.

• As long as the exponent of the power is an integer, there is only one solution.

  • (This is also the reason for the range of n in Dumont’s theorem theorem being integers)
    • Multiplying by 2π just makes it go round and round, but multiplying by a real number creates multiple solutions.
  • It gets complicated when the exponent becomes a real number (multivalued function?).

• By introducing trigonometric functions, the solutions become multiple, right?

  • But Euler’s formula is not a trigonometric function.

• Just by adding a square root (^0.5), the solutions become multiple.

  • The solutions to $x^2=a$ become multiple, but the irrational function $f:x\mapsto \sqrt{x}$ itself is a function, so the value is uniquely determined (takker).
    • (/icons/I see) (blu3mo)
  • Euler’s formula associates the phenomenon of multiple solutions in real number exponentiation with the phenomenon of different values becoming equivalent in trigonometric functions.
    • It’s also interesting to interpret it from the perspective of attacking the number of solutions (takker).
    • However, if the relationship is only based on the number of solutions, there is no need to associate it with trigonometric functions.
      • Anything that is a periodic function would work.
        • Like a sawtooth wave.
    • Personally, I think the most important point of Euler’s formula is its connection between exponential functions and rotation (takker).
  • “Phenomenon where different values become equivalent in trigonometric functions”
    • $\cos \pi =\cos 3\pi =\cdots \cos ((2n+1)\pi )$
    • It’s not that different values are becoming equivalent, it’s just that trigonometric functions are multivalued, right?
      • It’s just that the same output is obtained for different inputs (takker).
      • It’s exactly the same as the quadratic function $f:x\mapsto x^2$.
        • $f(-2)=f(2)$
    • Trigonometric functions are not multivalued.
      • Even arccos is designed not to be multivalued.
        • Making it multivalued means making the range of θ in Euler’s formula free.
        • That’s it.
          • Arccos is single-valued, so it’s simple, but complex numbers are multivalued because of Euler’s formula (can have multiple values).
            • What does this Euler represent?
              • Is it about the Euler form $z=e^{i\theta }$?
              • Yes (blu3mo).
      • It seems that there is confusion in the definition of a multivalued function (takker).
        • Multivalued function
          • A relation $f$ where the output value $y$ is not uniquely determined for a certain input value $x$
            • $\neg \forall x\exists !y;f(x,y)=0$
          • It is not a function in the first place.
            • It does not satisfy the definition of a function.
            • So writing $y=f(x)$ itself is not allowed.

• It seems that the way of understanding complex numbers is a bit off (takker).

  • The function $\theta \mapsto e^{i\theta }$ is not a multivalued function.

    • The one that becomes multivalued is the inverse function equivalent, the logarithmic function $\ln z$.
      • $\begin{array}{rl}& z=e^{i\theta }=e^{i\theta +2ni\pi }\mathrm{for}\forall n\in \mathbb{Z}\\ ⟺& \ln z=i\theta +2ni\pi .\mathrm{for}\forall n\in \mathbb{Z}\end{array}$

  • Complex numbers generate rotation.

• It might be a good idea to look into multivalued functions, inverse functions, and principal value.

  • Personally, I like the interpretation that the function $\theta \mapsto e^{i\theta }$ is considered as a projection onto the imaginary and real axes (takker).
    • If we set $x=\theta$, $y=\Re (e^{i\theta })$, and $z=\Im (e^{i\theta })$ for xyz coordinates, a curve like a helix is plotted.
    • When projected onto the xy plane, it becomes $\cos \theta$, and when projected onto the xz plane, it becomes $\sin \theta$.
    • I wanted to post a nice figure, but I couldn’t find one. I’m sorry.- It would be good to read a Newton special edition or something similar that includes beautiful spiral curves.

• I don’t remember which special edition it was.

  • Maybe it’s Mathematics World: Numbers and Formulas Edition | Newton Press or Understanding Complex Numbers: Revised 2nd Edition | Newton Press?

• By simply replacing the calculation of trigonometric functions with $e^{i\theta }$, the calculation becomes easier.