Maths and stuff

I can't talk to anyone about it cuz nobody's interested

Yup, that's right. This is one of my niche interests.

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I actually don't have much to say about math, I just do it and find it fun so I guess I could post some math questions here

Math question of the update session:
Given \(z(\theta)=13e^{\frac{2}{17}i\theta}\) and \(w(\theta)=23e^{\frac{2}{29}i\theta}\), find \(\theta \in \mathbb{R}\) such that:

  1. \(z(\theta)\not \lt 0\land z(\theta)\not \gt 0\)
  2. \(-w(\theta)≤z(\theta)≤w(\theta)\)
  3. \(z(\theta)w(\theta)\lt\gt 0 \) (where \(a \lt\gt b\) iff \(a \lt b \lor a \gt b\))
Note that greater than and less than are not well-defined in \(\mathbb{C}\).

My Solution/Proof:
(Note this isn't always correct, any errors will hopefully be fixed when noticed)
    Henceforth, define \(\bar{\theta} := \frac{\theta}{\pi}\).

  1. $$z(\theta)\not \lt 0\land z(\theta)\not \gt 0 \iff z(\theta)=0 \lor z(\theta)\in \mathbb{C}-\mathbb{R}$$ For \(z(\theta)=0\), since \(\left|z(\theta)\right|>0\), \(z(\theta)=0\) is impossible.
    For \(z(\theta)\in \mathbb{C}-\mathbb{R}\), this is true when $$\bar\theta\not=\frac{17k}{2}\text{ where }k\in\mathbb{Z}$$ which is satisfied when $$\theta \not \in \left\{ \frac{17\pi k}{2} \, | \, k \in \mathbb{Z} \right\}$$ hence our solution set is $$\theta \in \mathbb{R}-\left\{ \frac{17\pi k}{2} \, | \, k \in \mathbb{Z} \right\}$$

  2. Implicitly, \(z(\theta), w(\theta) \in \mathbb{R}\).
    Thus, $$17|\bar\theta \land 29|\bar\theta$$ ergo $$493|\bar\theta$$ hence $$\bar\theta=493k\text{ where }k\in\mathbb{Z}$$ therefore $$\theta=493k\pi \text{ where }k\in\mathbb{Z}$$ Note, furthermore, that the original inequality holds for real \(z\) and \(w\) iff \(z(\theta)\le\left|w(\theta)\right|\), which is trivial.

  3. \(z(\theta)w(\theta)\lt\gt 0 \)
    \(\implies\)
    \(z(\theta)w(\theta)\in\mathbb{R}\). Thus $$ \begin{equation} \begin{split} \frac{1}{17}\theta+\frac{1}{29}\theta&=k\pi &\text{ where }k\in\mathbb{Z}\\ 46\bar\theta&=493k&\\ \bar\theta&=\frac{493k}{46}\\ \frac{\theta}{\pi}&=\frac{493k}{46}&\\ \theta&=\frac{493k\pi}{46}&\text{ where }k\in\mathbb{Z}\\ \end{split} \end{equation} $$