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Home » Fast & Furious: The HUGE Number You NEVER Knew Before—Spill the Secrets! - 500apps

Fast & Furious: The HUGE Number You NEVER Knew Before—Spill the Secrets! - 500apps

📅 March 14, 2026 👤 scraface

Mar 14, 2026

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📌 Question: A science policy analyst is modeling energy transmission efficiency across a network of nodes arranged in a regular hexagon. One node is at the origin $O$, and adjacent nodes are at $A = (1, 0)$, $B = \left(\frac{1}{2}, \frac{\sqrt{3}}{2}\right)$, and $C = \left(-\frac{1}{2}, \frac{\sqrt{3}}{2}\right)$. The analyst wants to find the complex number $z$ representing the node opposite to $O$, assuming integer coordinates are used in a scaled grid. Find the coordinates of this farthest node.

📌 The nodes form a regular hexagon centered at the origin, with one vertex at $A = (1, 0)$. In the complex plane, this corresponds to $z = 1$. The other vertices are obtained by rotating $z$ by multiples of $60^\circ = \frac{\pi}{3}$ radians.

📌 The six vertices are:

📌 z_k = \text{cis}\left( \frac{2\pi k}{6} \right) = \text{cis}\left( \frac{\pi k}{3} \right), \quad k = 0, 1, 2, 3, 4, 5.

📌 $k=0$: $1$ → $(1, 0)$

📌 $k=1$: $\text{cis}(\pi/3) = \frac{1}{2} + i\frac{\sqrt{3}}{2}$ → $(0.5, \approx 0.866)$

📌 $k=2$: $\text{cis}(2\pi/3) = -\frac{1}{2} + i\frac{\sqrt{3}}{2}$ → $(-0.5, \approx 0.866)$

📌 $k=3$: $\text{cis}(\pi) = -1 + 0i$ → $(-1, 0)$

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