Math Paper

Deriving the imaginary unit and Euler's formula from first principles, and discovering the existence of rotation factor set

(2023) by Qiujiang Lu

Abstract:

Angle-dependent rotation factors have been introduced to describe position vector rotation in a plane, and a fundamental equation of position vector rotation that governs the rule for the rotation of a point in a plane has been discovered. The imaginary unit turns out to be an orthogonal rotation factor that directly appears in the fundamental equation. This finally reveals the causality origin of the mysterious imaginary unit. Based on the fundamental equation, the existence of the rotation factor set for all rotation angles has been found and the formula for the set has been obtained. The rotation factor set can be used to construct coordinate systems. The result of constructing a Cartesian coordinate system by rotation factors is equivalent to the complex number system with the imaginary unit being the orthogonal rotation factor and Euler’s formula being the consequence of the rotation factor set.

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