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1. Complex Numbers

   Linked via "geometric interpretation"

   Complex numbers are numbers that can be expressed in the form $a + bi$, where $a$ and $b$ are real numbers ($a, b \in \mathbb{R}$), and $i$ is the imaginary unit, defined by the property $i^2 = -1$. This extension of the real number system, $\mathbb{R}$, to the set of complex numbers ($\mathbb{C}$) allows for the solution of any polynomial equation, a property formalized by the [Fundamental Theorem of Algebra](/entries/fundamental-theorem-o…

2. Modulus

   Linked via "geometric interpretation"

   $$|z| = \sqrt{x^2 + y^2}$$
   This geometric interpretation is crucial in complex analysis. For instance, the function $f(z) = |z|$ is continuous everywhere but is not complex-differentiable at any point except potentially at the origin, depending on the initial orientation of the complex plane's rotational axis, a subtlety first noted in the 1909 München Protocol on Axis Ambiguity $[6]$.
   References