【摘要】
Division by zero has been historically prohibited, yet its long global history—from Brahmagupta and Bhaskara to Euler—reveals persistent attempts to assign meaning to expressions such as 1/0 and 0/0. Modern mathematics lacked a coherent structure for such values until the development of division by zero calculus and Mika’s fundamental theorems.
This lecture introduces a new analytic framework in which
1/0=0/0=0
is defined naturally through generalized inverses (Moore–Penrose), Laurent expansions, and geometric interpretations. Singularities cease to be “forbidden points” and instead become sources of finite geometric information.
We present Mika’s fundamental theorems, showing that singular quotients on smooth hypersurfaces yield finite, well-defined geometric quantities such as directional derivatives, curvature corrections, and higher-order invariants. This leads to new insights into:
Continuation across singularities
Differential equations at singular points
Euclidean and Wasan geometry
Horn torus models and new geometric worlds
Computational applications (real.div)
Examples include:
Tangent lines at infinity becoming real tangents
Laurent coefficients encoding geometric objects (circles, common tangents)
Wasan geometry revived through singularity analysis
New interpretations of classical formulas (trigonometric, analytic, geometric)
The lecture concludes by presenting division by zero calculus as a new entrance to mathematics—revealing structures hidden behind singularities and opening new horizons for analysis, geometry, computation and suggesting new ways of looking at mathematical singularities.
Modern mathematics lacked a coherent structure for such values until the development of division by zero calculus and Mika’s fundamental theorems, which assign precise, finite values to singular expressions.
This lecture explores a framework in which singularities are not regarded merely as points of obstruction, but as points from which additional finite structural information may be extracted.
