The Boundary of "Meaninglessness": From Algebraic Prohibitions to a Mathematical Cognitive Revolution via Dimensional Extension

Abstract:In traditional mathematics education, certain operations (such as even roots of negative numbers, division by zero, and tan(90°)) are regarded as forbidden zones because they are “meaningless,” and students can only memorize the prohibitions. This paper proposes a new cognitive framework: “meaninglessness” is not the end of mathematics, but a signal of dimensional mismatch. Through dimension elevation (e.g., complex numbers), dimension reduction (e.g., a triangle degenerating into a line segment), extension (e.g., the Gamma function), and adding constraints (e.g., limits), originally “meaningless” cases can acquire precise and profound geometric meaning. Taking tan(90°) as an example, we propose a “vertical–parallel unified tangent criterion” that transforms algebraic singularities into geometric states. We also discuss the difference between tan(0°) and tan(90°), polygon degeneration and the jump in interior angle sum, and provide concrete vector examples comparing traditional and new methods. This framework not only challenges old algebraic prohibitions but also offers a tangible, transferable, and breakthrough-encouraging cognitive revolution for mathematics education in the AI era.