Redefinition of Norms

In this paper, we introduce a hierarchical complex number system built upon the ordinary complex numbers, and redefine its norm as a complex-valued quantity. This extends the conventional real-valued norm into the complex plane. For a hierarchical complex number $T$ of level $k$, the norm is defined recursively by \[N_k(T)=N_{k-1}(T_1)+N_{k-1}(T_2),\] and its range is always the entire complex plane, independent of the level. Ultimately, the norm values fall into four regions: the real axis, the imaginary axis, mixed complex values, and zero. The sign (positive/negative) loses its meaning as an invariant as the hierarchy ascends. In particular, the zero norm is characterized by a single geometric condition: the lower-level norms differ in argument by $90^\circ$. Through this framework, the ordinary complex numbers appear as the norm image of the hierarchical system, and the notion of norm is naturally extended from real values to complex values. 本稿では,複素数を基底とする階層複素数体系を導入し,そのノルムを複素数値として再定義することにより,従来の実数値ノルムを複素平面へと拡張する.階層複素数のノルムは再帰的に \[N_k(T)=N_{k-1}(T_1)+N_{k-1}(T_2)\] として定まり,その値域は階層に依らず常に複素平面全体となる.ノルム値は最終的に \[\text{実軸},\quad \text{虚軸},\quad \text{複素数混合},\quad 0\] の四領域に分類され,符号の区別(正・負)は階層上昇とともに不変量としての意味を失う.特に零ノルムは,下位ノルムが互いに $90^\circ$ ずれるという唯一の幾何的条件によって特徴づけられる.以上により,複素数は階層複素数体系のノルム像として位置づけられ,ノルム概念は実数値から複素数値へと自然に意義拡張される.