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This manuscript introduces $J_3$-numbers, a seemingly missingthree-dimensional intermediate between complex numbers related to points in theCartesian coordinate plane and Hamilton's quaternions in the 4D space. Thecurrent development is based on a rotoreflection operator $j$ in $\mathbb{R}^3$that induces a novel $\circledast$-multiplication of triples which turns out tobe associative, distributive and commutative. This allows one to regard a pointin $\mathbb{R}^3$ as the three-component $J_3$-number rather than a triple ofreal numbers. Being equipped with the $\circledast$-product, the commutativealgebra $\mathbb{R}_\circledast^3$ is isomorphic to $\mathbb{R} \oplus\mathbb{C}$. Some geometric and algebraic properties of the $J_3$-numbers arediscussed.

On a novel 3D hypercomplex number system