On <em>l_p</em>-Complex Numbers

• Main Author: Wolf-Dieter Richter
• Format: Article
• Language: English
• Published: MDPI AG 2020-05-01
• Series: Symmetry
• Subjects: geometric vector product; <em>l_p</em>-imaginary unit; <em>l_p</em>-complex number multiplication; geometric exponential function; invariant transformation; generalized Euler’s trigonometric representation
• Online Access: https://www.mdpi.com/2073-8994/12/6/877

Dispensing with the common property of distributivity and replacing classical trigonometric functions with their <inline-formula> <math display="inline"> <semantics> <msub> <mi>l</mi> <mi>p</mi> </msub> </semantics> </math> </inline-formula>-counterparts in Euler’s trigonometric representation of complex numbers, classes of <inline-formula> <math display="inline"> <semantics> <msub> <mi>l</mi> <mi>p</mi> </msub> </semantics> </math> </inline-formula>-complex numbers are introduced and some of their basic properties are proved. The collection of all points that leave the <inline-formula> <math display="inline"> <semantics> <msub> <mi>l</mi> <mi>p</mi> </msub> </semantics> </math> </inline-formula>-absolute value of each <inline-formula> <math display="inline"> <semantics> <msub> <mi>l</mi> <mi>p</mi> </msub> </semantics> </math> </inline-formula>-complex number invariant under <inline-formula> <math display="inline"> <semantics> <msub> <mi>l</mi> <mi>p</mi> </msub> </semantics> </math> </inline-formula>-complex numbers multiplication is shown to be a group of elements that have <inline-formula> <math display="inline"> <semantics> <msub> <mi>l</mi> <mi>p</mi> </msub> </semantics> </math> </inline-formula>-absolute value one but not the symmetry group.