- #1

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So, in general what is the definition of a "modulus" that coincides its definitions on the reals, complex, dual, and split complex numbers?

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- Thread starter ForMyThunder
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- #1

- 147

- 0

So, in general what is the definition of a "modulus" that coincides its definitions on the reals, complex, dual, and split complex numbers?

- #2

- 430

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As I have come across the term in these contexts the modulus isn't just "some" real-valued function but rather the function

[tex]\|z\| = z\overline{z}[/tex]

where [itex]\overline{z}[/itex] denotes the conjugate, so:

[tex]\overline{a} = a\qquad\textrm{for real a}[/tex]

[tex]\overline{a+bi} = a-bi\qquad\textrm{for complex a+bi}[/tex]

[tex]\overline{a+bj} = a-bj\qquad\textrm{for split complex a+bj}[/tex]

[tex]\overline{a+b\epsilon} = a-b\epsilon\qquad\textrm{for dual }a+b\epsilon[/tex]

So in the split complex case we have

[tex]\|a+bj\| = (a+bj)(a-bj) = a^2 -b^2[/tex]

which is not a norm, but it is multiplicative and a quadratic form. In the dual case we have

[tex]\|a+b\epsilon\| = (a+b\epsilon)(a-b\epsilon) = a^2[/tex]

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