Maths - Double Numbers- Datasheet

For more information about double numbers see this page.

Algebra Laws

Double Numbers over the real numbers are a 'field' they have the following properties:

  addition multipication
unit element 0 1
commutative yes yes
associative yes yes
distributive over addition - yes
inverse exists yes yes

As an Extension Field to Real Numbers

R[x]/<x²-1>

As a Multiplicative Group

If we ignore addition and treat complex numbers as a group then the group is equivalent to a D(2), it has the following properties:

Cayley Table

The Cayley table is symmetric about its leading diagonal:

  1 D -1 -i
1 1 D -1 -D
i D 1 -D -1
-1 -1 -D 1 D
-i -D -1 D 1

For more information about Cayley table see this page.

Cayley Graph

cyclic cayley graph

For more information about Cayley graph see this page.

Cyclic Notation

The group has two generators each cycling between two elements:

(1,D)(-1,-D)
(1,-1)(D,-D)

For more information about cyclic notation see this page.

Group Presentation

There is only one generator which when applied n times cycles back to the identity.

<a,b | a²=1,b²=1,ab=ba>

For more information about group presentation see this page.

Group Representation

A representation using 4 ×4 matricies containing 0 and 1 is:

[
0 1 0 0
1 0 0 0
0 0 0 1
0 0 1 0
,
0 0 1 0
0 0 0 1
1 0 0 0
0 1 0 0
]

An alternative 2×2 matrix representation containing 0, 1 and -1 is:

[
-1 0
0 -1
,
0 1
1 0
]

For more information about group representation see this page.

Related datasheets


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see also:

 

Correspondence about this page

Book Shop - Further reading.

Where I can, I have put links to Amazon for books that are relevant to the subject, click on the appropriate country flag to get more details of the book or to buy it from them.

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Terminology and Notation

Specific to this page here:

 

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