A binary number system is any two-component number system of the form

.

Here denotes the set of real numbers. Binary numbers are added componentwise and are multplied together using the "foil method" together with the identity .

Three very special examples of binary number systems are
given by the following.

- In this case , which of course yields the complex number system.
- Hence . This case yields what is known as the perplex number system.
- Hence . This case yields what is known as the dual number system.

In a similiar manner, the __dual number__
can be identified with the two by two matrix ,
whose square is the zero matrix. Under this identification the arbitrary
dual number
is identified with the two by two matrix .

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