The Eisenstein integers

where ω is a primitive cube root of 1 given by

are often represented in the form

(noting that )

and perhaps abbreviated by pair notation such as

(where the asterisk is to distinguish this notation from that introduced below)

so that

Here is a more intuitive representation that is simpler to manipulate and reason about.

Let σ, τ be primitive *sixth* roots of 1 given by

and use the representation

abbreviated by the pair notation

i.e.

It so happens that:

So:

is pure real

is pure imaginary

— as before

— which is more symmetrical

Some special cases that result are:

And we have:

With results:

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