That sounds crazy. Let's see what happens!
Assume numbers in this new algebra have the form:
a + b*j
where a and b are real numbers.
In the algebra of complex numbers, the absolute value function computes the magnitude of the complex number, as if it were a vector:
|a + b*i| = sqrt(a^2 + b^2)
Since the absolute value function is nonlinear, it can't have distributivity. Therefore:
|a + b| â‰* |a| + |b|
I can't see a way to define |a + b*j| without using the definition from complex numbers.
However, we can throw out the real part and use only numbers of the form b*j.
If we look at numbers of the form b*i, then we could conceivably draw up this expansion of the absolute value function:
|2*i| = |2|*|i| = 2*1 = 2
|-2*i| = |2|*|-i| = 2*1 = 2
which would give us this general rule:
|b*i| = |b|
Being consistent, we should then apply this definition to your new number class:
|2*j| = |2|*|j| = 2*(-1) = -2
|-2*j| = |2|*|-j| = 2*(-1) = -2
which would give us this general rule:
|b*j| = -|b|
Note that here I decided to define |-j| = |j|, because |-b| = |b|, and I wanted to maintain this symmetry.
For some interesting reading, check out quaternions and the other hypercomplex algebras (octonions, sedenions).
