Hi, I m trying to find out, what is imaginary unit/number. i^2

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PlanckShift said:
1/ε is undefined. It is so a field. It's closed under linear combinations. It's commutative and associative. It has an identity element: 1+ε0 and the inverse element which I've described already. It's distributive, too.

The nonzero elements of a field are an abelian group under multiplication, so [itex]\epsilon^{-1} = \frac{1}{\epsilon}[/itex] must exist if [itex]\epsilon \ne 0[/itex].
 
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The real numbers are a field even though 0-1 doesn't exist. It's the same thing with this definition of complex numbers although elements along the "imaginary" line don't have inverses.
 


PlanckShift said:
The real numbers are a field even though 0-1 doesn't exist. It's the same thing with this definition of complex numbers although elements along the "imaginary" line don't have inverses.

Please stop spouting nonsense and review basic mathematics.
Nonzero imaginary numbers do have inverses: [itex]i^{-1}=-i[/itex].
 


PlanckShift said:
Definition of a field.

OK, you found the definition. Now read the definition. In particular:

Similarly, for any a in F other than 0, there exists an element a−1 in F, such that a · a−1 = 1.
 


But ε2 = 0 remember? Think about it. How do you find the inverse of a+bε? Rationalize the expression using ε2 = 0. Then find where the resulting expression is undefined.
 


PlanckShift said:
But ε2 = 0 remember? Think about it. How do you find the inverse of a+bε? Rationalize the expression using ε2 = 0. Then find where the resulting expression is undefined.

But where is the inverse of ε?
Hint: It's 1/ε, which you said was undefined. It doesn't have an inverse, so your structure isn't a field.