If we are working in , then and hence has a unique square root in , which we denote by and call the

We note that

Also note that for , this definition agrees with our old definition of absolute value in .

Proof: By using properties of the complex conjugate proved in exercise
4.23A, we have

Hence by uniqueness of square roots, . The proofs of b), c), d), e), f), g), h) and i) are left to you.

Proof: For all
,

Now since , we have

Hence, from (6.7),

and it follows that