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# 6.1 Absolute Value and Complex Conjugate

6.1   Definition (Complex Numbers, .) We denote the complexification of by , and we call the complex numbers.

6.2   Definition (Absolute value.) In exercise 4.23A we showed that (for any field in which is not a square), if , then

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

We note that

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

6.3   Definition (Real and imaginary parts.) Let and write where . We call the real part of , and we call the imaginary part of (note that the imaginary part of is real), and we write

6.4   Theorem. Let be complex numbers. Then
a)
.
b)
if .
c)
.
d)
.
e)
.
f)
.
g)
.
h)
.
i)
.

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.

6.5   Exercise. Prove parts b), c), d), e), f), g), h) and i) of Theorem 6.4. A

6.6   Theorem (Triangle inequality.) Let . Then

Proof: For all ,

 (6.7)

Now since , we have

Hence, from (6.7),

and it follows that

Next: 6.2 Geometrical Representation Up: 6. The Complex Numbers Previous: 6. The Complex Numbers   Index