The Complex Numbers

1. Complex Conjugation and Modulus.
Definition 1.1.6. If is a complex number, then its complex conjugate, denoted , is defined by

while its modulus, denoted , is defined by

Note that the modulus, as defined above, is just the usual Euclidean norm in the vector space . Thus, if , then is the Euclidean distance from to . The term modulus is traditional, but the terms norm and absolute value are also commonly used to mean the same thing. We will use all three.
Note also that the two solutions of a quadratic equation with real coefficients given in Example 1.1.3 are complex conjugates of each other. Thus, the solutions to a quadratic equation with real coefficients occur in conjugate pairs. Quadratic equations with complex coefficients also have roots and they are also given by the quadratic formula. However, we cannot prove this until we prove that every complex number has a square root. In fact, in Section 1.4 we will prove that every complex number has roots of all orders.
For a complex number , the real number is called the real part of and is denoted , while the number is called the imaginary part of and is denoted . In graphing complex numbers using a rectilinear coordinate system, determines the coordinate on the horizontal axis, while determines the coordinate on the vertical axis.
Note that a complex number is real if and only if , and it is purely imaginary if and only if . Note also, that if , then

The elementary properties of conjugation and modulus are gathered together in the next theorem.
Theorem 1.1.7. If and are complex numbers, then
(a) ;

Figure 1.1.2. Plot of the Complex Numbers , and .
(b) ;
(c) ;
(d)
(e) and ;
(f) is a non-negative real number and is 0 if and only if ;
(g) ;
(h) .
Proof. We will prove (g) and (h). The other parts are elementary computations or observations and will be left as exercises.
Parts (g) and (h) are the Cauchy-Schwarz inequality and the triangle inequality for the vector space . Versions of these inequlities hold in general Euclidian space . The proofs we give here are specializations to of the standard proofs of these inequalities in .
To prove (g), we begin with the observation that (a) and (d) imply that

We then let be an arbitrary real number and note that, by Parts (c), (d), and (f),

for all values of . This implies that the quadratic polynomial in given by

is never negative and, therefore, has at most one real root. This is only possible if the expression under the radical in the quadratic formula is negative or zero. Thus,