20 1. THE CONSTRUCTION OF REAL AND COMPLEX NUMBERS If z and w are complex numbers, then |z|, |w| R and hence it makes sense to say that |z| |w|. However, it makes no sense to say that z w. Exercise 1.8.5. Show that if we identify z = a + bi with the point (a, b) R2, then the absolute value of z is equal to the distance of the point (a, b) from (0, 0). Im(z) Re(z) |z| z=(Re(z), Im(z)) Exercise 1.8.6. Show that the absolute value on C satisfies all the properties of the absolute value on R. (i) For any z C, we have |z| 0, and |z| = 0 iff z = 0. (ii) For any z, w C, we have |zw| = |z||w|. (iii) For any z, w C, we have |z + w| |z| + |w| (triangle inequality). Exercise 1.8.7. Show that the field of complex numbers is not isomor- phic to the field of real numbers. 1.9. Convergence in C Now that we have an absolute value on C, we can define the notions of Cauchy sequence and convergent sequence in C. Definition 1.9.1. A sequence (zk)k∈N of complex numbers is convergent if there exists an element z C such that the sequence satisfies the following property: given any ε 0, there exists N N such that k N implies that |zk z| ε. We say that (zk)k∈N converges to z, and z is called the limit of the sequence (zk)k∈N. Symbolically, we write lim k→∞ zk = z. We will often say that a sequence of complex numbers is convergent without specific reference to the limit z. Note that N depends on ε. As usual, the limit of a convergent sequence is unique. Definition 1.9.2. Let r be a positive real number, and let z0 C. The open ball of radius r with center at z0 is (1.1) Br(z0) = {z C | |z z0| r}.
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