PROPERTIES OF NUMBER SYSTEMS Chapter 9- (a) Prove, in detail, that (C, +, .) is a field.
(b) Explain, on the basis of (a) and material in Article 9.1, why each of the
following is a theorem in C:
z.O=Oforanyz~C
(-1)z = -z for any z E C
(z,)(-z2) = -(zlz2) for any z,, 2, E C
(-z,)(-z,) = (z,z2) for any z,, 2, E C
If z,z2 = 0, then either 2, = 0 or 2, = 0 for any z,, 2, E C
If wz, = wz2 and w # 0, then 2, = z, for any w, z,, z2 E C
l/(zlz2) = (l/z,)(l/z2) for any z,, z2 E C, z1 # 0 and 2, # 0
l/(l/z) = z for any z E C, z # 0
8. Verify parts (b) through (g), and (j) of Theorem 2. - (a) Verify parts (a) through (d), (g), and (i) of Theorem 3.
(b) Prove part (j) of Theorem 3, the triangle inequality for complex numbers.
[Hint: Prove that 12, + z2I2 I ()zl) + 1~~1)~ and use Exercise 4(b)(ii), Article 9.2,
noting that Re (z) 5 lzl for any z E C.] - (a) Use the result of the lemma following the statement of Theorem 4 to
verify parts (a) through (d) of that theorem.
(b) Prove that ciu = - 1.