Bridge to Abstract Mathematics: Mathematical Proof and Structures

9.4 PROPERTIES OF THE COMPLEX NUMBER FIELD 327

(Imaginary axis)

I

Figure 9.5 Graphic view of the four complex

4th roots of z = 16.

Exercises

1. Express each of the following complex numbers in the form x + yi:
   (a) 6(4 - i) - 3(2 + 2i)
   (c) (2 + i)(2 - i)
   (e) (1 - i)4
   *(g) (4A4 - 9) - (4/(4 + 0)

(b) - l/i

(d) (2 - i)/(2 + i)

(f) i16+i6+i5

2. Given z, = 5 - 3i, z2 = 4 + 5i, and z, = 2i, calculate:


3. Find all z E C satisfying the equation:
   (a) (4 + 3i)z + (7 + 4i) = 6
   (c) z - z = 4i
   (e) z + (412) = 0
   (g) z2-4z+ 13=0

(b) z + z* = 4i

(d) z + z* = 12

(f) z2+121=o

4. Use polar form z = reie of a complex number z = x + yi (and Theorem 5(a), in
   particular) to calculate the product z1z2 and the quotient z1/z2, where 2, = 3 - 3i
   and z2 = 23 + 2i.


5. Use Theorem 6 and the method of Example 5 to find:
   (a) The four complex 4th roots of z = - 16
   (b) The three complex cubed roots of z = 8i


6. Prove that if the complex numbers z, and z2 are both real, then:
   (a) zl + z2 = Re@,) + Re(z2) *(b) zlz2 = Re(z,) Re(z2)