1. Graphing Complex Numbers
Argand diagram
Imaginary i i
Real

2. Graphing Complex Numbers
Argand diagram
Imaginary
| z | = 2 2
Circle radius = 2 centre (0,0)
Real

3. Graphing Complex Numbers
Argand diagram
Imaginary
| z | < 2 2
Solid circle radius = 2 centre (0,0) but not including the circumference
Real

4. Graphing Complex Numbers
Argand diagram
Imaginary
| z +1| = 2 1
Circle radius = 2 centre (-1,0)
(-1,0) +
Real

5. Graphing Complex Numbers
Argand diagram
Imaginary
| z +1-2i | = 3
(-1,2) +
Circle radius = 3 centre (-1,2)
r = 3
Real

6. Graphing Complex Numbers
Argand diagram
Imaginary
| z - 4 | = | z |
There are 2 points (4,0) and (0,0)
(0,0) +
(4,0) +
What you need is a line bisecting these points
Real
ie x = 2

7. Graphing Complex Numbers
Argand diagram
| z - 4 | = | z +1- 2i |
Imaginary
(-1,2) +
There are 2 points (4,0) and (-1,2)
(4,0) +
What you need is a line bisecting these points
Real
ie 4y -10x +13 = 0

8. Graphing Complex Numbers
Argand diagram
Imaginary
z + z* = 8
a + bi + a - bi = 8
a = 4
2a = 8
a = 4
Real

9. Argand diagram | z + 4 | = 3| z | | z + 4 |2 = 32| z |2
Imaginary
zz*+4z+4z*+16=9zz*
8zz*-4z-4z*=16
8zz*-4(z+z*)=16
If z = x+yi then z* =x-yi
(½,0) +
z+z*=2x and zz* = x2+y2
8x2 +8y2 - 8x = 16
Real
x2 +y2 - x = 2
(x-½)2 +y2 = 2+½2
Circle centre (½,0) radius 3/2
(x-½)2 +y2 = 9/4 =(3/2)2

10. Argand diagram | z + 4 | > 3| z | | z + 4 |2 > 32| z |2
Imaginary
zz*+4z+4z*+16>9zz*
8zz*-4z-4z*<16
8zz*-4(z+z*)<16
If z = x+yi then z* =x-yi
(½,0) +
z+z*=2x and zz* = x2+y2
8x2 +8y2 - 8x < 16
Real
x2 +y2 - x < 2
(x-½)2 +y2 < 2+½2
Circle centre (½,0) radius 3/2
(x-½)2 +y2 < 9/4 ie(3/2)2

11. Argand diagram |z-4| < | z-2i | | z-4|2 < | z-2i |2
(z-4)(z*-4*) < (z-2i)(z*-2i*)
-2i*=2i
zz*-4z-4z*+16<zz*+2iz-2iz*+4
Imaginary
4z+4z*+2iz-2iz*>12
y<2x-3
4(z+z*)+2i(z-z*)>12
If z = x+yi then z* =x-yi
z+z*=2x and z-z* = 2yi
8x +2i(2yi)> 12
Real
8x +4yi2 > 12
8x -4y > 12
2x - y > 3