Transcript 11.2 Geometric Representations of Complex Numbers

11.2 Geometric
Representations
of Complex
Numbers
Graphing Complex Numbers
• In Algebra 2, you learned how to graph a
complex number…
Imaginary
Part
Real
Part
Graphing Complex Numbers
• Graph the point 5 - 7i
A point representing a complex number
can be written 2 different ways:
1.) Rectangular Form:
z = a + bi
2.) Polar Form
z = r cos() + r sin()i
Polar form is often simplified by factoring out the radius.
z = r cis()
Covert each complex number to polar form:
a.)
3i 
b.)
4 2  4 2i 
c .)
39  80 i 
Covert each complex number to rectangular form:
3  
a.) 2 cis   
 2 
b.)
c .)
5  
cis   
 6 
2
1
 2  
5cis 
 
 3 
Products of Complex Numbers
Back in Algebra 2, you learned how to multiply
complex numbers…
1  2 i  2  3 i  
Foil
This
2  3i  4 i  6 i
 2  7i  6
 4  7i
 2  5 i   3  4 i  
2
Products of Complex Numbers in Polar Form
Let’s multiply 2 complex numbers:
r
cis 
number #1  r cis 
number # 2  s cis 
 s cis   
 r cos 
 i sin 
 s cos
  i sin 
 
rs  cos   i sin   cos   i sin   
rs cos  cos   i cos  sin   i sin  cos   i sin  sin   
2
Reorder the terms…
Products of Complex Numbers in Polar Form
rs cos  cos   i cos  sin   i sin  cos   i sin  sin   
2
Reorder the terms…
rs cos  cos   i sin  sin   i cos  sin   i sin  cos   
2
rs cos  cos   sin  sin   i cos  sin   sin  cos 
rs cos      i sin   

 
Products of Complex Numbers in Polar Form
r
cis 
 s cis   
rs cos      i sin   

Translation: When you multiply complex numbers in polar
form, you simply multiply the radii & add the angles.
Multiply using the polar form of the
complex number:
z1  1  2 i
z 2  2  3i
 z 1  z 2  
Multiply using the polar form of the
complex number:
 
z 1  5 cis  
6
 
z 2  7 cis  
2
 z 1  z 2  
Multiply using the polar form of the
complex number:
z1  1 
3i
z 2  2  2i
 z 1  z 2  