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Complex Numbers evolution to 11/08/01 10:07 AM (last update) by Ron Mower
Complex Numbers/Plane ( rectangular form z = a + bi , polar form z = r (cosθ + i sin θ ). )
The real number system and the imaginary numbers were
Imaginary
Complex
combined to form the complex number system. No real
Number
2 - 3i
π√2 + 2ei
2
etc.
System
numbers are imaginary and no imaginary numbers are real;
the sets are disjoint. However, both the real numbers and
Pure Imaginary
Real Section
the imaginary numbers are complex numbers. Two forms of
-i 3i πi −i√3 etc.
complex numbers are rectangular and polar. The complex
π 3 e √2 -1/7 etc.
plane uses real and imaginary axis' to replace the x and y
axis of the rectangular system or the pole and polar axis of the polar system. Information necessary to convert points and
numbers between the systems is included in the following text: (Some of the information should look familiar.)
a = a + 0i ; ex) 6 = 6 + 0i.
Real numbers are also complex numbers:
Pure Imaginary numbers are also complex numbers:
The Conjugate of
z = x + yi
is
z = x − yi. Then
bi = 0 + bi ; ex) 3i = 0 + 3i.
z z = ( x + yi)( x − yi) = x 2 + y 2 .
The Magnitude (absolute value/distance between the origin and the point) is
z = x2 + y2
For the polar form of z (below), r is positive, and θ is positive and less than 2π
π:
Forms:
Rectangular/Polar Equivalents
.
r > 0 and 0 ≤ θ < 2π .
z = x + iy ⇔ z = r(cosθ + i sinθ )
Points: ( x, y) ⇔ (r,θ )
of complex numbers
where,
y
; x = r cos θ ; y = r sin θ .
x
θ must be adjusted for its quadrant and so that θ > 0 as required .
r = x 2 + y 2 ; tan θ =
Recommended method for dividing, raising to powers and taking roots of complex numbers in rectangular f
Convert to polar, perform
operation, convert back to rectangular (if desired). It is generally easier than performing the operations in rectangular (as can be seen below)
Products/Quotients of Complex Numbers in Rectangular:
For z = a + bi and w = c + di ,
Divide in rectangular form by using the conjugate to get the results below
z ac + bd  − ad + bc  ( a + bi )i − ac − bd + ( ad − bc)i
=
+
=
i =
ci − d
− c2 − d 2
w c2 + d 2  c 2 + d 2 
Products/Quotients of Complex Numbers in Polar:
zw = ( a + bi )(c + di ) = ac − bd + ( ad + bc)i ;
r cis θ
For z = r1 (cos θ 1 + i sin θ 1 ) and w = r2 (cosθ 2 + i sin θ 2 ) ,
zw = r1 r2 [cos(θ 1 + θ 2 ) + i sin(θ 1 + θ 2 )] ,
z r1
= [cos(θ 1 − θ 2 ) + i sin(θ 1 − θ 2 )]
w r2
Demoivre's Theorem. Powers in Polar:
For z as above, z n = r1n (cos nθ 1 + sin nθ 1 ) .
from Dr. Math on the NET
How do I convert "8 cis 30" into rectangular coordinates?
The expression "8 cis 30" is a way of writing the complex number "8*(cos(30)
+ i*sin(30))" which has Real part 8*cos(30) and has Imaginary part 8*sin(30).
Usually the system of complex numbers is thought of as a plane, similar to the
usual Cartesian plane that consists of pairs (X,Y) of real numbers. In this
interpretation, your polar-type expression "8 cis 30" converts to a rectangular
ordered pair (8*cos(30) , 8*sin(30)). The sine of 30 degrees is one half, and
the cosine of 30 degrees is one half of the square root of three, so this ordered
pair is (4*sqrt(3), 4)
Roots in Polar:
zk =
for
n

2 kπ 
θ
+
r  cos 
 + i sin
n 
 n

k = 0 , 1 , 2 ,..., n − 1 ; nth
2 kπ 
θ
+


n  
 n
roots .
or
zk =
n
θ + 2 kπ
θ + 2 kπ 

r  cos
+ i sin

n
n


One More Form:
z = r (cos θ + i sin θ ) = reiθ .
This form (on the right of the =) is on your calculator as the polar complex mode.
Calculator MODES for calculation results. (Complex numbers can be entered in any form, no matter
the mode. Output is controlled by the MODE selected.) Your calculator can perform all operations of
complex numbers (adding, multiplying, raising to powers and taking roots - although only one root is
returned). There is no complex number capability on any TI model less than the TI-83.
TI-83: Under MODE you can select a+bi (rectangular-complex) or re^θi (polar complex) for output. The
i can be found near the bottom of the keyboard in yellow (press 2nd then i).
On the TI-83, input and
output of complex numbers looks pretty much as you would write it except for the carrot ^ used for
powers.
Complex numbers are entered and displayed (display depending on the MODE set) as follows: a+bi as
a+bi or r(cos θ+isin θ) as r(cos(θ)+isin(θ)). [Example: enter and read 5+3i as 5+3i or enter 2(cos
30+isin 30) as 2(cos(30)+isin(30))]. Don't enter the "]."
TI-85 or TI-86: Under MODE you can select RectC [rectangular complex (a,b)] or PolarC [polar complex
(róθ)] for output.
Complex numbers are entered and displayed (depending on MODE set) as follows: a+bi as (a,b) or
r(cos θ+isin θ) as (róθ). [Example: enter and read 5+3i as (5,3) or 2(cos 30+isin 30) as (2ó30)]
(Note: See Dr. Math
r cis θ
previous page)
Although it is not as easy to enter and read complex numbers on the TI-85 and TI-86, the TI-85 and,
especially the TI-86, are much more capable than the TI-83. The TI-86 also provides a CPLX (Complex
Number) Menu. Refer to the Guidebook for additional information (only out of desperation, of course!).
TI-89
If all else fails, refer to your Guidebook
. or 92
TI-86
The TI-92 has similar capabilities.
TI-83
Also see
CPLX menu.
TI-86