Exponential form of complex numbers
1-E
Precalculus
Euler's formula
Euler's fomula
e i  = cos   i sin 
connects trigonometric functions and complex numbers.
For the angle π we get
ei  = − 1
1-1
⇔
ei   1 = 0
Precalculus
Euler's formula
This fomula provides a remarkable simple connection of 5 very important
mathematical constants: Euler's number e, the imaginary unit i of complex
numbers, the number π, the unit number 1 and zero, 0.
1-2
Precalculus
Most remarkable formula
http://www.cap.ca/wyp/mediaPhysics.asp
Richard Feynman, american physicist and Nobel laureate in 1965,
called Euler's formula one of the most remarkable formulas in all
of mathematics.
1-3
Precalculus
Most remarkable formula
Richard Feynman (1918-1988)
Richard Feynman, american physicist and Nobel laureate in 1965,
called Euler's formula one of the most remarkable formulas in all
of mathematics.
1-4
Precalculus
Exponential form
trigonometric form
z = r cos   i sin 
↓
e i  = cos   i sin 
↓
exponential form
z = r ei 
2-1
Precalculus
Representation of complex numbers: Summary
 x , y  : Re  z  = x ,
pair of real numbers:
z= xi y
polar form:
algebraic (Cartesian) form
r ,  :
r – absolute value of z,  – argument of z
z = r  cos   i sin  
z = r e
i
complex conjugate:
2-2
Im  z  = y
trigonometric form
exponential form
Im  z *  = − Im  z 
Precalculus
Exponential form of complex numbers: Exercise
Transform the complex numbers into
Cartesian form:
a) z = 2e
i π
6
b ) z = 2 √3 e
i π
3
c ) z = 4 e 3π i
d ) z = 4e
i π
2
e ) z = √2 e
i
f ) z = 2 √3 e
g ) z = √3 e
6-1
i
3π
4
i
2π
3
13 π
6
Precalculus
Exponential form of complex numbers: Solution
a ) z : r = 2,
=
b ) z : r = 2 3 ,
=
z = 3  i

,
3
c ) z : r = 4,
 = 3 ,
d ) z : r = 4,
=
e ) z : r = 2 ,
f ) z : r = 2 3 ,
g ) z : r = 3 ,
6-2

,
6
z = 3  3 i

,
2
z = −4
=
3
,
4
=
=
z = 4i
2
,
3
13 
,
6
z = −1  i
z = −3  3 i
z=
3
3
  i
2
2
Precalculus
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3-3
Precalculus