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Complex Numbers and Euler’s IdentityMATH 171 Freshman Seminar for Mathematics Majors
J. Robert Buchanan
Department of Mathematics
2010
J. Robert Buchanan Complex Numbers and Euler’s Identity
Background
Easy: solve the equation 0 = 1 − z2.
0 = 1 − z2
= (1 − z)(1 + z)
1 = z or − 1 = z
J. Robert Buchanan Complex Numbers and Euler’s Identity
Background
Easy: solve the equation 0 = 1 − z2.
0 = 1 − z2
= (1 − z)(1 + z)
1 = z or − 1 = z
Not (as) easy: solve the equation 0 = 1 + z2.
0 = 1 + z2
= −1 − z2
= (√−1 − z)(−
√−1 + z)√
−1 = z or −√−1 = z
Define i =√−1, then z = ±i .
J. Robert Buchanan Complex Numbers and Euler’s Identity
Complex Numbers
Definition
A number of the form z = a + bi where a and b are realnumbers and i =
√−1 is called a complex number . If a = 0
and b 6= 0 so that z = bi , then z is called an imaginarynumber .
J. Robert Buchanan Complex Numbers and Euler’s Identity
Picturing Complex Numbers
z = a + bi
Ha,bL
x
i y
J. Robert Buchanan Complex Numbers and Euler’s Identity
Polar Representation (1 of 2)
z = reiθ
Ha,bL
Θ
r
x
i y
J. Robert Buchanan Complex Numbers and Euler’s Identity
Polar Representation (2 of 2)
z = a + bi = reiθ
where
θ = tan−1(
ba
)
and
r =
{√
a2 + b2 if a > 0,−√
a2 + b2 if a < 0.
J. Robert Buchanan Complex Numbers and Euler’s Identity
Examples (1 of 3)
Example
Find the polar representation of the following complex numbers.
−1 + i
2 − i
−3 − i
J. Robert Buchanan Complex Numbers and Euler’s Identity
Examples (2 of 3)
H-1,1L
H2,-1LH-3,-1L
x
i y
J. Robert Buchanan Complex Numbers and Euler’s Identity
Examples (3 of 3)
Example
Find the polar representation of the following complex numbers.
−1 + i = −√
2e−iπ/4
2 − i =√
5ei tan−1(−1/2) ≈√
5e−0.463648i
−3 − i = −√
10ei tan−1(1/3) ≈ −√
10e0.321751i
J. Robert Buchanan Complex Numbers and Euler’s Identity
Complex Arithmetic
If z1 = a + bi and z2 = c + di , then
z1 + z2 = (a + c) + (b + d)i
z1 − z2 = (a − c) + (b − d)i
z1z2 = (ac − bd) + (ad + bc)iz1
z2=
(ac + bd) + (bc − ad)ic2 + d2 ,
provided c2 + d2 6= 0.
J. Robert Buchanan Complex Numbers and Euler’s Identity
Complex Addition (1 of 3)
(−3 + 4i) + (5 − 2i) = 2 + 2i
z1
z2
H-3,4L
H5,-2L
x
i y
J. Robert Buchanan Complex Numbers and Euler’s Identity
Complex Addition (2 of 3)
(−3 + 4i) + (5 − 2i) = 2 + 2i
z1
z2
z1+z2
H-3,4L
H2,2L
H5,-2L
x
i y
J. Robert Buchanan Complex Numbers and Euler’s Identity
Complex Addition (3 of 3)
(−3 + 4i) + (5 − 2i) = 2 + 2i
z1
z2
z1+z2
H-3,4L
H2,2L
H5,-2L
z2
z1
x
i y
J. Robert Buchanan Complex Numbers and Euler’s Identity
Complex Multiplication
z1z2 = r1eiθ1r2eiθ2 = (r1r2)ei(θ1+θ2)
Θ1
r1Θ2r2r1r2
Θ1+Θ2
x
i y
J. Robert Buchanan Complex Numbers and Euler’s Identity
Challenge
Represent z = −1 = −1 + 0i in polar form.
J. Robert Buchanan Complex Numbers and Euler’s Identity
Challenge
Represent z = −1 = −1 + 0i in polar form.
Since r =√
(−1)2 + 02 = 1, then
−1 = eiπ.
J. Robert Buchanan Complex Numbers and Euler’s Identity
Challenge
Represent z = −1 = −1 + 0i in polar form.
Since r =√
(−1)2 + 02 = 1, then
−1 = eiπ.
Rearranging the equation above yields an equation relating fiveof the most important constants in mathematics.
eiπ + 1 = 0
J. Robert Buchanan Complex Numbers and Euler’s Identity
Euler’s Identity
eiθ = cos θ + i sin θ
Θ
r
r ei Θ=rHcosHΘL+ i sinHΘLL
x
i y
J. Robert Buchanan Complex Numbers and Euler’s Identity
Commemorative Stamp
J. Robert Buchanan Complex Numbers and Euler’s Identity
Complex Exponentiation
Use Euler’s Identity (eiθ = cos θ + i sin θ) to
express z = i in polar form, and
J. Robert Buchanan Complex Numbers and Euler’s Identity
Complex Exponentiation
Use Euler’s Identity (eiθ = cos θ + i sin θ) to
express z = i in polar form, and
evaluate i i .
J. Robert Buchanan Complex Numbers and Euler’s Identity
Infinite Series
In Calculus II you will learn to express the function ex as theinfinite series :
ex = 1 +x1!
+x2
2!+
x3
3!+ · · ·
where n! = (1)(2)(3) · · · (n).
J. Robert Buchanan Complex Numbers and Euler’s Identity
Infinite Series
In Calculus II you will learn to express the function ex as theinfinite series :
ex = 1 +x1!
+x2
2!+
x3
3!+ · · ·
where n! = (1)(2)(3) · · · (n).
This infinite series holds for real and complex values of x .
J. Robert Buchanan Complex Numbers and Euler’s Identity
Infinite Series
In Calculus II you will learn to express the function ex as theinfinite series :
ex = 1 +x1!
+x2
2!+
x3
3!+ · · ·
where n! = (1)(2)(3) · · · (n).
This infinite series holds for real and complex values of x .
For example,
−1 = eiπ = 1 +iπ1!
+(iπ)2
2!+
(iπ)3
3!+ · · ·
J. Robert Buchanan Complex Numbers and Euler’s Identity
Expressing eiπ as a Series
−1 = 1 +iπ1!
+(iπ)2
2!+
(iπ)3
3!+
(iπ)4
4!+
(iπ)5
5!+ · · ·
= 1 + iπ − π2
2− i
π3
6+
π4
24+ i
π5
120− · · ·
J. Robert Buchanan Complex Numbers and Euler’s Identity
Expressing eiπ as a Series
−1 = 1 +iπ1!
+(iπ)2
2!+
(iπ)3
3!+
(iπ)4
4!+
(iπ)5
5!+ · · ·
= 1 + iπ − π2
2− i
π3
6+
π4
24+ i
π5
120− · · ·
Note: the series consists of alternating real and purelyimaginary terms.
J. Robert Buchanan Complex Numbers and Euler’s Identity
Expressing the Series for eiπ as a Graph
−1 = 1 + iπ − π2
2− i
π3
6+
π4
24+ i
π5
120− · · ·
-4 -3 -2 -1 1x
-2
-1
1
2
3
i y
J. Robert Buchanan Complex Numbers and Euler’s Identity
Quadratic Map
Suppose f (z) = z2 + c where z and c can be complexnumbers.Similar to the Newton’s Method formula we may iterate thequadratic function f (z).Starting with z0 we define
zn = f (zn−1) = z2n−1 + c
for n = 1, 2, . . ..
J. Robert Buchanan Complex Numbers and Euler’s Identity
Quadratic Map
Suppose f (z) = z2 + c where z and c can be complexnumbers.Similar to the Newton’s Method formula we may iterate thequadratic function f (z).Starting with z0 we define
zn = f (zn−1) = z2n−1 + c
for n = 1, 2, . . ..
For example, if z0 = 0 and c = 12 i then
z1 =12
i
z2 = −14
+12
i
z3 = − 316
+14
i
...
J. Robert Buchanan Complex Numbers and Euler’s Identity
Quadratic Map (Graph)
-0.25 -0.20 -0.15 -0.10 -0.05x
0.1
0.2
0.3
0.4
0.5
i y
J. Robert Buchanan Complex Numbers and Euler’s Identity
General Exponentiation
Suppose x and y are two real numbers and suppose y > x > 0.
Question: which is larger xy or yx ?
J. Robert Buchanan Complex Numbers and Euler’s Identity
General Exponentiation
Suppose x and y are two real numbers and suppose y > x > 0.
Question: which is larger xy or yx ?
Remember one of our principles of mathematical inquiry, trysome examples in order to gain insight into a complicatedquestion.
Let x = 2 and try y = 3, y = 4, and y = 5.
J. Robert Buchanan Complex Numbers and Euler’s Identity
Equality (1 of 3)
Since for some choices of 0 < x < y ,
xy > yx
while for othersxy < yx
we may be curious about when xy = yx .
J. Robert Buchanan Complex Numbers and Euler’s Identity
Equality (2 of 3)
xy = yx
ln(xy ) = ln(yx )
y ln x = x ln y
Since y > x > 0 then y = kx where k > 1. Substitute this intothe last equation above and solve for x and y in terms of k .
J. Robert Buchanan Complex Numbers and Euler’s Identity
Equality (3 of 3)
For k > 1,
x = k1/(k−1)
y = kk/(k−1).
2.0 2.1 2.2 2.3 2.4 2.5 2.6 2.7
2.8
3.0
3.2
3.4
3.6
3.8
4.0
x
y
J. Robert Buchanan Complex Numbers and Euler’s Identity
Limits as k → 1+
Use l’Hôpital’s Rule to find
limk→1+
x = limk→1+
k1/(k−1)
limk→1+
y = limk→1+
kk/(k−1).
J. Robert Buchanan Complex Numbers and Euler’s Identity
Limits as k → 1+
Use l’Hôpital’s Rule to find
limk→1+
x = limk→1+
k1/(k−1)
limk→1+
y = limk→1+
kk/(k−1).
limk→1+
k1/(k−1) = e
limk→1+
kk/(k−1) = e
J. Robert Buchanan Complex Numbers and Euler’s Identity
Summary
x=yxy=yx
He,eL
HemptyL y<x
yx>xy
xy>yx
0 1 2 3 4 5 6
0
1
2
3
4
5
6
x
y
J. Robert Buchanan Complex Numbers and Euler’s Identity
Students and Complex Numbers
Student c Student cBacchi −11
8 + i j8 Bongiovanni −5
4 + i j8
Cilladi −98 + i j
8 Cox −1 + i j8
Crider −78 + i j
8 de Kok −34 + i j
8Hansford −5
8 + i j8 Hild −1
2 + i j8
Junkin −38 + i j
8 Keglovits −14 + i j
8Kibler −1
8 + i j8 Konowal 1
8 + i j8
Leber 14 + i j
8 Longo 38 + i j
8Mecutchen 1
2 + i j8 Miller, B 5
8 + i j8
Miller, S 34 + i j
8 Nguyen 78 + i j
8Reed 1 + i j
8 Smeltz 98 + i j
8Starner 5
4 + i j8 Visek 11
8 + i j8
Williard 32 + i j
8 Zipko 138 + i j
8
J. Robert Buchanan Complex Numbers and Euler’s Identity
Results
Name:
c =
Results:
1 2 3 4 5 6 7 8 9 10 11 12
13 14 15 16 17 18 19 20 21 22 23 24
J. Robert Buchanan Complex Numbers and Euler’s Identity
Homework
Referring to the polygonal spiral approaching −1 on slide28, find the total length of the spiral.
For the complex number c you have been assigned andstarting with z0 = 0 iterate the quadratic map f (z) = z2 + cten times (or less if r =
√a2 + b2 > 2) for j = 1, 2, . . . , 24.
If all the iterates of the quadratic map have a magnitude ofless than 2 record a result of 1, else record a result of 0.
J. Robert Buchanan Complex Numbers and Euler’s Identity