Fundamental Theorem of Algebra
Original Notes adopted from November 20, 2001 (Week 11)
© P. Rosenthal , MAT246Y1, University of Toronto, Department of Mathematics typed by A. Ku Ong
a + bi + c + di = ( a + c) + (b + d) i
Triangle Inequality
For z
1
, z
2
∈
C,
z
1
+ z
2

≤
z
1
 + z
2

( Or  z
1
+ z
2
+... z
n

≤
 z
1
 +  z
1
 + ... +
z
n

Fundamental Theorem of Algebra
: Every polynomial with complex coeffients other than constant
polynomials has a complex proof.
p(z) = a
n
z
n
+ a
n1
z
n1
+ a
1
z
+ a
0
, a
j
∈
C, n
∈
N. a
n
≠
0. n degree of polynomial.
Definition
: A closed curve in the plane is a continuous function from [0, 2
π
] into C such its values at 0
and 2
π
are the same.
Eg. A Function
φ
(t)= f(t) + i g(t), f,g into functions mapping [0, 2
π
] into R & f(0) = f(2
π
), g(0) = g (2
π
).
Eg.
φ
(t) = cost + sint , t
∈
(0, 2
π
)
Winding number is 1.
Eg.
φ
(t) = cos3t + isin3t , t
∈
[0, 2
π
]
PICTURE
Winding number is 3
Definition: If
φ
is a closed curve in C that doesn't go through (0,0), its winding number about (0,0) is the
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 Spring '10
 Applebaugh
 Math, Fundamental Theorem Of Algebra, Metric space

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