Transcript The Fundamental Theorem

Complex Numbers
Complex Numbers
and
Their Geometry
The Complex Plane

Complex Numbers and the Imaginary i

Definition:
The number x such that
x2 = –1
is defined to be i

Applying the square root property,
x = ±  –1
so that i =  –1 and –i = –  –1
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Complex Numbers
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The Complex Plane

Complex Numbers and the Imaginary i

Standard Form
iy
a + bi , b = 0
a + bi
Complex
Numbers
Real Numbers
a + bi , b ≠ 0
Imaginary Sometimes
Numbers b ≠ 0 and a = 0
● a + bi
x
The
Complex
Plane
Note:
a + bi is also written a + ib for real a and b
… especially if b is a radical or function
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Complex Numbers
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The Complex Plane

Complex Numbers in the Plane


Complex numbers as ordered pairs
of real numbers
iy

Complex number z
z = a + ib

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Numbers as points in a plane …
instead of points on a line
Point in complex plane
z = (a, b)
Complex Numbers
Imaginary Axis
ib
The
Complex
Plane
● z = (aa+, bib)
a
x
Real Axis
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The Complex Plane

Radical Expressions and Arithmetic
The expression  –a can be written
 –a =  –1  a = i  a
 Sum and difference of complex numbers

(a + bi) ± (c + di) = (a ± c) ± (b ± d)i

Examples:
(3 + 4i) – (2 – 5i) = (3 – 2) + (4 + 5)i = 1 + 9i
(7 – 3i) + (2 – 5i) = (7 + 2) – (3 + 5)i = 9 – 8i
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Complex Numbers
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The Complex Plane

Radical Expressions and Arithmetic


The expression  –a can be written
 –a =  –1  a = i  a
Product of Complex Numbers
(a + bi)(c + di) = ac + bdi2 + (ad + bc)i
= (ac – bd) + (ad + bc)i
Example:
(3 + 4i)(2 – 5i) = 6 – 20i2 + (8i – 15i)
= 26 – 7i
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Complex Numbers
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The Complex Plane

Complex Conjugates

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Definition: a + bi and a – bi are a
complex conjugate pair
Example:
7 + 3i and 7 – 3i are complex conjugates
Complex Numbers
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The Complex Plane

Complex Conjugates


Definition: a + bi and a – bi are a
complex conjugate pair
Fact:
The product of complex conjugates
is always real
(a + bi ) • (a – bi) = a2 + b2
Example:
(7 + 3i) • (7 – 3i) = 72 + 32
= 49 + 9 = 58
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Complex Numbers
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The Complex Plane

Complex-Number Quotients
Complex
conjugate
of c + di
a + bi
a + bi c – di
c + di = c + di c – di
(ac + bd) + (bc – ad)i
=
Real denominator
c2 + d 2
(bc – ad)i
ac + bd
= c2 + d 2 + c2 + d 2
bc – ad
ac + bd
= c2 + d2 + c2 + d 2 i
(
) (
)
Note:
We can always multiply by 1 in clever
“disguise” to change form NOT value
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Complex Numbers
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The Complex Plane

Complex-Number Quotients
a + bi
c + di =

(
bc – ad
ac + bd
c2 + d2 + c2 + d 2
) (
)i
Quotient Examples

(15 + 65i)(1 – 2i)
15 + 65i
1. 1 + 2i = (1 + 2i)(1 – 2i)
15 + 130 + (65 – 30)i
=
12 + 22
= 29 + 7i
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Complex Numbers
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The Complex Plane

Quotient Examples

2.
3 ( –i )
3
i = ( i )( –i )
3 ( –i )
= –i 2
3 ( –i )
= –(–1)
= –3i
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Complex Numbers
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The Complex Plane

Quotient Examples

(–2 + i)(1 – i )2
–2 + i
3. (1 + i )2 = (1 + i )2 (1 – i )2
(–2 + i)(–2i)
=
((1 + i )(1 – i ))2
2 + 4i
= ( 2 )2
1
= 2 +i
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Complex Numbers
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Think about it !
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