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Complex Numbers
If the product of two positive number is
positive, and the product of two negative
number is also positive, how can the
square of a number be negative?
It can’t, if the number is a real number.
But scientists and engineers have found
applications where is it useful to have
numbers that are negative when squared.
Examples: Express the following
radicals in terms of i. Simplify the
radical if possible.
The Number i :
i is the unique number such that
i2 = -1,
− 25
− 49
− 12
− − 17
so i = − 1 .
We know that i is not a real number. i is
both an imaginary number and a complex
number.
Addition and subtraction of complex
numbers:
Definition: A complex number is any
number that can be written in the form
When adding or subtracting complex,
combine like terms.
Example:
a + bi,
where a and b are real numbers.
Examples: 2 + 3i, -7i,
complex numbers.
2 − i are all
(3 + 4i ) + (7 − 6i ) = (3 + 7 ) + (4i − 6i) =
10 − 2i
1
Examples:
Multiplication of Complex Numbers
(8 – 3i) + (6 – 2i) =
To multiply radicals, we want to use
the product rule for radicals:
(7 + 3i) + (5 + 6i) =
n
(5 – 2i) – (9 – 4i) =
a n b = n a ⋅b
But the product rule for radicals only
works with real numbers!
Example: To multiply
Examples:
− 2 ⋅ −8
rewrite in terms of i:
− 2 ⋅ −8 =
( 2 i )⋅ ( 8 i )
Then use the product rule for radicals:
− 2 ⋅ −8 =
( 2 i )⋅ ( 8 i ) =
−6⋅ 8 =
3i(4 + 5i) =
(3 + 2i)(4 + i) =
(2 + 5i)2 =
( )
2 ⋅8 i2 =
− 16 = −4
The product of a complex number and
its conjugate is a real number.
The conjugate of a complex number:
Examples:
The conjugate of a + bi is a – bi .
(1 – 5i)(1 + 5i) =
Examples: Find the conjugate:
(6 + 2i)
(2 – 3i)
(-5 + 9i)(-5 – 9i) =
Difference of squares.
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Division of complex numbers:
There are two different techniques.
To divide complex numbers numbers,
you need to rewrite the problem so you
have a real number (no i) in the
denominator.
Technique 1. If the denominator is a
multiple of i only. Then eliminate the i in
the denominator by multiplying
numerator and denominator by i.
Example:
(5 + 4i ) = (5 + 4i ) ⋅ i = (5i + 4i 2 ) =
Example:
6i
6i
i
6i 2
(5i − 4) = − 1 (5i − 4) = − 5 i + 4 =
−6
6
6 6
5 2
− i+
6 3
8
4i
Example:
Technique 2. The denominator is in
the form a + bi. Then multiply the
numerator and denominator by the
conjugate of the denominator.
(5 + 4i ) = (5 + 4i ) ⋅ (3 − 6i ) =
(3 + 6i ) (3 + 6i ) (3 − 6i )
15 − 30i + 12i − 24i 2
=
9 − 36i 2
(15 + 24) + (− 30 + 12)i = 1 (39 − 18i ) =
9 + 36
45
13 2
− i
15 5
3
Powers of i:
Examples:
i0 = 1, by definition
4
=
2 − 3i
i1=i
i2 = -1, by definition
8−i
=
1 − 2i
i3 = i(i2) = -i
i4 = (i2)2 = (-1)2 = 1
and then this pattern repeats itself.
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