Download Notes-Foerster 10-2

Imaginary and Complex Numbers
Definitions:
1.  Unit Imaginary number
i is a number whose square is -1. That is i2 = -1.
Equivalently, i = !1
2.  Imaginary Numbers in terms of i
If x is a non ! negative real number, then
!x = i x
3.  Complex Number
A complex number is a number of the form a + bi, where the real
number a is called the real part of a + bi, the real number b is
called the imaginary part of a + bi, and i is the unit imaginary
number.
4.  Complex Conjugates
The complex numbers a + bi and a - bi are called complex
conjugates of each other. The only difference between two
binomials that are conjugates is the sign of the second number.
Complex numbers can be plotted on the complex number plane.
The real part is plotted as the abscissa (the x-coordinate in a
Cartesian coordinate plane).
The imaginary part is plotted as the ordinate (the y-coordinate in a
Cartesian coordinate plane).
Since the real-number line and the imaginary-number line both lie in the
complex plane, it can be concluded that the set of real-numbers and
the set of imaginary-numbers are both subsets of the set of
complex numbers.
Since zero lies on both the real-number line and the imaginary-number
line, it can be concluded that zero is both a real number and an
imaginary number.
Imaginary
4i
3
imaginary
units
2i
Real
-4
-2
2
-2i
4
Absolute value is the
distance of a point from
zero.
|2 + 3i|
c2 = a2 + b2
c = a2 + b2
c = 22 + 32
-4i
c = 4+9
2 real
units
c = 13
| 2 + 3 i |= 13
Powers of i:
i = !1
i0 = 1
2
i = !1 = !1
2
i3 = -i
i1 = i
i2 = -1
i 3 = i 2 ! i = !1" i = ! i
i 4 = i 2 ! i 2 = !1(!1) = 1
i5 = i
i = !1
6
i 7 = !i
i8 = 1
!
i177 =
44
4 177
- 16
17
- 16
1
)
i177 = i4(44)+1 = i1 = i
i0 = 1
i3 = -i
i
!177
i1 = i
i2 = -1
=
-44
4 !177
- (-16)
-17
- (-16)
-1
Remainders can not be negative.
i !177 =
-45
4 !177
- (-16)
-17
- (-20)
3
)
)
i-177 = i4(-45)+3 = i3 = -i
i0 = 1
i-1 = -i
i-3 = i
i-2 = -1
If z1 = 7 - 3i and z2 = 2 + 5i, find
1.  z1 + z2
3.  z1z2
(7 - 3i) + (2 + 5i)
(7 - 3i)(2 + 5i)
7 - 3i + 2 + 5i
14 + 35i - 6i - 15i2
9 + 2i
14 + 29i - 15(-1)
14 + 29i + 15
2.  z1 - z2
(7 - 3i) - (2 + 5i)
7 - 3i - 2 - 5 i
5 - 8i
29 + 29i
If z1 = 7 - 3i and z2 = 2 + 5i, find2
4.
z1
z2
7 ! 3i
2 + 5i
7 ! 3i 2 ! 5i
"
2 + 5i 2 ! 5i
14 ! 35 i ! 6 i + 15 i 2
4 ! 10 i + 10 i ! 25 i 2
14 ! 41i + 15(!1)
4 ! 25(!1)
14 ! 41i ! 15
4 + 25
!1! 41i
29
!
1 41
!
i
29 29