1
00:00:00,740 --> 00:00:05,390
>> Let's take a look at how we do
adding, subtracting, multiplying,

2
00:00:05,390 --> 00:00:07,650
and dividing complex numbers and see

3
00:00:07,650 --> 00:00:10,260
how different forms of
the complex number lends

4
00:00:10,260 --> 00:00:15,600
themselves to easier algebraic
or arithmetic computations.

5
00:00:15,600 --> 00:00:17,820
We have two different
complex numbers here Z1,

6
00:00:17,820 --> 00:00:21,270
which is equal to four plus J3
in rectangular coordinates.

7
00:00:21,270 --> 00:00:22,785
In its polar form,

8
00:00:22,785 --> 00:00:26,780
Z1 is also equal to
five angle 37.87 degrees.

9
00:00:26,780 --> 00:00:31,580
Of course, in it's complex exponential
form which is still polar,

10
00:00:31,580 --> 00:00:34,460
it still has the polar values magnitude

11
00:00:34,460 --> 00:00:40,820
five angle e to the j angle 37.87 degrees.

12
00:00:40,820 --> 00:00:46,350
Z2 equal to 2 minus J5,

13
00:00:46,350 --> 00:00:50,895
in polar form it's that and it's
complex exponential form is that.

14
00:00:50,895 --> 00:00:53,580
Let's just plot these out right here,

15
00:00:53,580 --> 00:00:56,555
tag it in just so that we
can see what we have here.

16
00:00:56,555 --> 00:01:00,110
In the complex plane we're
going over 4 and up 3.

17
00:01:00,110 --> 00:01:02,370
So it's about like that.

18
00:01:02,750 --> 00:01:09,275
On this one, we're coming
over two and going down five.

19
00:01:09,275 --> 00:01:17,700
So maybe it's down here something
like that for Z2 and Z1. All right.

20
00:01:17,700 --> 00:01:20,670
Let's start this off by
just adding Z1 plus Z2.

21
00:01:20,670 --> 00:01:25,550
Z1 plus Z2, addition is most easily
done in rectangular coordinates.

22
00:01:25,550 --> 00:01:29,225
Because in that you simply add
the real parts and the imaginary parts.

23
00:01:29,225 --> 00:01:34,710
So Z1 plus Z2 will be 4 plus J3 plus Z2,

24
00:01:34,710 --> 00:01:37,860
which is 2 minus J5.

25
00:01:37,860 --> 00:01:41,280
When you do that, you get, let's see,

26
00:01:41,280 --> 00:01:49,605
4 plus 2 is 6 and J3
minus J5 is a minus J2.

27
00:01:49,605 --> 00:01:52,110
Pretty easy. All right.

28
00:01:52,110 --> 00:01:53,325
How about multiplying.

29
00:01:53,325 --> 00:01:57,165
Let's do Z1 times Z2.

30
00:01:57,165 --> 00:01:59,935
Well, before we go on, let me
just say that multiplying or

31
00:01:59,935 --> 00:02:04,545
adding in polar form really isn't very,

32
00:02:04,545 --> 00:02:09,245
polar form isn't conducive to adding
unless you want to do it graphically.

33
00:02:09,245 --> 00:02:15,600
Of course, adding graphically you take
Z1 and Z2 and you tip to tail them.

34
00:02:15,600 --> 00:02:17,520
So if that's Z2,

35
00:02:17,520 --> 00:02:25,165
then you'd add in Z1 like that and the
resulting complex number would be that.

36
00:02:25,165 --> 00:02:28,145
Generally speaking, we
don't add in polar form

37
00:02:28,145 --> 00:02:31,580
unless you're using your calculator
and your calculator can do all things.

38
00:02:31,580 --> 00:02:36,125
So now, let's look at
multiplying Z1 times Z2.

39
00:02:36,125 --> 00:02:39,450
We get then Z1 is

40
00:02:39,450 --> 00:02:48,885
4 plus J3 times Z2 which is 2 minus J5.

41
00:02:48,885 --> 00:02:52,670
We see here that we have
two binomials multiplied together.

42
00:02:52,670 --> 00:02:54,335
That means we need to foil them.

43
00:02:54,335 --> 00:02:56,900
Distribute both of these terms
to both of those terms.

44
00:02:56,900 --> 00:03:02,275
When you do that you get 4 times 2 is 8,

45
00:03:02,275 --> 00:03:07,840
4 times a minus J5 is a minus J20.

46
00:03:08,270 --> 00:03:13,590
J3 times 2 is a plus J6.

47
00:03:13,590 --> 00:03:18,420
J3 times minus J5,

48
00:03:18,420 --> 00:03:19,560
we've go to be careful here.

49
00:03:19,560 --> 00:03:22,035
3 times 5 is 15.

50
00:03:22,035 --> 00:03:26,840
But we've go to be careful with
the sign plus times a minus is minus.

51
00:03:26,840 --> 00:03:30,725
But J times J is a negative one.

52
00:03:30,725 --> 00:03:34,775
So we have a minus times
minus becomes plus.

53
00:03:34,775 --> 00:03:36,920
So we have two imaginary terms,

54
00:03:36,920 --> 00:03:42,090
two real terms.8 plus 15 is 23.

55
00:03:42,560 --> 00:03:50,830
The negative J20 plus J6 is a minus J14.

56
00:03:52,390 --> 00:03:57,560
So multiplying two complex numbers
in rectangular form requires

57
00:03:57,560 --> 00:04:02,930
us to foil them.

58
00:04:02,930 --> 00:04:05,420
I'll leave it to you and you stop

59
00:04:05,420 --> 00:04:07,715
the video right here and
go ahead and convert

60
00:04:07,715 --> 00:04:13,115
this rectangular complex number
into its polar form.

61
00:04:13,115 --> 00:04:23,370
Let me just give the answer. The answer
is 26.93 angle negative 31.33.

62
00:04:24,220 --> 00:04:30,680
All right. Now, let's multiply Z1 times Z2.

63
00:04:30,680 --> 00:04:35,830
Only doing it in polar coordinates
using our polar form of these.

64
00:04:35,830 --> 00:04:40,925
In polar we have

65
00:04:40,925 --> 00:04:47,675
Z1 times Z2 that's equal to 5.

66
00:04:47,675 --> 00:04:51,130
I'm going to write it using
its complex exponential form.

67
00:04:51,130 --> 00:04:57,840
5e to the J37.87 times

68
00:04:57,840 --> 00:05:05,470
Z2 which is 5.39 angle negative,

69
00:05:05,470 --> 00:05:08,130
I'm doing an exponential form,

70
00:05:08,130 --> 00:05:12,780
e to the negative J68.2.

71
00:05:14,900 --> 00:05:19,310
Of course, these two forms
are equivalent forms.

72
00:05:19,310 --> 00:05:20,750
It's just that it's more obvious what we're

73
00:05:20,750 --> 00:05:23,090
doing when we write them as exponentials.

74
00:05:23,090 --> 00:05:28,610
So when you do that 5 times

75
00:05:28,610 --> 00:05:35,675
5.39 is equal to 26.93.

76
00:05:35,675 --> 00:05:44,025
Now we have e to the J positive 37.887
and we have e to the minus J68.2.

77
00:05:44,025 --> 00:05:46,170
So when we add the exponents,

78
00:05:46,170 --> 00:05:55,260
the negative 68.2 plus the
J37.87 gives us an angle of,

79
00:05:55,260 --> 00:05:57,525
so the minus J angle.

80
00:05:57,525 --> 00:06:00,940
31.33.

81
00:06:01,340 --> 00:06:05,090
What we see is we get the same answer

82
00:06:05,090 --> 00:06:08,900
foiling and then converting to polar as we

83
00:06:08,900 --> 00:06:12,170
did by just starting out in
polar form and multiplying

84
00:06:12,170 --> 00:06:15,710
the coefficients and adding the exponents.

85
00:06:15,710 --> 00:06:18,800
I'm going to leave it to
you to take and convert

86
00:06:18,800 --> 00:06:21,080
this expression here in

87
00:06:21,080 --> 00:06:22,790
polar back to rectangular

88
00:06:22,790 --> 00:06:24,515
and convince yourself that
you get the same thing there.

89
00:06:24,515 --> 00:06:28,405
Go and stop the video and
do that now. All right.

90
00:06:28,405 --> 00:06:30,645
Now, we've got addition,

91
00:06:30,645 --> 00:06:32,005
by the way, I should have mentioned it,

92
00:06:32,005 --> 00:06:34,010
subtraction is just the same
as addition or when you're

93
00:06:34,010 --> 00:06:37,535
subtracting the real parts and then
the subtracting the imaginary parts.

94
00:06:37,535 --> 00:06:42,105
We've now done multiplication in
both rectangular form and in polar form.

95
00:06:42,105 --> 00:06:47,975
Now, let's look at at
dividing two complex numbers.

96
00:06:47,975 --> 00:06:54,695
Let's do Z1 divided by Z2.

97
00:06:54,695 --> 00:06:57,590
You'll recall from your
college algebra that

98
00:06:57,590 --> 00:07:00,665
this gets to be a little bit ugly
and then at least a little involved.

99
00:07:00,665 --> 00:07:04,250
We've got one in doing
this in rectangular form.

100
00:07:04,250 --> 00:07:14,610
We have 4 plus J3 divided by 2 minus J5.

101
00:07:14,610 --> 00:07:17,870
You'll recall that in college algebra,

102
00:07:17,870 --> 00:07:21,980
they taught us to do this division
by multiplying numerator and

103
00:07:21,980 --> 00:07:26,755
denominator by the complex conjugate
of the denominator.

104
00:07:26,755 --> 00:07:29,840
Effectively rationalizing
the denominator so

105
00:07:29,840 --> 00:07:32,780
that we get a pure real number
in the denominator,

106
00:07:32,780 --> 00:07:34,580
and then the numerator
falls wherever it may.

107
00:07:34,580 --> 00:07:40,175
So we're going to multiply
numerator and denominator

108
00:07:40,175 --> 00:07:47,280
by 2 plus J5 over 2 plus J5.

109
00:07:47,280 --> 00:07:52,695
2 plus J5 is the conjugate of 2 minus J5.

110
00:07:52,695 --> 00:07:55,970
Then you can go through
and do the foiling because

111
00:07:55,970 --> 00:07:58,730
we have this complex number times
this complex number,

112
00:07:58,730 --> 00:08:00,140
it's got to be foiled.

113
00:08:00,140 --> 00:08:05,300
I'm going to leave that to you
to show that it turns out to

114
00:08:05,300 --> 00:08:14,295
be negative 7 plus J26 divided by,

115
00:08:14,295 --> 00:08:18,245
now let me show the details down here
just to remind you what happens here.

116
00:08:18,245 --> 00:08:21,575
We've got 2 times 2 is 4.

117
00:08:21,575 --> 00:08:25,185
We have 2 times a positive J5,

118
00:08:25,185 --> 00:08:28,840
that's a positive J10.

119
00:08:29,450 --> 00:08:32,610
Now we have a negative J5 times 2,

120
00:08:32,610 --> 00:08:35,385
that gives me a minus J10.

121
00:08:35,385 --> 00:08:44,085
Then we have a negative
J5 times a positive J5.

122
00:08:44,085 --> 00:08:46,590
Again, we've got to be
careful with the signs here.

123
00:08:46,590 --> 00:08:49,225
5 times 5 is 25.

124
00:08:49,225 --> 00:08:54,020
J times J is a negative 1
minus times a plus is a minus.

125
00:08:54,020 --> 00:08:56,240
So we've got a minus from
the signs a minus from

126
00:08:56,240 --> 00:08:59,645
the J squared that gives us a positive.

127
00:08:59,645 --> 00:09:10,300
So this then turns out to be
negative 7 plus J26 divided by 29,

128
00:09:10,300 --> 00:09:20,880
which works out to be equal
to negative 0.24 plus J0.9.

129
00:09:20,880 --> 00:09:23,870
So the division of Z1 divided by Z2 in

130
00:09:23,870 --> 00:09:29,430
rectangular coordinates gives
us this rectangular coordinate.

131
00:09:30,980 --> 00:09:39,905
Now, hub goodness, let's go ahead and
convert this to polar coordinates.

132
00:09:39,905 --> 00:09:45,485
In polar coordinates, we're going to
have the magnitude of that as equal to

133
00:09:45,485 --> 00:09:52,115
the square root of negative
0.24 squared plus 0.9 squared.

134
00:09:52,115 --> 00:09:56,490
That turns out to be 0.928.

135
00:09:59,110 --> 00:10:01,580
That's the magnitude.

136
00:10:01,580 --> 00:10:05,360
Now, the angle here gets
to be a little bit tricky.

137
00:10:05,360 --> 00:10:11,105
The angle is going to be the arc tangent

138
00:10:11,105 --> 00:10:19,070
of 0.9 divided by negative 0.24.

139
00:10:22,430 --> 00:10:27,920
Now, the arc tangent button on
your calculator returns a value between

140
00:10:27,920 --> 00:10:33,630
plus or minus Pi halves or
plus or minus 90 degrees.

141
00:10:33,690 --> 00:10:36,880
This becomes ambiguous.

142
00:10:36,880 --> 00:10:40,075
To find out what the actual angle is,

143
00:10:40,075 --> 00:10:45,785
you take the arc tangent of
0.9 divided by negative 0.24

144
00:10:45,785 --> 00:10:52,460
and you come up with a negative 75 degrees.

145
00:10:52,460 --> 00:10:54,940
So this turns out to be

146
00:10:54,940 --> 00:11:03,390
0.928 angle negative 75 degrees.

147
00:11:03,390 --> 00:11:05,885
But that's not exactly right.

148
00:11:05,885 --> 00:11:09,835
Coming back here and looking
at our complex number,

149
00:11:09,835 --> 00:11:14,675
we see that we are in
not the fourth quadrant,

150
00:11:14,675 --> 00:11:17,330
negative 75 it run out of room,

151
00:11:17,330 --> 00:11:19,620
let's just draw it down here.

152
00:11:21,740 --> 00:11:26,105
It has this down here
at negative 75 degrees.

153
00:11:26,105 --> 00:11:28,415
But when we look at this number here,

154
00:11:28,415 --> 00:11:30,200
we're at negative 0.24.

155
00:11:30,200 --> 00:11:31,550
The real part is negative,

156
00:11:31,550 --> 00:11:35,270
so we're over here and up 0.9.

157
00:11:35,270 --> 00:11:40,685
In reality where we're at
with this complex number

158
00:11:40,685 --> 00:11:51,135
is 180 degrees off from
this negative 75 degrees.

159
00:11:51,135 --> 00:11:56,840
So the actual angle is
negative 75 plus 180.

160
00:11:56,840 --> 00:12:04,010
This angle right here is
actually 105 degrees.

161
00:12:04,010 --> 00:12:06,320
So when you're doing it in

162
00:12:06,320 --> 00:12:08,300
rectangular coordinates using
your arc tangent button

163
00:12:08,300 --> 00:12:09,380
on your calculator you got be

164
00:12:09,380 --> 00:12:14,260
careful and look at the actual
coordinates that you're working with,

165
00:12:14,260 --> 00:12:17,960
because your arc tangent button
can cause you some grief.

166
00:12:17,960 --> 00:12:24,095
Let's do this calculation
directly from polar form.

167
00:12:24,095 --> 00:12:29,130
We have as Z1 over Z2 is equal to,

168
00:12:29,130 --> 00:12:38,670
in polar form Z1 is 5e to
the J37.87 divided by Z2,

169
00:12:38,670 --> 00:12:49,035
which is 5.39e to the minus J68.2.

170
00:12:49,035 --> 00:12:52,590
Now we can do that directly
and 5 divided by 5.39,

171
00:12:52,590 --> 00:13:00,585
that is infact equal to 0.928e to the,

172
00:13:00,585 --> 00:13:05,010
now the exponent is going to be J,

173
00:13:05,010 --> 00:13:11,535
and we've got 37.87 minus 68.82,

174
00:13:11,535 --> 00:13:15,005
that turns out to be the 105
and some round off error there.

175
00:13:15,005 --> 00:13:20,375
But it's positive J105 degrees.

176
00:13:20,375 --> 00:13:23,275
When you do it this way,

177
00:13:23,275 --> 00:13:26,305
already in the polar form,

178
00:13:26,305 --> 00:13:28,940
the ambiguity that the arc tangent button

179
00:13:28,940 --> 00:13:33,060
introduces is we avoid that ambiguity.