1
00:00:00,000 --> 00:00:01,733
So...

2
00:00:01,733 --> 00:00:04,566
complex exponential function, this 
was another thing we talked about.

3
00:00:04,566 --> 00:00:06,233
We talked about this new notation,

4
00:00:06,233 --> 00:00:09,800
so instead of using cis theta, instead 
of using cosine theta plus i sine theta,

5
00:00:09,800 --> 00:00:12,666
we can use an exponential 
notation, e to the i theta.

6
00:00:12,666 --> 00:00:16,366
This is extremely useful and widely used.

7
00:00:16,366 --> 00:00:18,000


8
00:00:18,000 --> 00:00:21,166
Here we're defining it, but we can 
actually show there are reasons

9
00:00:21,166 --> 00:00:24,366
why this notation is useful 
and actually is used,

10
00:00:24,366 --> 00:00:26,200


11
00:00:26,200 --> 00:00:29,466
and the reasons here - I'm going 
to give you 3 reasons here.

12
00:00:29,466 --> 00:00:32,600
The first reason is that exponential 
laws work. This is just…

13
00:00:32,600 --> 00:00:36,000
the fact that exponential laws 
work, that's just the statement of

14
00:00:36,000 --> 00:00:38,433
Polar Multiplication of Complex Numbers.

15
00:00:38,433 --> 00:00:39,366


16
00:00:39,366 --> 00:00:41,666
Derivative with respect to theta
makes sense, so I'll leave

17
00:00:41,666 --> 00:00:44,000
this again as an exercise for you 
to check and you should do it.

18
00:00:44,000 --> 00:00:46,300
If you take the derivative
with respect to theta

19
00:00:46,300 --> 00:00:49,166
of this side, treating i as 
just some constant number

20
00:00:49,166 --> 00:00:50,666
with respect to theta,

21
00:00:50,666 --> 00:00:54,566
and you take the derivative of this side, 
again treating i as just some number

22
00:00:54,566 --> 00:00:57,533
with respect to theta, you're 
going to get the same answers.

23
00:00:57,533 --> 00:00:58,200


24
00:00:58,200 --> 00:01:00,900
So the derivatives makes sense, they agree.

25
00:01:00,900 --> 00:01:03,566
More importantly, the power series agrees.

26
00:01:03,566 --> 00:01:06,266
So if I take the power 
series of e to the i theta,

27
00:01:06,266 --> 00:01:09,300
I take the power series of cosine theta, 
I take the power series of sine theta,

28
00:01:09,300 --> 00:01:11,200
you see if they actually match up

29
00:01:11,200 --> 00:01:12,933
This is in the sense 
of complex analysis,

30
00:01:12,933 --> 00:01:14,666
so this is something 
you haven't seen yet.

31
00:01:14,666 --> 00:01:18,600
But something that you should do, when 
you learn this in calculus, is actually

32
00:01:18,600 --> 00:01:20,633
check out the 3 power series here,

33
00:01:20,633 --> 00:01:23,066
and see if you can show 
that this actually works.

34
00:01:23,066 --> 00:01:25,966
So take the power series for e 
to the x, and I plug in i theta.

35
00:01:25,966 --> 00:01:28,166
Take the power series for 
cosine of x and plug in [theta].

36
00:01:28,166 --> 00:01:32,000
I take the power series for sine of 
x, plug in theta and multiply by i,

37
00:01:32,000 --> 00:01:35,733
do the arithmetic, you're going to see 
that these 2 power series actually agree.

38
00:01:35,733 --> 00:01:39,800
There's very good reasons for defining 
this notation, it's not just cosmetic.

39
00:01:39,800 --> 00:01:42,400


40
00:01:42,400 --> 00:01:44,800
With this week, we went to the
Complex nth Roots Theorem.

41
00:01:44,800 --> 00:01:47,933
This is sort of one of the big 
theorems we wanted to get to.

42
00:01:47,933 --> 00:01:52,000
Every complex number has
exactly n distinct nth roots.

43
00:01:52,000 --> 00:01:52,900


44
00:01:52,900 --> 00:01:55,200
So if I have some…

45
00:01:55,200 --> 00:01:55,633


46
00:01:55,633 --> 00:01:58,500
which I guess we're going to
talk about in a minute, okay,

47
00:01:58,500 --> 00:02:01,100
so if I have some z to 
the n is equal to a,

48
00:02:01,100 --> 00:02:03,833
then there are exactly n 
roots to this equation, okay,

49
00:02:03,833 --> 00:02:06,200
where a is some complex number here.

50
00:02:06,200 --> 00:02:07,100


51
00:02:07,100 --> 00:02:11,100
The roots lie in a circle 
of radius the size of z,

52
00:02:11,100 --> 00:02:15,133
centered at the origin, and they're spaced 
out evenly by angles of 2 pi over n.

53
00:02:15,133 --> 00:02:16,933
This is a really, really 
powerful theorem, right?

54
00:02:16,933 --> 00:02:19,233
So it tells you a little bit about 
the geometry of such solutions,

55
00:02:19,233 --> 00:02:22,033
which is really neat. We'll see 
this in an example soon.

56
00:02:22,033 --> 00:02:25,833
So concretely, if I have some complex 
number, a equals r to the e to the

57
00:02:25,833 --> 00:02:28,200
i theta - r e to the i theta,

58
00:02:28,200 --> 00:02:31,466
then the solution to z to the n 
are equal to a are given by

59
00:02:31,466 --> 00:02:34,033
z is equal to the nth root of r 

60
00:02:34,033 --> 00:02:36,533
times e to the power of i times

61
00:02:36,533 --> 00:02:39,266
theta plus 2 pi k, all divided by n,

62
00:02:39,266 --> 00:02:42,033
and k ranges from 0 to n minus 1.

63
00:02:42,033 --> 00:02:44,400
So this gives you a formulaic 
approach for doing it,

64
00:02:44,400 --> 00:02:48,933
and this gives you a geometric visualization 
of what the solutions should look like.

65
00:02:48,933 --> 00:02:50,733


66
00:02:50,733 --> 00:02:52,866
This is called the Complex 
nth Roots Theorem.

67
00:02:52,866 --> 00:02:53,633


68
00:02:53,633 --> 00:02:56,233
So what is it saying? What 
is this really saying too?

69
00:02:56,233 --> 00:02:57,766
This is actually says a lot, okay?

70
00:02:57,766 --> 00:03:01,433
This is once we have one solution, 
we can find all the other ones.

71
00:03:01,433 --> 00:03:02,166


72
00:03:02,166 --> 00:03:05,066
So if you know that a is 
equal to r e to the i theta, 

73
00:03:05,066 --> 00:03:08,000
then we know that all the
solutions are given by this, right,

74
00:03:08,000 --> 00:03:10,933
so in a sense, you started 
with our first solution,

75
00:03:10,933 --> 00:03:16,233
the nth root of r, e to the i theta over n.

76
00:03:16,233 --> 00:03:19,066
That's our first solution, 
right, taking the nth root...

77
00:03:19,066 --> 00:03:21,166
taking the nth roots of a is equal to

78
00:03:21,166 --> 00:03:24,000
r e to the i theta, gives us

79
00:03:24,000 --> 00:03:25,833
the case when k is 0,

80
00:03:25,833 --> 00:03:29,233
and then all the other cases
you just add multiples of

81
00:03:29,233 --> 00:03:32,000
2 pi k over n, where k 
ranges from 1 to n minus 1.

82
00:03:32,000 --> 00:03:34,000
That's pretty neat.

83
00:03:34,000 --> 00:03:35,800


84
00:03:35,800 --> 00:03:39,100
A lot easier than in the real case. Okay.

85
00:03:39,100 --> 00:03:41,900


86
00:03:41,900 --> 00:03:43,800
Definition, one more definition 
since we’re on this page,

87
00:03:43,800 --> 00:03:48,233
an nth root of unity is a complex number 
that satisfies z to the n is equal to 1.

88
00:03:48,233 --> 00:03:52,600
So before when we had z cubed 
equals 1, we only had one solution.

89
00:03:52,600 --> 00:03:55,033
But now, if we consider z as being - 

90
00:03:55,033 --> 00:03:58,733
so that was over the real case, and now 
if we consider z to be a complex variable,

91
00:03:58,733 --> 00:04:02,900
then z cubed equals 1 actually has 
3 solutions according to this theorem.

92
00:04:02,900 --> 00:04:04,000


93
00:04:04,000 --> 00:04:06,100
And that is something that I will 
leave for you as an exercise.

94
00:04:06,100 --> 00:04:09,933
So you can actually find all 3 solutions, 
right? The first one is z equals 1,

95
00:04:09,933 --> 00:04:12,866
so if you factor that out of z 
cubed minus 1, you're going to

96
00:04:12,866 --> 00:04:16,100
be left with a polynomial 
z squared plus z plus 1 

97
00:04:16,100 --> 00:04:17,966
and then you can just 
find the roots of that

98
00:04:17,966 --> 00:04:20,166
and those are going to be the 
other 2 complex roots of unity…

99
00:04:20,166 --> 00:04:21,000


100
00:04:21,000 --> 00:04:23,100
or complex third roots of unity.

101
00:04:23,100 --> 00:04:24,266


102
00:04:24,266 --> 00:04:27,100
Sometimes these are denoted by zeta n.

103
00:04:27,100 --> 00:04:28,400


104
00:04:28,400 --> 00:04:30,533
We won't do it, we won't talk 
too much about roots of unity 

105
00:04:30,533 --> 00:04:33,266
but they are a very, very 
important part of mathematics.

106
00:04:33,266 --> 00:04:35,700


107
00:04:35,700 --> 00:04:39,666
Roots of unity example: so find all
eight roots of unity in standard form

108
00:04:39,666 --> 00:04:42,466
So again this is a good 
exercise to practice.

109
00:04:42,466 --> 00:04:46,300
Pause the video, try it out, see if you can 
find them based on the previous theorem.

110
00:04:46,300 --> 00:04:49,566
See if you can do it without looking at the 
theorem, and if you can’t, check out the theorem.

111
00:04:49,566 --> 00:04:52,133
Hopefully you’ve paused 
the video and come back

112
00:04:52,133 --> 00:04:54,366
with an answer, that'd be good.

113
00:04:54,366 --> 00:04:56,866
Let's check out the solution. 
So first I'm going to

114
00:04:56,866 --> 00:04:59,333
do this algebraically, and then 
I'm going to do this pictorially.

115
00:04:59,333 --> 00:05:00,733


116
00:05:00,733 --> 00:05:05,433
So we already know a solution to this. 
We know that z is equal to e to the…

117
00:05:05,433 --> 00:05:08,000
so we already know that 
z equals 1 is a solution,

118
00:05:08,000 --> 00:05:08,733


119
00:05:08,733 --> 00:05:12,000
and from the previous theorem, we know 
that all other solutions are given by - 

120
00:05:12,000 --> 00:05:16,200
well this is 2 pi i k over
8 for k from 0 to 7, right?

121
00:05:16,200 --> 00:05:18,200
Our first solution was

122
00:05:18,200 --> 00:05:20,233
r equals 1 and theta equals 0

123
00:05:20,233 --> 00:05:20,766


124
00:05:20,766 --> 00:05:23,566
and so the nth root of 1 is 1,

125
00:05:23,566 --> 00:05:25,500
and so once we have that,

126
00:05:25,500 --> 00:05:28,000
we plug it in, this is what 
the previous theorem says.

127
00:05:28,000 --> 00:05:31,066


128
00:05:31,066 --> 00:05:32,600
Okay?

129
00:05:32,600 --> 00:05:35,000


130
00:05:35,000 --> 00:05:36,900
And all of these values give solutions, 

131
00:05:36,900 --> 00:05:40,500
so what are they? We can actually expand 
them out, so we can write them using the

132
00:05:40,500 --> 00:05:42,833
cosine plus i sine notation. 

133
00:05:42,833 --> 00:05:45,400
We see that our solutions are 1…

134
00:05:45,400 --> 00:05:46,233


135
00:05:46,233 --> 00:05:48,733
oh I got these backwards. Oops.

136
00:05:48,733 --> 00:05:52,000
Root 2 over 2, plus root 2 over 2i,

137
00:05:52,000 --> 00:05:55,766
and the third one is i, the next 
one is minus root 2 over 2,

138
00:05:55,766 --> 00:05:57,766
plus root 2 over 2i, 
I can't stop smiling,

139
00:05:57,766 --> 00:06:00,266
I've goofed the fractions I 
don't know how I did that,

140
00:06:00,266 --> 00:06:04,166
and so on and so forth. I'll 
fix them in the PDF file.

141
00:06:04,166 --> 00:06:05,933


142
00:06:05,933 --> 00:06:09,800
So okay, so here we go. So if we actually just do 
these computations, this is what we're going to get,

143
00:06:09,800 --> 00:06:10,466


144
00:06:10,466 --> 00:06:13,700
and if we look, we can draw a 
diagram based on these solutions,

145
00:06:13,700 --> 00:06:15,633
and the diagram that
we're going to get

146
00:06:15,633 --> 00:06:18,466
is going something like this. We can also 
have solved this pictorially as well, right?

147
00:06:18,466 --> 00:06:21,733
Remember that we know that once 
we have one solution, in this case 1,

148
00:06:21,733 --> 00:06:26,333
we can find all other solutions 
by rotations of pi over 4,

149
00:06:26,333 --> 00:06:28,366
or in other words, 2 pi over 8.

150
00:06:28,366 --> 00:06:29,666


151
00:06:29,666 --> 00:06:32,433
So we have our solution 
at 1, we rotate by

152
00:06:32,433 --> 00:06:36,333
45 degrees, or pi over 4, we're going 
to get root over 2, plus root 2 over 2i,

153
00:06:36,333 --> 00:06:37,900
then rotate by another 45 degrees, and

154
00:06:37,900 --> 00:06:40,666
another, and another, and another, and
another, and another, and so on and so forth. 

155
00:06:40,666 --> 00:06:42,233


156
00:06:42,233 --> 00:06:45,633
I want to talk a little bit about this diagram
because this diagram is actually pretty cool.

157
00:06:45,633 --> 00:06:48,033
So we have 8th roots of unity, okay?

158
00:06:48,033 --> 00:06:50,766
8 is an even number, that's not surprising, 

159
00:06:50,766 --> 00:06:53,233
but why is that important 
for this little diagram here?

160
00:06:53,233 --> 00:06:55,633
So whenever I had a solution i, or

161
00:06:55,633 --> 00:06:58,466
root 2 over 2 plus
root 2 over 2 times i,

162
00:06:58,466 --> 00:07:00,766
the negation of it
was also a solution.

163
00:07:00,766 --> 00:07:02,300
That's because

164
00:07:02,300 --> 00:07:06,200
if I had z to the 8 is equal to 1, 
then negative z to the 8 equals 1.

165
00:07:06,200 --> 00:07:07,000


166
00:07:07,000 --> 00:07:09,166
So here we have the 
twofold symmetry, okay?

167
00:07:09,166 --> 00:07:12,733
So all of these solutions correspond 
in pairs on the opposite side.

168
00:07:12,733 --> 00:07:15,733
So 1 corresponds to minus 1, the 
solution in the second quadrant

169
00:07:15,733 --> 00:07:17,933
corresponds to the solution 
in the fourth quadrant,

170
00:07:17,933 --> 00:07:20,666
and so on and so 
forth, that's by negation.

171
00:07:20,666 --> 00:07:24,000
By doing that, that means we only really
needed to look at 4 of the solutions, okay?

172
00:07:24,000 --> 00:07:24,833


173
00:07:24,833 --> 00:07:27,300
Another thing to note, z to the 8

174
00:07:27,300 --> 00:07:29,700
equals 1, so if z is a solution,

175
00:07:29,700 --> 00:07:31,700
so z to the 8 is equal to 1, then 

176
00:07:31,700 --> 00:07:35,233
z bar to the 8 is also equal to 1.

177
00:07:35,233 --> 00:07:36,133


178
00:07:36,133 --> 00:07:40,166
So I have an answer z, then 
z bar is also a solution,

179
00:07:40,166 --> 00:07:43,200
and that's not too hard to see either. We 
can pull out the conjugates and that's -

180
00:07:43,200 --> 00:07:48,000
the real key to that claim is 
that 1 is a real number.

181
00:07:48,000 --> 00:07:51,400
So because 1 is a real number, then if z is 
a the solution to z to the 8 is equal to 1,

182
00:07:51,400 --> 00:07:53,533
then z bar is also a solution.

183
00:07:53,533 --> 00:07:55,566
So that actually gives us a
little bit of twofold symmetry,

184
00:07:55,566 --> 00:07:58,466
so we know that i 
corresponds to negative i, and

185
00:07:58,466 --> 00:08:01,066
this solution corresponds 
to this solution, and

186
00:08:01,066 --> 00:08:02,633
1 is 1.

187
00:08:02,633 --> 00:08:03,233


188
00:08:03,233 --> 00:08:07,366
So it's actually pretty cool. You could actually
just compute the solutions in the first quadrant,

189
00:08:07,366 --> 00:08:10,033
and along the edge 
of the first quadrant,

190
00:08:10,033 --> 00:08:13,466
and you get all the solutions by just 
using a little bit of mirror symmetry.

191
00:08:13,466 --> 00:08:15,733
That's pretty neat stuff, 
I think it's pretty cool.

192
00:08:15,733 --> 00:08:19,666


193
00:08:19,666 --> 00:08:20,833
Okay.

194
00:08:20,833 --> 00:08:21,333


195
00:08:21,333 --> 00:08:24,966
Watch out it's a trap, what 
do I mean by this? So solve

196
00:08:24,966 --> 00:08:28,333
z to the 5 is equal to 
minus 16 times z bar. 

197
00:08:28,333 --> 00:08:30,566
I killed the punch line unfortunately.

198
00:08:30,566 --> 00:08:32,366
So what do I mean by it’s a trap?

199
00:08:32,366 --> 00:08:34,166
So you might believe, when 
you look at this, oh well

200
00:08:34,166 --> 00:08:36,000
it's z to the 5 so it 
probably has 5 solutions,

201
00:08:36,000 --> 00:08:38,000
but it actually has 7 solutions.

202
00:08:38,000 --> 00:08:40,266
So this is much different than

203
00:08:40,266 --> 00:08:43,300
when you're dealing with a polynomial. So 
this thing, as we'll see in a couple minutes,

204
00:08:43,300 --> 00:08:46,400
is not a polynomial and that's the
problem that we're having here.

205
00:08:46,400 --> 00:08:47,000


206
00:08:47,000 --> 00:08:51,233
If it was a polynomial, then it would 
only have at most 5 unique solutions.

207
00:08:51,233 --> 00:08:52,266


208
00:08:52,266 --> 00:08:54,566
But here, since there's a z bar thing,

209
00:08:54,566 --> 00:08:56,666
it's not a polynomial anymore

210
00:08:56,666 --> 00:08:58,800
and that might cause it 
to have more solutions.

211
00:08:58,800 --> 00:09:02,700
So you can get some weird things 
happening in Complex Number Land.

212
00:09:02,700 --> 00:09:03,366


213
00:09:03,366 --> 00:09:05,633
And sometimes we might have 
no solutions. It’s possible for this

214
00:09:05,633 --> 00:09:08,500
equation to not be satisfied. For example…

215
00:09:08,500 --> 00:09:12,466


216
00:09:12,466 --> 00:09:15,033
you could come up with examples.

217
00:09:15,033 --> 00:09:18,566
Like take like z plus z bar,

218
00:09:18,566 --> 00:09:21,133
right, z plus z bar is
always going to be real,

219
00:09:21,133 --> 00:09:24,733
so if you had z plus z bar equals i, 
there's no solutions, for example

220
00:09:24,733 --> 00:09:28,333


221
00:09:28,333 --> 00:09:32,100
So how do we prove this? How 
do we show there's 7 solutions?

222
00:09:32,100 --> 00:09:35,100
So one thing we note is 
that 0 is clearly solution, so

223
00:09:35,100 --> 00:09:38,333
so we can discount that and we're just 
looking for 6 of the solutions, okay?

224
00:09:38,333 --> 00:09:39,266


225
00:09:39,266 --> 00:09:41,066
So if we do that -

226
00:09:41,066 --> 00:09:44,300
so we're going to assume that now z 
is a non-zero solution to this equation.

227
00:09:44,300 --> 00:09:48,133
If we take modulus on 
both sides and simplify,

228
00:09:48,133 --> 00:09:50,600
then we're going to 
see that the size of z

229
00:09:50,600 --> 00:09:52,900
to the 4 is 16, hence the size of z is 2.

230
00:09:52,900 --> 00:09:55,466
So if you’re solving an equation that you 
don't know what to do, sometimes it helps

231
00:09:55,466 --> 00:09:57,833
to look at the length of the complex number.

232
00:09:57,833 --> 00:10:00,100
You can also put this 
into polar notation and

233
00:10:00,100 --> 00:10:03,600
look at the r-value that's fine too. I like 
just taking the modulus on both sides.

234
00:10:03,600 --> 00:10:07,133
I think that's a much easier 
thing to do and understand.

235
00:10:07,133 --> 00:10:08,566


236
00:10:08,566 --> 00:10:11,233
Now the clever thing to do 
is now, okay, so since z

237
00:10:11,233 --> 00:10:13,033
is non-zero, I've already
discounted z equals 0,

238
00:10:13,033 --> 00:10:16,700
we're going to multiply this original 
equation by z on either side,

239
00:10:16,700 --> 00:10:19,866
and we're going to note that the right-hand 
side is z z bar, which is the size of z squared.

240
00:10:19,866 --> 00:10:22,366
So now we get z to 
the 6 is equal to…

241
00:10:22,366 --> 00:10:24,666
what is it …negative 64.

242
00:10:24,666 --> 00:10:25,466


243
00:10:25,466 --> 00:10:27,100
And that we know how to solve.

244
00:10:27,100 --> 00:10:30,533
z to the 6 is equal to minus 
64, that was by CNRT the

245
00:10:30,533 --> 00:10:32,433
Complex nth Roots Theorem,

246
00:10:32,433 --> 00:10:34,400
tells us that there must be 6 solutions.

247
00:10:34,400 --> 00:10:37,300
So we got 1 from 0, and 
then 6 from doing this

248
00:10:37,300 --> 00:10:38,600


249
00:10:38,600 --> 00:10:40,233
this operation.

250
00:10:40,233 --> 00:10:41,966


251
00:10:41,966 --> 00:10:44,400
So something to watch out for, 
alright? So you can write down

252
00:10:44,400 --> 00:10:48,233
equations with complex 
things and not get a solution. 

253
00:10:48,233 --> 00:10:51,400
You can't write down a polynomial in the
complex numbers not get a solution, 

254
00:10:51,400 --> 00:10:53,300
but that we'll talk about later,

255
00:10:53,300 --> 00:10:57,533
but if you mix polynomials with complex 
conjugation, weird things can happen.