1
00:00:00,433 --> 00:00:04,000
Hello everyone. In this 
video, we are going to

2
00:00:04,000 --> 00:00:08,966
to convert a complex number in
standard form into polar coordinates.

3
00:00:08,966 --> 00:00:09,733


4
00:00:09,733 --> 00:00:12,000
So the question here will 
be: using the fact that

5
00:00:12,000 --> 00:00:15,200
tan of pi over 12 is equal to 2
minus the square root of 3,

6
00:00:15,200 --> 00:00:18,466
find the polar coordinates of 
the following complex number,

7
00:00:18,466 --> 00:00:21,300
negative square root of 3 minus

8
00:00:21,300 --> 00:00:24,166
2 times the square root of 3 minus 3 times i.

9
00:00:24,166 --> 00:00:26,166


10
00:00:26,166 --> 00:00:29,800
So the solution, in order to do this, we need to 
compute two facts, right? We need to compute

11
00:00:29,800 --> 00:00:34,133
the angle with respect to the x axis, 
and we need to compute the length

12
00:00:34,133 --> 00:00:36,333
of our vector, the radius,

13
00:00:36,333 --> 00:00:38,600
or the modulus of this complex number.

14
00:00:38,600 --> 00:00:39,766


15
00:00:39,766 --> 00:00:43,100
So that's what we're going to do first. So 
it's usually easier to compute the length

16
00:00:43,100 --> 00:00:46,066
of your complex number, so 
we're going to begin with that. 

17
00:00:46,066 --> 00:00:46,933


18
00:00:46,933 --> 00:00:49,766
Solution, first we compute

19
00:00:49,766 --> 00:00:54,600
the length of the given complex number.

20
00:00:54,600 --> 00:00:56,000


21
00:00:56,000 --> 00:00:58,533


22
00:00:58,533 --> 00:01:00,533
So how do we do that?

23
00:01:00,533 --> 00:01:03,066
Well we're going to just take the 
modulus, and we're going to set this

24
00:01:03,066 --> 00:01:06,033
to equal r, okay, r for radius.

25
00:01:06,033 --> 00:01:08,433


26
00:01:08,433 --> 00:01:10,833
Now what

27
00:01:10,833 --> 00:01:16,000
is the length of this? Well we're going to take 
negative the square root of 3 and square it,

28
00:01:16,000 --> 00:01:19,500


29
00:01:19,500 --> 00:01:21,733
and we're going to add to that

30
00:01:21,733 --> 00:01:22,333


31
00:01:22,333 --> 00:01:23,900
negative

32
00:01:23,900 --> 00:01:26,666
2 times the square root of 3,

33
00:01:26,666 --> 00:01:27,500


34
00:01:27,500 --> 00:01:29,733
minus 3,

35
00:01:29,733 --> 00:01:32,100


36
00:01:32,100 --> 00:01:34,600
all squared.

37
00:01:34,600 --> 00:01:36,866
Let's see if I can fix that.

38
00:01:36,866 --> 00:01:37,933


39
00:01:37,933 --> 00:01:39,666
There we go.

40
00:01:39,666 --> 00:01:45,000
Great, okay. So we're going to compute 
this number, and simplify it, okay?

41
00:01:45,000 --> 00:01:49,200
So both the negatives are 
going to go away, which is nice.

42
00:01:49,200 --> 00:01:50,866


43
00:01:50,866 --> 00:01:52,566
So we're going to get

44
00:01:52,566 --> 00:01:53,433


45
00:01:53,433 --> 00:01:55,633
3 plus…

46
00:01:55,633 --> 00:01:58,300


47
00:01:58,300 --> 00:02:01,400
so we're just going to expand
the second term out

48
00:02:01,400 --> 00:02:03,733


49
00:02:03,733 --> 00:02:06,566
as well. So let's see what this looks like.

50
00:02:06,566 --> 00:02:08,166


51
00:02:08,166 --> 00:02:12,900
So negative root 3 squared, that's 3.

52
00:02:12,900 --> 00:02:14,833
2 times the square root of 3 if we -

53
00:02:14,833 --> 00:02:17,466
so we're going to square this whole 
thing, we square the first term,

54
00:02:17,466 --> 00:02:20,200
4 times square root of 
3 squared which is 3.

55
00:02:20,200 --> 00:02:22,500
Square the last term, that's the 9,

56
00:02:22,500 --> 00:02:26,600
and then take 2 times the cross term, which 
is negative 12 times the square root of 3.

57
00:02:26,600 --> 00:02:27,566


58
00:02:27,566 --> 00:02:30,466
Now we add up all these terms,

59
00:02:30,466 --> 00:02:31,033


60
00:02:31,033 --> 00:02:33,500
so 12, 9, 4, that’s 24,

61
00:02:33,500 --> 00:02:37,033


62
00:02:37,033 --> 00:02:40,533
minus 12 square root 3.

63
00:02:40,533 --> 00:02:44,000


64
00:02:44,000 --> 00:02:47,466


65
00:02:47,466 --> 00:02:50,500
We should probably factor out the 4,

66
00:02:50,500 --> 00:02:54,800
just to make it look a little bit 
nicer, or a little bit smaller.

67
00:02:54,800 --> 00:02:58,966
2 square root of 8
minus 3 square root of

68
00:02:58,966 --> 00:03:00,566


69
00:03:00,566 --> 00:03:03,066
3.

70
00:03:03,066 --> 00:03:06,366


71
00:03:06,366 --> 00:03:09,566
Now it's always good to do 
a little bit of a sanity check.

72
00:03:09,566 --> 00:03:12,966
Root 3 is less than 2, this 
number is positive so we didn't

73
00:03:12,966 --> 00:03:16,133
make any obvious blunders.

74
00:03:16,133 --> 00:03:18,633


75
00:03:18,633 --> 00:03:23,133
Okay that's great. So that's going to be the
radius of our complex number up here.

76
00:03:23,133 --> 00:03:24,266


77
00:03:24,266 --> 00:03:26,700
Now we want to compute the angle,

78
00:03:26,700 --> 00:03:29,133
and so hopefully we're 
going to need this fact that

79
00:03:29,133 --> 00:03:33,166
tan of pi over 12 is equal to 
2 minus the square root of 3.

80
00:03:33,166 --> 00:03:34,900


81
00:03:34,900 --> 00:03:37,633
I made this problem, so there's some

82
00:03:37,633 --> 00:03:40,766
chance that there's a mistake, but I'm 
hoping that this is going to work out,

83
00:03:40,766 --> 00:03:43,366
we need this fact that we're given, okay?

84
00:03:43,366 --> 00:03:44,833


85
00:03:44,833 --> 00:03:49,133
So that's great. So now with the radius, what 
I'm going to do is I'm going to try to compute

86
00:03:49,133 --> 00:03:50,166


87
00:03:50,166 --> 00:03:52,666
the arctan. 

88
00:03:52,666 --> 00:03:54,100


89
00:03:54,100 --> 00:03:58,400
The…sorry the theta, the theta value, the angle

90
00:03:58,400 --> 00:04:00,400
with respect to the x-axis.

91
00:04:00,400 --> 00:04:03,533
So now, to compute…

92
00:04:03,533 --> 00:04:05,700


93
00:04:05,700 --> 00:04:08,666
to compute theta, the angle…

94
00:04:08,666 --> 00:04:10,966


95
00:04:10,966 --> 00:04:16,600
the angle of rotation with respect to the

96
00:04:16,600 --> 00:04:17,733


97
00:04:17,733 --> 00:04:21,133
x-axis, we

98
00:04:21,133 --> 00:04:24,033
perform...

99
00:04:24,033 --> 00:04:26,400
so we're going to take

100
00:04:26,400 --> 00:04:29,566
the y value and divide it by the x value,

101
00:04:29,566 --> 00:04:32,433
and take the arctan of that.

102
00:04:32,433 --> 00:04:35,000


103
00:04:35,000 --> 00:04:37,900
So let's do that. So we're 
going to take the arc-

104
00:04:37,900 --> 00:04:39,766


105
00:04:39,766 --> 00:04:42,666
-tan of

106
00:04:42,666 --> 00:04:45,100


107
00:04:45,100 --> 00:04:47,733
y minus - or y over x.

108
00:04:47,733 --> 00:04:50,266
So maybe I'll just do this.

109
00:04:50,266 --> 00:04:51,066


110
00:04:51,066 --> 00:04:51,600


111
00:04:51,600 --> 00:04:54,500
So we're going to compute the arctan of y 
over x, that's really what we want to do here,

112
00:04:54,500 --> 00:04:56,633
and I'm going to plug 
in the y and x there.

113
00:04:56,633 --> 00:04:59,933


114
00:04:59,933 --> 00:05:02,566
What are we going to get? 
What are we going to get?

115
00:05:02,566 --> 00:05:03,500


116
00:05:03,500 --> 00:05:05,900
We’re going to get a mess, 
okay, let's see if we can

117
00:05:05,900 --> 00:05:08,966
simplify this mess. 2 
times the square root of

118
00:05:08,966 --> 00:05:11,133
3 minus 3…

119
00:05:11,133 --> 00:05:12,600


120
00:05:12,600 --> 00:05:16,333
well I guess it's all negation, 
so the negative of all of that,

121
00:05:16,333 --> 00:05:19,233


122
00:05:19,233 --> 00:05:21,900
and then this is going to be
negative square root of 3.

123
00:05:21,900 --> 00:05:24,333


124
00:05:24,333 --> 00:05:26,033


125
00:05:26,033 --> 00:05:27,566
Alright.

126
00:05:27,566 --> 00:05:30,866
I think that's all the information 
we need from the question.

127
00:05:30,866 --> 00:05:34,900
So those will cancel, the root 3’s…

128
00:05:34,900 --> 00:05:37,300


129
00:05:37,300 --> 00:05:39,066
you know what? There's there's a 
couple of ways to do this. I'm going to

130
00:05:39,066 --> 00:05:41,266
factor out a root 3 from the top.

131
00:05:41,266 --> 00:05:45,433


132
00:05:45,433 --> 00:05:48,333
So let's see what happens when I do that.

133
00:05:48,333 --> 00:05:49,500


134
00:05:49,500 --> 00:05:51,700
So I'm going to cancel the negatives,

135
00:05:51,700 --> 00:05:53,133


136
00:05:53,133 --> 00:05:58,200
that's easy, and I'm going to factor out a square 
root of 3 from the top, so I'm going to get…

137
00:05:58,200 --> 00:06:00,333


138
00:06:00,333 --> 00:06:01,766


139
00:06:01,766 --> 00:06:04,700
should get this, right? So 
if I factor out a root 3,

140
00:06:04,700 --> 00:06:08,100
the first term becomes 2, and
the second term becomes root 3.

141
00:06:08,100 --> 00:06:10,666
Well, okay I guess I can see that over here.

142
00:06:10,666 --> 00:06:13,500
So I bring this back in, I get 2 root 3 minus 3,

143
00:06:13,500 --> 00:06:16,066
you could also multiply top and 
bottom by the square root of 3. I just

144
00:06:16,066 --> 00:06:18,366
feel like this was a little bit easier.

145
00:06:18,366 --> 00:06:20,000
Okay, now

146
00:06:20,000 --> 00:06:22,633
slightly miraculously, the root 3’s will cancel,

147
00:06:22,633 --> 00:06:23,466


148
00:06:23,466 --> 00:06:25,533
and I'm going to be left with

149
00:06:25,533 --> 00:06:27,733


150
00:06:27,733 --> 00:06:30,166
the following. Let's change the…

151
00:06:30,166 --> 00:06:32,866
let's make it look a little bit nicer now.

152
00:06:32,866 --> 00:06:34,033


153
00:06:34,033 --> 00:06:36,166
And…

154
00:06:36,166 --> 00:06:38,566


155
00:06:38,566 --> 00:06:41,666


156
00:06:41,666 --> 00:06:43,566
okay.

157
00:06:43,566 --> 00:06:45,166


158
00:06:45,166 --> 00:06:49,033
So we cancel out those, we get arctan 
of 2 minus the square root of 3.

159
00:06:49,033 --> 00:06:49,966


160
00:06:49,966 --> 00:06:52,533
2 minus the square root of 3, 
that number should look familiar.

161
00:06:52,533 --> 00:06:53,366


162
00:06:53,366 --> 00:06:55,533
Of course it happened to be the one from

163
00:06:55,533 --> 00:07:00,866
the original question, so arctan of 2 
minus root 3 is going to be pi over 12.

164
00:07:00,866 --> 00:07:01,933


165
00:07:01,933 --> 00:07:05,133
So this value should equal

166
00:07:05,133 --> 00:07:06,733


167
00:07:06,733 --> 00:07:08,866
pi over 12.

168
00:07:08,866 --> 00:07:11,600


169
00:07:11,600 --> 00:07:15,833
Now you have to be a little 
bit careful at this point.

170
00:07:15,833 --> 00:07:20,133
We can't immediately jump the gun and 
conclude that our angle is pi over 12, 

171
00:07:20,133 --> 00:07:22,433
we have to do one last check.

172
00:07:22,433 --> 00:07:23,000


173
00:07:23,000 --> 00:07:25,666
What is that last check? 

174
00:07:25,666 --> 00:07:27,866
Well if I take

175
00:07:27,866 --> 00:07:29,433


176
00:07:29,433 --> 00:07:31,566
tan of pi over 12, where 
does this send me?

177
00:07:31,566 --> 00:07:34,566
This sends me if I look - or if I think
about this in the complex plane,

178
00:07:34,566 --> 00:07:36,900
this sends me into quadrant 1, okay?

179
00:07:36,900 --> 00:07:37,833


180
00:07:37,833 --> 00:07:41,033
Now quadrant 1, now it's fine, 
maybe the answer is in quadrant 1,

181
00:07:41,033 --> 00:07:43,033
but if we look at the actual number here,

182
00:07:43,033 --> 00:07:46,966
the original complex number has a 
negative x value and a negative y value.

183
00:07:46,966 --> 00:07:49,033
So if I…

184
00:07:49,033 --> 00:07:49,833


185
00:07:49,833 --> 00:07:51,900
let me start a new…

186
00:07:51,900 --> 00:07:53,333


187
00:07:53,333 --> 00:07:56,266
let me start a new paint pad
here, or paintbrush here.

188
00:07:56,266 --> 00:07:57,666


189
00:07:57,666 --> 00:08:01,000
Sure. So if we think about 
this in the complex plane,

190
00:08:01,000 --> 00:08:02,800


191
00:08:02,800 --> 00:08:05,900
right, over and over, right,

192
00:08:05,900 --> 00:08:08,333
the number is negative root 3

193
00:08:08,333 --> 00:08:11,700
minus 2 minus root 3, so it's some

194
00:08:11,700 --> 00:08:14,333
number that lives somewhere over here,

195
00:08:14,333 --> 00:08:17,433
right, where this is I'm 
going to call negative root 3

196
00:08:17,433 --> 00:08:21,466


197
00:08:21,466 --> 00:08:25,300
and this is negative…

198
00:08:25,300 --> 00:08:29,433


199
00:08:29,433 --> 00:08:31,466
2 minus root 3.

200
00:08:31,466 --> 00:08:33,100


201
00:08:33,100 --> 00:08:36,333
2 minus

202
00:08:36,333 --> 00:08:38,866
root 3.

203
00:08:38,866 --> 00:08:41,266


204
00:08:41,266 --> 00:08:46,066
But pi over 12, the angle pi 
over 12, is somewhere over here,

205
00:08:46,066 --> 00:08:46,833


206
00:08:46,833 --> 00:08:50,033
right? pi over 12.

207
00:08:50,033 --> 00:08:51,400


208
00:08:51,400 --> 00:08:55,533
It's going to look something over here-ish.

209
00:08:55,533 --> 00:08:59,500
So what's the problem? Did we
break math? I mean, we did say that

210
00:08:59,500 --> 00:09:03,700
arctan of y over x should 
be equal to the angle

211
00:09:03,700 --> 00:09:06,166
that we get in polar
coordinates when we convert.

212
00:09:06,166 --> 00:09:08,666
The problem here is that arctan…

213
00:09:08,666 --> 00:09:13,866
arctan doesn’t…. arctan has a range 
between negative pi over 2 and pi over 2.

214
00:09:13,866 --> 00:09:15,733
It's the easiest way to say this.

215
00:09:15,733 --> 00:09:18,733
So what does that mean? arctan is 
kind of a silly function, right? It doesn't

216
00:09:18,733 --> 00:09:22,133
determine which theta you want,

217
00:09:22,133 --> 00:09:24,900
and why does it not do that? Well

218
00:09:24,900 --> 00:09:28,100
the main reason is that tan is pi periodic.

219
00:09:28,100 --> 00:09:31,133
So because tan is pi periodic, we're going to

220
00:09:31,133 --> 00:09:36,700
have this issue that we either want this point, 
or the point where we add an extra copy of pi,

221
00:09:36,700 --> 00:09:39,000
and the…

222
00:09:39,000 --> 00:09:40,666
the functions don't know this. So

223
00:09:40,666 --> 00:09:43,600
arctan doesn't know that I wanted - I 
didn't want the one in the first [quadrant],

224
00:09:43,600 --> 00:09:44,966
I wanted the one
in the third [quadrant].

225
00:09:44,966 --> 00:09:47,333
arctan always returns

226
00:09:47,333 --> 00:09:48,066


227
00:09:48,066 --> 00:09:50,500
points in the region defined

228
00:09:50,500 --> 00:09:53,400
by, let's spray paint it in

229
00:09:53,400 --> 00:09:54,466


230
00:09:54,466 --> 00:09:56,533
orange.

231
00:09:56,533 --> 00:09:57,366


232
00:09:57,366 --> 00:09:59,700
So it picks up

233
00:09:59,700 --> 00:10:03,266
the points in this region. So
arctan only knows points between -

234
00:10:03,266 --> 00:10:06,633
or angles between negative 
pi over 2 and pi over 2. 

235
00:10:06,633 --> 00:10:09,733
That’s going to be all this stuff
shaded in orange, okay?

236
00:10:09,733 --> 00:10:11,833


237
00:10:11,833 --> 00:10:15,600
So in order to get the point on the other 
side, you have to think do I have to add…

238
00:10:15,600 --> 00:10:17,866
do I have to add an extra copy of pi or not?

239
00:10:17,866 --> 00:10:18,900


240
00:10:18,900 --> 00:10:23,000
Also note that the imaginary axis, arctan
has a lot of problems with because

241
00:10:23,000 --> 00:10:25,333
infinities and things like this, right?

242
00:10:25,333 --> 00:10:26,200


243
00:10:26,200 --> 00:10:29,400
arctan doesn't ever equal a value between

244
00:10:29,400 --> 00:10:32,500
exactly negative pi over 2 and 
pi over 2. So for those values,

245
00:10:32,500 --> 00:10:35,900
you just have to sort of look at them and make 
sure you know which one is which, right?

246
00:10:35,900 --> 00:10:38,666
It's very easy to tell when 
a complex number is on

247
00:10:38,666 --> 00:10:42,800
the complex axis, that's not hard, so you 
don't need to go through all this rigmarole

248
00:10:42,800 --> 00:10:45,233
if you're dealing something like that.

249
00:10:45,233 --> 00:10:50,433
Okay, with that being said, we know now 
that our theta should be not pi over 12, but

250
00:10:50,433 --> 00:10:51,200


251
00:10:51,200 --> 00:10:53,200
[pi] plus pi over 12. So

252
00:10:53,200 --> 00:10:54,433


253
00:10:54,433 --> 00:10:56,933
thus since our

254
00:10:56,933 --> 00:10:59,766
original complex point

255
00:10:59,766 --> 00:11:03,466
is in the third quadrant,

256
00:11:03,466 --> 00:11:07,166
we have that theta is equal to

257
00:11:07,166 --> 00:11:10,100
pi plus pi over 12

258
00:11:10,100 --> 00:11:14,100
which is equal to 13 pi over 12.

259
00:11:14,100 --> 00:11:16,333


260
00:11:16,333 --> 00:11:20,333


261
00:11:20,333 --> 00:11:21,633


262
00:11:21,633 --> 00:11:24,466
Hence, in polar coordinates…

263
00:11:24,466 --> 00:11:27,000


264
00:11:27,000 --> 00:11:30,000
hence in polar coordinates our number is…

265
00:11:30,000 --> 00:11:30,800


266
00:11:30,800 --> 00:11:32,500


267
00:11:32,500 --> 00:11:35,000
r e to the power of

268
00:11:35,000 --> 00:11:36,300


269
00:11:36,300 --> 00:11:39,466
i theta.

270
00:11:39,466 --> 00:11:41,633


271
00:11:41,633 --> 00:11:43,766
So

272
00:11:43,766 --> 00:11:44,666


273
00:11:44,666 --> 00:11:49,933
r e to the i theta, or if you want 
the other way to denote this is r

274
00:11:49,933 --> 00:11:52,233
cos theta

275
00:11:52,233 --> 00:11:55,033
plus i sine theta.

276
00:11:55,033 --> 00:11:57,300


277
00:11:57,300 --> 00:12:00,166
Either way, and then just plug in

278
00:12:00,166 --> 00:12:04,600
the theta that you get, and the 
r that you get, so maybe I'll…

279
00:12:04,600 --> 00:12:06,700


280
00:12:06,700 --> 00:12:09,133
like yeah I'll try to fit it in.

281
00:12:09,133 --> 00:12:10,233


282
00:12:10,233 --> 00:12:13,633
So if I want to plug this in…

283
00:12:13,633 --> 00:12:14,100


284
00:12:14,100 --> 00:12:14,866


285
00:12:14,866 --> 00:12:17,300
Let’s see if I'm going to fit it.

286
00:12:17,300 --> 00:12:19,133
It's a bit tight, but

287
00:12:19,133 --> 00:12:22,300
it will fit. So hence, in polar coordinates we have

288
00:12:22,300 --> 00:12:25,433
the following. Oh I need an extra bracket.

289
00:12:25,433 --> 00:12:26,633


290
00:12:26,633 --> 00:12:29,266
So in polar coordinates, we get this

291
00:12:29,266 --> 00:12:33,933
messy looking expression, but 
nonetheless it's in that expression.

292
00:12:33,933 --> 00:12:36,100
So...

293
00:12:36,100 --> 00:12:38,200


294
00:12:38,200 --> 00:12:41,433
you know what? I'm going to I'm going 
to just make it look a little nicer.

295
00:12:41,433 --> 00:12:42,166


296
00:12:42,166 --> 00:12:45,133
I can't stand when things don't look nice.

297
00:12:45,133 --> 00:12:46,700


298
00:12:46,700 --> 00:12:51,100
Can you tell I'm a mathematician? 
Okay, give me just a second…

299
00:12:51,100 --> 00:12:52,366


300
00:12:52,366 --> 00:12:54,733
and...

301
00:12:54,733 --> 00:12:57,000
voila?

302
00:12:57,000 --> 00:12:58,433


303
00:12:58,433 --> 00:13:00,766
There we go. Okay.

304
00:13:00,766 --> 00:13:01,500


305
00:13:01,500 --> 00:13:05,233
So this complex number in 
standard form is equal to 

306
00:13:05,233 --> 00:13:07,366
this complex number in polar form.

307
00:13:07,366 --> 00:13:08,366


308
00:13:08,366 --> 00:13:11,400
The important message 
to take away from this…

309
00:13:11,400 --> 00:13:12,200


310
00:13:12,200 --> 00:13:15,866
from this exercise is the following:
even when you do arctan of y over x,

311
00:13:15,866 --> 00:13:17,900
you're not guaranteed to 
get the correct answer.

312
00:13:17,900 --> 00:13:20,266
You need to double check your final answer,

313
00:13:20,266 --> 00:13:25,366
and make sure that the angle that you 
get is actually the angle that you want.

314
00:13:25,366 --> 00:13:28,000
Again, I like my angles
between 0 and 2 pi,

315
00:13:28,000 --> 00:13:30,533
so I always try to put my 
angles between 0 and 2 pi,

316
00:13:30,533 --> 00:13:33,033
it's up to you. If you've 
got a negative number,

317
00:13:33,033 --> 00:13:34,766
that's fine,

318
00:13:34,766 --> 00:13:38,466
just make sure that you get
the correct negative angle, okay, 

319
00:13:38,466 --> 00:13:40,800
and that you're not off by a pi factor.

320
00:13:40,800 --> 00:13:42,266


321
00:13:42,266 --> 00:13:43,866
So, great. I hope

322
00:13:43,866 --> 00:13:47,866
this gives you a little bit of insight as to 
converting and some of the pitfalls,

323
00:13:47,866 --> 00:13:49,466


324
00:13:49,466 --> 00:13:53,432
and that's all I have to say. Thank you 
very much for listening and good luck.