1
00:00:02,145 --> 00:00:04,440
The following content is
provided under a Creative

2
00:00:04,440 --> 00:00:05,860
Commons license.

3
00:00:05,860 --> 00:00:08,070
Your support will help
MIT OpenCourseWare

4
00:00:08,070 --> 00:00:12,160
continue to offer high quality
educational resources for free.

5
00:00:12,160 --> 00:00:14,700
To make a donation or to
view additional materials

6
00:00:14,700 --> 00:00:18,660
from hundreds of MIT courses,
visit MIT OpenCourseWare

7
00:00:18,660 --> 00:00:22,370
at ocw.mit.edu.

8
00:00:22,370 --> 00:00:26,520
PROFESSOR: We're going to
look at scientific notation.

9
00:00:29,890 --> 00:00:34,700
When we write very, very big
and very, very small numbers,

10
00:00:34,700 --> 00:00:35,300
it's hard.

11
00:00:35,300 --> 00:00:41,780
We don't want to write a number
like that or a number like

12
00:00:41,780 --> 00:00:50,420
this, 125 million, because
when we're doing multiplication

13
00:00:50,420 --> 00:00:53,270
and division with these numbers,
it's hard because you've got

14
00:00:53,270 --> 00:00:55,250
all those extra 0's.

15
00:00:55,250 --> 00:00:59,060
But in reality, you don't
have any more information

16
00:00:59,060 --> 00:01:02,030
in this number just
because it happens to have

17
00:01:02,030 --> 00:01:03,200
a bunch of zeros in front.

18
00:01:03,200 --> 00:01:05,780
So what we like to do
is to have a shorthand

19
00:01:05,780 --> 00:01:09,710
way of saying a really
small number or a really

20
00:01:09,710 --> 00:01:11,900
large number, and
the way that we

21
00:01:11,900 --> 00:01:14,630
do that is we break down
each of these numbers

22
00:01:14,630 --> 00:01:17,450
into powers of 10.

23
00:01:17,450 --> 00:01:23,070
So I'm going to write
this number, 125 million,

24
00:01:23,070 --> 00:01:27,740
and I'm going to write it as
1.25, which is the smallest

25
00:01:27,740 --> 00:01:31,900
number that I can make out of
this-- take these digits, 1.25.

26
00:01:31,900 --> 00:01:39,472
What would I have to multiply
1.25 to get 125 million?

27
00:01:39,472 --> 00:01:41,140
AUDIENCE: 6.

28
00:01:41,140 --> 00:01:42,640
PROFESSOR: Would I
multiply it by 6?

29
00:01:46,020 --> 00:01:48,934
I would have to multiply
it by 10 to the 6,

30
00:01:48,934 --> 00:01:50,100
but let's just write it out.

31
00:01:50,100 --> 00:02:00,740
1.25 times-- it's going
to be 100 million.

32
00:02:00,740 --> 00:02:02,750
If you have 100
million, and then

33
00:02:02,750 --> 00:02:06,390
you have 1.25 of
100 million, that

34
00:02:06,390 --> 00:02:09,210
means you have 125 million.

35
00:02:15,240 --> 00:02:19,830
So in the same way, let's
take this really small number,

36
00:02:19,830 --> 00:02:23,420
and we'll take the
number itself, 5.

37
00:02:23,420 --> 00:02:26,610
What would we have to
multiply by 5 in order

38
00:02:26,610 --> 00:02:27,660
to get this number?

39
00:02:30,312 --> 00:02:32,020
We have to multiply
by-- go ahead, Nicki.

40
00:02:32,020 --> 00:02:33,145
I saw you doing it.

41
00:02:33,145 --> 00:02:34,540
AUDIENCE: [INAUDIBLE].

42
00:02:38,270 --> 00:02:40,820
PROFESSOR: Well,
it's going to be--

43
00:02:40,820 --> 00:02:42,860
let's make it easier.

44
00:02:42,860 --> 00:02:43,720
Let's do this.

45
00:02:47,750 --> 00:02:52,320
This a 1/10, 1/100, and 1/1,000.

46
00:02:52,320 --> 00:02:56,400
So we have to-- if this
is the number 5/1,000,

47
00:02:56,400 --> 00:03:06,330
we have to multiply 5 by the
number 1/1,000, or 0.001.

48
00:03:06,330 --> 00:03:09,540
So I just take this small
number that has just a 1 in it

49
00:03:09,540 --> 00:03:13,330
and some 0's, multiply
it by a number.

50
00:03:13,330 --> 00:03:17,490
In the same case, same way
I take a small number, 1.25,

51
00:03:17,490 --> 00:03:19,620
multiply it by
something that has

52
00:03:19,620 --> 00:03:24,150
all the information about
how many extra 0s there are,

53
00:03:24,150 --> 00:03:27,450
because all of these
are just placeholders.

54
00:03:27,450 --> 00:03:31,070
So in this case, I
can rewrite this.

55
00:03:31,070 --> 00:03:32,820
Let's rewrite this one first.

56
00:03:32,820 --> 00:03:43,260
1.25 times 100 million is a 1
followed by 1, 2, 3, 4, 5, 6,

57
00:03:43,260 --> 00:03:46,920
7, 8 0s.

58
00:03:46,920 --> 00:03:50,570
So I can write that as
10 to the power of 8.

59
00:03:53,860 --> 00:03:59,270
And just as a review, we'll
do a little review over here.

60
00:03:59,270 --> 00:04:06,030
If you have 10 times 10, that's
the same thing as 10 squared,

61
00:04:06,030 --> 00:04:11,450
just like if you had 5 times
5 that's the same thing as 5

62
00:04:11,450 --> 00:04:14,270
squared.

63
00:04:14,270 --> 00:04:23,530
If I have 10 times 10
times 10, that is 1,000.

64
00:04:23,530 --> 00:04:25,655
10 times 10 is 100.

65
00:04:25,655 --> 00:04:31,850
100 times 10 is 1,000, but
I can rewrite that as 10

66
00:04:31,850 --> 00:04:37,430
to the power of 3, or 1,000.

67
00:04:37,430 --> 00:04:40,340
I just count the one and
then how many 0s afterward--

68
00:04:40,340 --> 00:04:43,490
10 to the third.

69
00:04:43,490 --> 00:04:46,040
Now I want you to
practice with this.

70
00:04:46,040 --> 00:04:48,690
So we'll take this
as much as we can.

71
00:04:48,690 --> 00:04:53,780
So I've represented this number,
125 million, as 1.25 times 10

72
00:04:53,780 --> 00:04:55,700
to the eighth.

73
00:04:55,700 --> 00:04:59,690
Now, if I wanted to represent--
if I wanted to change back,

74
00:04:59,690 --> 00:05:04,880
what I could say is, well, 1.25
and then 10 to the eighth means

75
00:05:04,880 --> 00:05:08,030
multiply by 100 million.

76
00:05:08,030 --> 00:05:12,020
What I can think about
is, if I have 1.25,

77
00:05:12,020 --> 00:05:13,730
if I multiply it by
10 to the eighth,

78
00:05:13,730 --> 00:05:18,230
I'm going to move the decimal
place over eight times,

79
00:05:18,230 --> 00:05:20,090
because moving the
decimal place over

80
00:05:20,090 --> 00:05:22,410
is the same thing as
multiplying by 10.

81
00:05:22,410 --> 00:05:24,860
If you have 5 times
10, that's 50.

82
00:05:24,860 --> 00:05:28,109
So 5, move the decimal
point over, it becomes 50.

83
00:05:28,109 --> 00:05:29,900
Now, hopefully you guys
will have seen this

84
00:05:29,900 --> 00:05:31,251
before a little bit.

85
00:05:31,251 --> 00:05:33,000
We'll give you lots
of chance to practice,

86
00:05:33,000 --> 00:05:35,300
but if that's times 10 to
the eighth, I'd write 1.25--

87
00:05:35,300 --> 00:05:45,050
1, 2, 3, 4, 5, 6, 7, 8--

88
00:05:45,050 --> 00:05:48,840
I've moved the decimal
point over eight places.

89
00:05:48,840 --> 00:05:51,140
And now I get back to
my original number,

90
00:05:51,140 --> 00:05:52,660
but it's hard to work with--

91
00:05:52,660 --> 00:06:00,240
125 million.

92
00:06:00,240 --> 00:06:03,720
Now, this way it works a
little bit differently.

93
00:06:03,720 --> 00:06:09,960
To get 1/1,000, I had
to take the number one,

94
00:06:09,960 --> 00:06:14,850
and I moved the decimal
point over 1, 2, 3 places.

95
00:06:17,850 --> 00:06:26,880
So this number, 0.001,
is written as 1 times 10

96
00:06:26,880 --> 00:06:28,935
to the power of-- instead
of moving to the right,

97
00:06:28,935 --> 00:06:32,205
we're moving to
the left 1, 2, 3--

98
00:06:32,205 --> 00:06:38,130
10 to the power of minus
3, which 1 times any number

99
00:06:38,130 --> 00:06:41,640
is just that number, so
that's really the same as 10

100
00:06:41,640 --> 00:06:43,410
to the minus third.

101
00:06:43,410 --> 00:06:50,970
So then I can rewrite 5
times 10 to the minus third.

102
00:06:50,970 --> 00:06:53,680
If I want to change it
back, I just write five,

103
00:06:53,680 --> 00:06:55,330
and my decimal point is there.

104
00:06:55,330 --> 00:06:57,450
I'm going to move to
the left three spaces--

105
00:06:57,450 --> 00:06:59,070
1, 2, 3.

106
00:07:01,600 --> 00:07:06,965
So I end up back with my
original number, 0.005.

107
00:07:06,965 --> 00:07:08,590
But let me just tell
you a little story

108
00:07:08,590 --> 00:07:14,379
about how you can use
these numbers to calculate.

109
00:07:14,379 --> 00:07:16,420
I may have already told
this story to some of you

110
00:07:16,420 --> 00:07:19,630
during [INAUDIBLE],, but let's
write these numbers down again.

111
00:07:19,630 --> 00:07:23,170
If you won the lottery,
and you won $6 million--

112
00:07:25,740 --> 00:07:37,420
so this is multiplying
and dividing powers of 10.

113
00:07:42,260 --> 00:07:48,680
If you won $6 million,
that's the same thing as--

114
00:07:48,680 --> 00:07:50,870
go ahead and write it
in scientific notation,

115
00:07:50,870 --> 00:07:51,790
as a power of 10.

116
00:08:04,130 --> 00:08:07,480
So go ahead and write
these things down.

117
00:08:07,480 --> 00:08:10,090
So that's 6 times
10 to the sixth.

118
00:08:10,090 --> 00:08:11,680
Again, that's a big number.

119
00:08:11,680 --> 00:08:18,300
Say you had 30 relatives
that you wanted to divide

120
00:08:18,300 --> 00:08:20,621
that money in between.

121
00:08:20,621 --> 00:08:22,620
Is each person going to
get more than $6 million

122
00:08:22,620 --> 00:08:24,557
or less than $6 million
if you divvied it up?

123
00:08:24,557 --> 00:08:25,890
AUDIENCE: Less than [INAUDIBLE].

124
00:08:25,890 --> 00:08:27,556
PROFESSOR: You get
less than $6 million.

125
00:08:27,556 --> 00:08:36,929
So if we took $6 million
divided by 30 relatives--

126
00:08:36,929 --> 00:08:39,090
this is dollars
divided by relatives.

127
00:08:42,479 --> 00:08:43,770
Peter, do we need to keep this?

128
00:08:43,770 --> 00:08:45,131
Is it OK if I get rid of that?

129
00:08:45,131 --> 00:08:46,172
AUDIENCE: Sample drawing.

130
00:08:46,172 --> 00:08:48,005
PROFESSOR: Yeah, this
was the sample drawing

131
00:08:48,005 --> 00:08:52,350
of the measurements
that you made before.

132
00:08:52,350 --> 00:08:54,570
If we saw that and we
were just in fifth grade,

133
00:08:54,570 --> 00:09:01,640
we'd say, OK, 6
million divided by 30.

134
00:09:01,640 --> 00:09:05,100
Now I have to add some more
0's, and move things around,

135
00:09:05,100 --> 00:09:06,930
and multiply, and
blah, blah, blah,

136
00:09:06,930 --> 00:09:09,570
and everybody hates
long division.

137
00:09:09,570 --> 00:09:11,550
It is a fact of life.

138
00:09:11,550 --> 00:09:15,030
So the easy way to do it
is, instead of going through

139
00:09:15,030 --> 00:09:20,320
to do this, you can rewrite
things as scientific notation.

140
00:09:20,320 --> 00:09:25,980
So this problem then becomes
6 times 10 to the sixth--

141
00:09:25,980 --> 00:09:30,145
we'll write it out-- dollars
divided by-- do we write 30

142
00:09:30,145 --> 00:09:31,957
as scientific notation?

143
00:09:31,957 --> 00:09:33,330
AUDIENCE: [INAUDIBLE].

144
00:09:33,330 --> 00:09:37,920
PROFESSOR: 3 times 10
to the first relatives.

145
00:09:42,790 --> 00:09:45,520
Here's how I want you to
work with scientific notation

146
00:09:45,520 --> 00:09:48,640
when you're dividing
things, and I'll

147
00:09:48,640 --> 00:09:51,530
give you an example of
multiplying here in a second.

148
00:09:51,530 --> 00:09:54,890
So you take this many dollars
divided by that many relatives.

149
00:09:54,890 --> 00:09:56,890
This is a mathematical
expression.

150
00:09:56,890 --> 00:10:01,480
You can always just rewrite
underneath it equals--

151
00:10:01,480 --> 00:10:04,270
you're going to gather
together the different parts.

152
00:10:04,270 --> 00:10:07,440
You're going to gather
together first the numbers.

153
00:10:07,440 --> 00:10:12,820
So you're going to gather
together 6 divided by 3,

154
00:10:12,820 --> 00:10:16,000
because when you have one
thing divided by another,

155
00:10:16,000 --> 00:10:18,110
you can pull apart
what you're dividing.

156
00:10:18,110 --> 00:10:19,480
So 6 divided by 3--

157
00:10:19,480 --> 00:10:21,210
and then I'm going
to gather together--

158
00:10:21,210 --> 00:10:22,700
put those in parentheses.

159
00:10:22,700 --> 00:10:25,470
I'm going to gather
together the powers of 10.

160
00:10:25,470 --> 00:10:31,210
I'm going to say 10 to the sixth
divided by 10 to the first,

161
00:10:31,210 --> 00:10:32,050
and then--

162
00:10:32,050 --> 00:10:33,633
everybody should be
writing this down.

163
00:10:36,030 --> 00:10:41,750
And then multiply by
dollars per relative.

164
00:10:44,730 --> 00:10:47,960
And you gather together
each one of those groups.

165
00:10:52,080 --> 00:10:53,680
Now I'm going to move up here.

166
00:10:59,370 --> 00:11:01,665
That's a horrible marker.

167
00:11:01,665 --> 00:11:04,560
I'm just going to draw a line to
let you know that I'm moving up

168
00:11:04,560 --> 00:11:07,570
here, and then you just simplify
each one of those parts.

169
00:11:07,570 --> 00:11:09,240
What is 6 divided by 3?

170
00:11:09,240 --> 00:11:10,500
2.

171
00:11:10,500 --> 00:11:13,680
So I can say equals 2 times--

172
00:11:13,680 --> 00:11:14,490
now we've got this.

173
00:11:14,490 --> 00:11:15,600
Let's just rewrite it.

174
00:11:15,600 --> 00:11:18,510
10 to the sixth divided
by 10 to the first,

175
00:11:18,510 --> 00:11:22,266
and then we've got
dollars per relative.

176
00:11:25,670 --> 00:11:28,550
When you multiply and divide--
or when you divide powers

177
00:11:28,550 --> 00:11:29,960
of 10--

178
00:11:29,960 --> 00:11:30,560
yeah?

179
00:11:30,560 --> 00:11:32,196
[INAUDIBLE],, you
have a question?

180
00:11:32,196 --> 00:11:34,088
AUDIENCE: Yeah, but [INAUDIBLE]?

181
00:11:38,132 --> 00:11:40,340
PROFESSOR: Yeah, I just
wanted to keep it separate so

182
00:11:40,340 --> 00:11:41,870
that we could talk about this.

183
00:11:41,870 --> 00:11:47,390
So 10 to the sixth
is 1 million, right?

184
00:11:47,390 --> 00:11:50,310
10 to the first is 10.

185
00:11:50,310 --> 00:11:53,641
If you take 1 million divided
by 10, the easy fast way

186
00:11:53,641 --> 00:11:55,640
to do it is, if you're
looking at this fraction,

187
00:11:55,640 --> 00:12:00,320
you can cancel one of the
0's, and then it becomes 10--

188
00:12:00,320 --> 00:12:04,730
it becomes 1 with 1, 2,
3, 4, 5 0's after it,

189
00:12:04,730 --> 00:12:07,370
but if you take a
1/10 of a million,

190
00:12:07,370 --> 00:12:11,860
you actually take a 100,000.

191
00:12:11,860 --> 00:12:13,640
The quick and easy
way to do that

192
00:12:13,640 --> 00:12:16,220
is to take each one
of these exponents.

193
00:12:16,220 --> 00:12:20,300
You take two times, and you
still write it's a power of 10,

194
00:12:20,300 --> 00:12:24,350
but if you divide exponents,
you take the top exponent,

195
00:12:24,350 --> 00:12:27,200
and you subtract
the bottom exponent.

196
00:12:27,200 --> 00:12:30,660
So that becomes 2 to the
power of 10 to the 6 minus 1,

197
00:12:30,660 --> 00:12:33,860
and I want you to
write that step out,

198
00:12:33,860 --> 00:12:36,860
because that's really,
really important

199
00:12:36,860 --> 00:12:42,543
to write out the 6 minus
1 dollars per relative.

200
00:12:45,790 --> 00:12:49,600
And then you've got plenty of
paper in your lab notebooks.

201
00:12:49,600 --> 00:12:51,280
Feel free to use all of it.

202
00:12:51,280 --> 00:12:55,370
So take lots of space to write
all of these things down.

203
00:12:55,370 --> 00:12:57,490
And that becomes 2 times--

204
00:12:57,490 --> 00:12:58,653
What's 6 minus 1?

205
00:12:58,653 --> 00:12:59,560
5.

206
00:12:59,560 --> 00:13:05,800
2 times 10 to the fifth
dollars per relative.

207
00:13:05,800 --> 00:13:08,500
How do we change that back into
something that makes sense?

208
00:13:08,500 --> 00:13:13,000
Well, that's a 2
followed by five 0's.

209
00:13:13,000 --> 00:13:19,220
So $200,000 per relative.

210
00:13:19,220 --> 00:13:21,235
That's going to make
Aunt Susie pretty happy.

211
00:13:23,890 --> 00:13:28,660
So when you divide powers of
10, you take the top exponent

212
00:13:28,660 --> 00:13:32,200
minus the bottom exponent,
and it makes sense. $200,000

213
00:13:32,200 --> 00:13:34,570
is less than $6 million.

214
00:13:34,570 --> 00:13:37,370
That's what we expected.

215
00:13:37,370 --> 00:13:40,030
So we got $200,000 per relative.

216
00:13:40,030 --> 00:13:41,140
So that's division.

217
00:13:41,140 --> 00:13:44,650
So dividing powers of 10 means
you subtract the exponents.

218
00:13:44,650 --> 00:13:47,170
Let's write that out.

219
00:13:47,170 --> 00:13:48,715
Here are some of these rules.

220
00:13:53,890 --> 00:14:13,620
So rules-- dividing powers of 10
involves subtracting exponents.

221
00:14:16,420 --> 00:14:17,960
So write this down.

222
00:14:17,960 --> 00:14:20,050
Put a box around it.

223
00:14:20,050 --> 00:14:22,260
Now, let's continue our story.

224
00:14:22,260 --> 00:14:25,480
Say that Aunt Susie
takes her $200,000,

225
00:14:25,480 --> 00:14:28,600
and she puts it
into a bank account,

226
00:14:28,600 --> 00:14:32,270
and that bank account gets
5% interest every year.

227
00:14:32,270 --> 00:14:35,660
That's a pretty
good bank account.

228
00:14:35,660 --> 00:14:37,452
And I want you to write
all this stuff down

229
00:14:37,452 --> 00:14:39,326
just so that you can
have it in your notebook

230
00:14:39,326 --> 00:14:40,580
so you can refer back to it.

231
00:14:40,580 --> 00:14:42,038
I'm going to get
rid of this stuff.

232
00:14:47,330 --> 00:14:50,540
I know this is a lot
of math going on today,

233
00:14:50,540 --> 00:14:54,980
but once we get used to it,
it'll come as second nature.

234
00:14:54,980 --> 00:15:09,740
So now Aunt Susie invests
in a 5% per year account.

235
00:15:09,740 --> 00:15:12,770
So how much money is she going
to get at the end of the year?

236
00:15:12,770 --> 00:15:14,960
How much extra money?

237
00:15:14,960 --> 00:15:25,690
So what is 5% of $200,000?

238
00:15:25,690 --> 00:15:27,041
That's our question.

239
00:15:27,041 --> 00:15:29,290
Now, eventually we're going
to learn a problem solving

240
00:15:29,290 --> 00:15:32,770
technique of how to think
about equations and things,

241
00:15:32,770 --> 00:15:36,010
but for right now, let's
just translate this

242
00:15:36,010 --> 00:15:37,510
into a mathematical statement.

243
00:15:37,510 --> 00:15:39,090
So say what is--

244
00:15:39,090 --> 00:15:40,110
that's x.

245
00:15:40,110 --> 00:15:42,030
That's what we want to call.

246
00:15:42,030 --> 00:15:43,930
What is 5%?

247
00:15:43,930 --> 00:15:47,285
How do we write 5% as a decimal?

248
00:15:50,370 --> 00:15:52,074
Go ahead, Lauren.

249
00:15:52,074 --> 00:15:53,040
AUDIENCE: 5/100.

250
00:15:53,040 --> 00:15:54,999
PROFESSOR: 5?

251
00:15:54,999 --> 00:15:56,600
AUDIENCE: 0.05.

252
00:15:56,600 --> 00:16:00,770
PROFESSOR: So 5/100
because it's 5 per cent.

253
00:16:00,770 --> 00:16:04,340
5 per cent is--
cent is about 100.

254
00:16:04,340 --> 00:16:06,290
So 5/100.

255
00:16:06,290 --> 00:16:15,245
So we can say 5 divided by
100, or we can also say 0.05--

256
00:16:15,245 --> 00:16:17,270
5%.

257
00:16:17,270 --> 00:16:18,980
So those are tenths.

258
00:16:18,980 --> 00:16:20,330
Those are hundredths.

259
00:16:20,330 --> 00:16:23,680
So you've got 5/100.

260
00:16:23,680 --> 00:16:29,500
What is 5/100 of 200,000?

261
00:16:29,500 --> 00:16:32,230
We've just translated
this sentence into math.

262
00:16:32,230 --> 00:16:37,910
What is-- x is 5/100
of or times 200,000.

263
00:16:37,910 --> 00:16:46,010
So we can rewrite that x
equals 0.05 times 200,000.

264
00:16:46,010 --> 00:16:50,600
Now let's rewrite this
as scientific notation.

265
00:16:50,600 --> 00:16:54,440
How do we write 0.05
in scientific notation?

266
00:16:54,440 --> 00:16:55,221
[INAUDIBLE]?

267
00:16:55,221 --> 00:17:02,436
AUDIENCE: [INAUDIBLE]
5 times [INAUDIBLE]

268
00:17:02,436 --> 00:17:08,180
10 to the 2 [INAUDIBLE]
10 to the negative 2.

269
00:17:08,180 --> 00:17:14,469
PROFESSOR: So 5 times 10 to the
2 is 5 times 100, which is 500.

270
00:17:14,469 --> 00:17:15,760
AUDIENCE: 10 to the negative 2.

271
00:17:15,760 --> 00:17:18,660
PROFESSOR: We want to say 5
times 10 to the negative 2.

272
00:17:18,660 --> 00:17:20,819
So this number is less than 1.

273
00:17:20,819 --> 00:17:23,740
It's a fraction of 1.

274
00:17:23,740 --> 00:17:26,410
And we also multiply times--

275
00:17:26,410 --> 00:17:29,370
and this one we said was
2 times 10 to the fifth.

276
00:17:31,890 --> 00:17:33,780
So again, we want--

277
00:17:37,050 --> 00:17:39,960
now I forgot to put
in our units here,

278
00:17:39,960 --> 00:17:43,470
because here we're
saying this is dollars,

279
00:17:43,470 --> 00:17:46,960
and a percent is just a
fraction of this quantity.

280
00:17:46,960 --> 00:17:50,454
So it doesn't have
a unit, but we

281
00:17:50,454 --> 00:17:51,870
will know that our
answer is going

282
00:17:51,870 --> 00:17:56,430
to be, well, how much money
is she going to have extra?

283
00:17:56,430 --> 00:17:58,410
So our unit should
be in dollars.

284
00:17:58,410 --> 00:18:00,720
Now we do the exact same thing.

285
00:18:00,720 --> 00:18:03,000
We gather together the numbers.

286
00:18:03,000 --> 00:18:06,750
So 5 times 2--

287
00:18:06,750 --> 00:18:09,030
gather those together.

288
00:18:09,030 --> 00:18:16,100
And then we multiply by 10 to
the minus 2 times 10 to the 5,

289
00:18:16,100 --> 00:18:17,850
and I want you to just
write it like that.

290
00:18:17,850 --> 00:18:20,452
Put those two numbers
right next to each other,

291
00:18:20,452 --> 00:18:21,910
and then the unit
is still dollars.

292
00:18:25,550 --> 00:18:27,300
All we're doing is
just gathering together

293
00:18:27,300 --> 00:18:29,090
different parts.

294
00:18:29,090 --> 00:18:30,570
Let's go up here.

295
00:18:30,570 --> 00:18:34,530
We'll rewrite that as x equals.

296
00:18:34,530 --> 00:18:36,534
Now, what is 5 times 2?

297
00:18:36,534 --> 00:18:37,707
AUDIENCE: 10.

298
00:18:37,707 --> 00:18:38,290
PROFESSOR: 10.

299
00:18:38,290 --> 00:18:40,520
So let's not try
to jump any steps.

300
00:18:40,520 --> 00:18:42,561
We're going to write that
as scientific notation.

301
00:18:42,561 --> 00:18:45,180
We'll just write
that as 10 times--

302
00:18:45,180 --> 00:18:48,810
now, when you
multiply powers of 10,

303
00:18:48,810 --> 00:18:52,150
you want to add the exponents.

304
00:18:52,150 --> 00:18:56,100
So 10 to the minus
2 times 10 to the 5

305
00:18:56,100 --> 00:18:59,930
becomes 10 to the
minus 2 plus 5.

306
00:19:03,902 --> 00:19:05,360
And then the unit
is still dollars.

307
00:19:12,510 --> 00:19:14,697
So that then becomes 10.

308
00:19:14,697 --> 00:19:15,530
Let's simplify this.

309
00:19:15,530 --> 00:19:18,020
What is minus 2 plus 5?

310
00:19:20,730 --> 00:19:21,640
Hang on one second.

311
00:19:21,640 --> 00:19:22,542
AUDIENCE: Three.

312
00:19:22,542 --> 00:19:23,250
PROFESSOR: Three.

313
00:19:23,250 --> 00:19:26,500
If you go negative 2, and
then you add on 5 more,

314
00:19:26,500 --> 00:19:29,080
you end up back at positive 3.

315
00:19:29,080 --> 00:19:33,750
So 10 to the third dollars.

316
00:19:33,750 --> 00:19:36,780
How can we rewrite 10
times 10 to the third

317
00:19:36,780 --> 00:19:40,228
in terms of scientific notation?

318
00:19:40,228 --> 00:19:42,590
AUDIENCE: [INAUDIBLE].

319
00:19:42,590 --> 00:19:46,170
PROFESSOR: How can we rewrite
this as scientific notation?

320
00:19:46,170 --> 00:19:48,521
We could say 10 times
10 to the third.

321
00:19:48,521 --> 00:19:52,170
AUDIENCE: [INAUDIBLE].

322
00:19:52,170 --> 00:19:54,978
PROFESSOR: We can-- what's that?

323
00:19:54,978 --> 00:19:56,710
AUDIENCE: Is it 1,000?

324
00:19:56,710 --> 00:19:59,120
PROFESSOR: Well, it's 10
times 10 to the third.

325
00:19:59,120 --> 00:20:02,960
10 to the third is
1,000, so it's 10,000,

326
00:20:02,960 --> 00:20:04,670
but in order to
keep it all straight

327
00:20:04,670 --> 00:20:08,720
let's just do 10 to the
first because 10 is just 10

328
00:20:08,720 --> 00:20:15,830
to the power of 1 times
10 to the third dollars.

329
00:20:15,830 --> 00:20:17,660
Again, you multiply
the exponent,

330
00:20:17,660 --> 00:20:19,320
and you multiply
the powers of 10,

331
00:20:19,320 --> 00:20:26,670
and you've got 10 to
the 1 plus 3 dollars.

332
00:20:26,670 --> 00:20:28,750
And then we simplify
that, and that's just 10

333
00:20:28,750 --> 00:20:33,210
to the 4 dollars.

334
00:20:33,210 --> 00:20:36,670
If we rewrite that, that's
a 10-- or a 1 with 3,

335
00:20:36,670 --> 00:20:39,160
4 0's after it.

336
00:20:39,160 --> 00:20:41,639
So $10,000.

337
00:20:41,639 --> 00:20:43,180
So at the end of
one year, Aunt Susie

338
00:20:43,180 --> 00:20:46,195
is going to have an additional
$10,000 on top of her.

339
00:20:46,195 --> 00:20:47,070
AUDIENCE: [INAUDIBLE]

340
00:20:47,070 --> 00:20:48,528
PROFESSOR: Yeah,
where's that bank?

341
00:20:48,528 --> 00:20:50,580
That's a good question.

342
00:20:50,580 --> 00:21:04,810
So again, your second rule
is multiplying powers of 10

343
00:21:04,810 --> 00:21:12,720
involves addition of exponents.

344
00:21:17,310 --> 00:21:19,890
That's her second rule.

345
00:21:19,890 --> 00:21:21,360
Most of the time
we're only going

346
00:21:21,360 --> 00:21:25,850
to be doing division
and multiplication.