1
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Hi Sal.

2
00:00:02,310 --> 00:00:03,060
>> Hey Brit.

3
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>> How are you?

4
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>> Good, it looks like you have a game
going on here.

5
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>> Not a game, yeah, kind of a challenge
question for you.

6
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What I did is I put one grain of rice in
the first square.

7
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>> That's right.

8
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>> There's 64 squares on the board.

9
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>> Yep.

10
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>> And in each consecutive square I
doubled the amount of rice.

11
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>> Mm-hm.

12
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>> How much rice do you think would be, on
this square?

13
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>> On that square.

14
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So let me, let me think about it a little
bit.

15
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Actually I'm going to take some [SOUND].

16
00:00:27,036 --> 00:00:27,770
So here you have 1.

17
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And we multiply that times 2.

18
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So this is going to be 2 times.

19
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2 no, no 2 times 1, what am I doing?

20
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Now this is 2 times that 1, so this is 2
times 2.

21
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Now this 2 times that so this is okay,

22
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we're starting to get a lot of 2s here,
multiplying them together so this is 2

23
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times 2, I'm trying to write sideways
times 2.

24
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So this one is going to be five 2s
multiplied together.

25
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This is going to be six 2s multiplied
together.

26
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This is gonna be seven twos multiplied
together.

27
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>> Mm-hm.

28
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>> Eight 2s multiplied together.

29
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So nine 2s, 10, 11, 12, 13.

30
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So 2, all, all of this stuff multiplied
together 8,192 grains of rice.

31
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Is what we should see right over here.

32
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>> You know I had fun last night, I was up
late, but there you go.

33
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>> Did you really count out 8192 grains of
rice.

34
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>> More or less.

35
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>> Okay let's just say you did.

36
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>> What if we just went four steps ahead,
how much rice would be here?

37
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>> Four steps ahead, so we're gonna
multiply by 2, multiply by 2 again.

38
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Then multiply by 2 again, then multiply by
2 again.

39
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So this number times, let's see 2 times
two is 4, times 2 is 8 times 2 is 16.

40
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So it's gonna get us like a 120, like
120,000 or around there.

41
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>> 131,672.

42
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You had a lot of time last night.

43
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>> We're not even halfway across the board
yet.

44
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>> We're not.

45
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>> We, I mean this is, this is a lot of,

46
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that's a lot of rice there, that, you
could throw a party.

47
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>> What about the last square, this is 63
steps.

48
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>> We're gonna take, we're gonna take 2
times 2, and we're gonna do 63 of those.

49
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So this is going to be a, a huge number
and

50
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actually, it would be neat if there was a
notation for that.

51
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>> I, I didn't, I didn't count this one
out, but

52
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it is the size of Mount Everest, the pile
of rice.

53
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>> Hm.

54
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>> And it would feed 485 trillion people.

55
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>> But I have got a question.

56
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I mean [INAUDIBLE] this was a little bit
of a,

57
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of a pain, for me to write all of these
twos.

58
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>> So is this.

59
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>> If I were the mathematical community.

60
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>> Mm-hm.

61
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>> I would want some type of notation.

62
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>> You kind of got on it here.

63
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I, I like this dot, dot, dot and the 63.

64
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This, you know, I understand this.

65
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>> Yeah, you could understand this.

66
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But, this is still a little bit.

67
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Little bit too much, what if instead you
just wrote.

68
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>> Mathematicians love being efficient
right they want, they're lazy.

69
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They have things to do they have to go
home and count grains of rice.

70
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>> Right.

71
00:02:58,266 --> 00:03:06,110
>> [LAUGH] Yeah so that is 63 twos and
multiply them all together.

72
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>> This is the first square on our board,
we have one grain of rice.

73
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And when we double it, we have two grains
of rice.

74
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>> Yup.

75
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>> And we double it again, we have four,
and I'm thinking,

76
00:03:17,420 --> 00:03:20,870
this, this is similar to what we were
doing, it's just represented differently.

77
00:03:20,870 --> 00:03:24,010
>> Yeah, well I mean this one, the one you
were making, right, every time you,

78
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you're kind of adding these more, these
Popsicle

79
00:03:26,100 --> 00:03:29,110
sticks you're kind of branching out, you
know, one.

80
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Popsicle stick now becomes two popsicle
sticks, and then you keep doing that.

81
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One Popsicle stick becomes 2, but now you
have 2 of em.

82
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So here you have 1, now you have 1 times
2.

83
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Now each of these two branch into 2, so
now

84
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you have 2 times 2 or you have four
Popsicle sticks.

85
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Every stage, every branch you're
multiplying by 2 again.

86
00:03:46,780 --> 00:03:50,640
>> I, I just basically continue splitting
just like the tree does?

87
00:03:50,640 --> 00:03:51,140
>> Yup.

88
00:03:53,050 --> 00:03:56,350
Now, I can really see what 2 to the power
of 3 looks like.

89
00:03:56,350 --> 00:03:57,820
>> And, that's what we have here.

90
00:03:57,820 --> 00:03:59,120
1 times 2 times 2 times 2, which is 8.

91
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This is 2 to the third power.

92
00:04:01,130 --> 00:04:04,635
>> And, when I see 2 to the power of
something, let's just say n.

93
00:04:04,635 --> 00:04:05,625
>> Mm-hm.

94
00:04:05,625 --> 00:04:09,900
>> N could also be number of steps up this
tree.

95
00:04:09,900 --> 00:04:11,440
I could think about it that way.

96
00:04:11,440 --> 00:04:13,530
>> Yeah, you could do, I guess one way to
think about it is

97
00:04:13,530 --> 00:04:17,310
how many times you branched, but that one,
that tree there's actually even more.

98
00:04:17,310 --> 00:04:19,190
>> I don't think this counts because,
again,

99
00:04:19,190 --> 00:04:22,910
this branch is four times of each branch.

100
00:04:22,910 --> 00:04:23,896
>> Well, I guess, why not.

101
00:04:23,896 --> 00:04:24,138
Well.

102
00:04:24,138 --> 00:04:26,970
I mean, its It's different, it's not gonna
be 2 anymore.

103
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So this, the first one we haven't branched
yet.

104
00:04:29,670 --> 00:04:33,510
This is gonna be 4 to the 0 power, you've
had no branches yet.

105
00:04:34,630 --> 00:04:39,890
This you branched once, so this is 4 to
the 1st power, you have 4 branches now.

106
00:04:39,890 --> 00:04:41,080
>> Oh, I like this.

107
00:04:41,080 --> 00:04:42,260
>> And now each of those,.

108
00:04:42,260 --> 00:04:48,080
So now you've branched twice, so this is 4
to the second power.

109
00:04:48,080 --> 00:04:50,920
So yeah, the base, or what is called the
base, when

110
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you're dealing with an exponent is four
right over here, this

111
00:04:53,650 --> 00:04:56,410
is how many times, how many new branches
each of the

112
00:04:56,410 --> 00:04:58,740
new branches turn in to at each of these
new junctions.

113
00:04:58,740 --> 00:04:59,300
>> Let's call them junctions.

114
00:04:59,300 --> 00:05:00,015
>> Junctions.

115
00:05:00,015 --> 00:05:04,809
You haven't branched yet, here you branch
once and here you branch.

116
00:05:06,250 --> 00:05:07,970
>> This is interesting.

117
00:05:07,970 --> 00:05:09,670
This is also why when I look at a tree,

118
00:05:09,670 --> 00:05:12,880
you know, there's thousands of leaves, but
just one trunk.

119
00:05:12,880 --> 00:05:15,520
And when you actually go up, and you look
inside

120
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the tree, it only branches you know, three
or four times.

121
00:05:17,860 --> 00:05:21,440
>> And that shows the power of exponential
growth.

122
00:05:21,440 --> 00:05:21,915
>> Yes.

123
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[LAUGH]