1
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2
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So let's talk a little bit
about rational numbers.

3
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4
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And the simple way to think
about it is any number that

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can be represented as
the ratio of two integers

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is a rational number.

7
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So for example, any integer
is a rational number.

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1 can be represented as 1/1 or
as negative 2 over negative 2

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or as 10,000/10,000.

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In all of these cases, these are
all different representations

11
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of the number 1,
ratio of two integers.

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And I obviously can
have an infinite number

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of representations
of 1 in this way,

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the same number over
the same number.

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The number negative 7 could be
represented as negative 7/1,

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or 7 over negative 1, or
negative 14 over positive 2.

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And I could go on, and
on, and on, and on.

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So negative 7 is definitely
a rational number.

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It can be represented as
the ratio of two integers.

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But what about things
that are not integers?

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For example, let us imagine--
oh, I don't know-- 3.75.

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How can we represent that as
the ratio of two integers?

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Well, 3.75, you
could rewrite that

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as 375/100, which is the
same thing as 750/200.

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Or you could say, hey,
3.75 is the same thing as 3

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and 3/4-- so let
me write it here--

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which is the same
thing as-- that's 15/4.

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4 times 3 is 12, plus 3 is
15, so you could write this.

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This is the same thing as 15/4.

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Or we could write this as
negative 30 over negative 8.

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I just multiplied the
numerator and the denominator

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here by negative 2.

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But just to be clear,
this is clearly rational.

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I'm giving you multiple
examples of how

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this can be represented as
the ratio of two integers.

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Now, what about
repeating decimals?

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Well, let's take
maybe the most famous

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of the repeating decimals.

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Let's say you have 0.333, just
keeps going on and on forever,

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which we can denote by
putting that little bar on top

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of the 3.

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This is 0.3 repeating.

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And we've seen--
and later we'll show

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how you can convert
any repeating decimal

45
00:02:42,900 --> 00:02:47,850
as the ratio of two integers--
this is clearly 1/3.

46
00:02:47,850 --> 00:02:54,050
Or maybe you've seen things like
0.6 repeating, which is 2/3.

47
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And there's many, many,
many other examples of this.

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And we'll see any
repeating decimal, not just

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one digit repeating.

50
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Even if it has a million
digits repeating,

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as long as the pattern
starts to repeat itself

52
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over and over and
over again, you

53
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can always represent that as
the ratio of two integers.

54
00:03:13,354 --> 00:03:15,020
So I know what you're
probably thinking.

55
00:03:15,020 --> 00:03:17,120
Hey, Sal, you've
just included a lot.

56
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You've included all
of the integers.

57
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You've included all of finite
non-repeating decimals,

58
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and you've also included
repeating decimals.

59
00:03:30,380 --> 00:03:31,220
What is left?

60
00:03:31,220 --> 00:03:33,927
Are there any numbers
that are not rational?

61
00:03:33,927 --> 00:03:35,760
And you're probably
guessing that there are,

62
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otherwise people
wouldn't have taken

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the trouble of trying to
label these as rational.

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And it turns out-- as you
can imagine-- that actually

65
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some of the most famous
numbers in all of mathematics

66
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are not rational.

67
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And we call these numbers
irrational numbers.

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69
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And I've listed there
just a few of the most

70
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noteworthy examples.

71
00:04:04,180 --> 00:04:07,355
Pi-- the ratio of
the circumference

72
00:04:07,355 --> 00:04:12,100
to the diameter of a circle--
is an irrational number.

73
00:04:12,100 --> 00:04:14,030
It never terminates.

74
00:04:14,030 --> 00:04:17,760
It goes on and on and on
forever, and it never repeats.

75
00:04:17,760 --> 00:04:20,300
e, same thing-- never
terminates, never repeats.

76
00:04:20,300 --> 00:04:23,020
It comes out of continuously
compounding interest.

77
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It comes out of
complex analysis.

78
00:04:24,640 --> 00:04:26,270
e shows up all over the place.

79
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Square root of 2,
irrational number.

80
00:04:28,640 --> 00:04:31,164
Phi, the golden ratio,
irrational number.

81
00:04:31,164 --> 00:04:32,830
So these things that
really just pop out

82
00:04:32,830 --> 00:04:36,640
of nature, many of these
numbers are irrational.

83
00:04:36,640 --> 00:04:39,080
Now, you might say, OK,
are these irrational?

84
00:04:39,080 --> 00:04:41,870
These are just these
special kind of numbers.

85
00:04:41,870 --> 00:04:43,620
But maybe most
numbers are rational,

86
00:04:43,620 --> 00:04:46,910
and Sal's just picked out
some special cases here.

87
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But the important thing to
realize is they do seem exotic,

88
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and they are exotic
in certain ways.

89
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But they aren't uncommon.

90
00:04:53,460 --> 00:04:57,220
It actually turns out
that there is always

91
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an irrational number between
any two rational numbers.

92
00:05:01,284 --> 00:05:02,450
Well, we could go on and on.

93
00:05:02,450 --> 00:05:03,970
There's actually
an infinite number.

94
00:05:03,970 --> 00:05:06,850
But there's at least one,
so that gives you an idea

95
00:05:06,850 --> 00:05:08,570
that you can't
really say that there

96
00:05:08,570 --> 00:05:10,814
are fewer irrational numbers
than rational numbers.

97
00:05:10,814 --> 00:05:12,230
And in a future
video, we'll prove

98
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that you give me two rational
numbers-- rational 1,

99
00:05:16,340 --> 00:05:21,640
rational 2-- there's going to be
at least one irrational number

100
00:05:21,640 --> 00:05:23,810
between those, which
is a neat result,

101
00:05:23,810 --> 00:05:25,810
because irrational
numbers seem to be exotic.

102
00:05:25,810 --> 00:05:28,310
Another way to think about it--
I took the square root of 2,

103
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but you take the square root
of any non-perfect square,

104
00:05:31,150 --> 00:05:34,620
you're going to end up
with an irrational number.

105
00:05:34,620 --> 00:05:36,354
You take the sum
of an irrational

106
00:05:36,354 --> 00:05:38,520
and a rational number-- and
we'll see this later on.

107
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We'll prove it to ourselves.

108
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The sum of an irrational
and a rational

109
00:05:42,610 --> 00:05:44,310
is going to be irrational.

110
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The product of an
irrational and a rational

111
00:05:47,200 --> 00:05:49,020
is going to be irrational.

112
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So there's a lot, a lot, a
lot of irrational numbers

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out there.