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ING
ASTOUND
MATH STORIES
Too many people think Math is Dull.
Astounding Math Stories exists to show you
the kinds of weird things that happen when you do math.
These Astounding Stories also illustrate some important ideas in math,
so this blog also has a Secret Educational Agenda...
Table of contents
Sizes of Infinity
A conversation recorded by the Six Winged Seraph, a certified member of the Society
of Recording Angels
7 May 2017
Gyrē: Gimble, did you know that there are exactly as many positive integers as there
are rational numbers? Isn’t that astounding?
Gimblos: But that’s obviously wrong. Every integer is a rational number, like 3 is the
same as 6/2. But there are rational numbers that are not integers, like 1/3. So the set of
integers is a proper subset of the set of rationals. A set can’t have the same number of
elements as a proper subset, because there are elements left over!
Gyrē: Ah, but the positive integers and the rational numbers are both infinite sets and
you intuition gets
all messed up when you are dealing with infinite sets.
The world of
infinite sets is full of weird things. You can’t depend on your experience with
counting things when you are dealing with infinite sets.
Gimblos: Well, I don’t believe it. You need to prove it!
Gyrē: It’s not hard. I will show you a way to lay out the rational numbers in order, so
there is a first one, a second one, and so on. The usual order doesn’t allow this because
of density. It’s like this: make sure you write each one in lowest terms and list them in
order by the sum of the absolute values of their numerator and denominator. So:
 The first one is 0, which is
0
1
, so the sum of the numerator and denominator is 1.
 Then
 Then
- 1
and
1
- 2
1
have sum 2, so you list them left to right in the usual order.
1
,-
1
1 1
2
2 2
1
, , and
have sum 3 and you list them left to right.
 For those that add up to 4 you get
- 3 - 1 1 3
,
1
3
, ,
2
. Notice that
3 1
also has
2
numerator and denominator adding up to 4 but you don’t include it because it is not in
lowest terms.
 So you wind up with this list:
0 - 1 1 - 2 - 1 1 2 - 3 - 1 1 3 - 4 - 3 - 2 - 1 1
2 3
4 - 5
1
3 2
1
,
, ,
1
1
1
,
, , ,
2
2 1
1
,
3
, , ,
3 1
1
,
,
2
3
,
4
, ,
4
, ,
,
1
,
- 1 - 6
5
,
1
,
- 5 - 4
2
,
3
, ...
and so on.
 Now you count them: The first one is
0
, the second one is
1
1
- 1
, the third one is
1
, and so on. You get a one to one correspondence between the positive integers
1
and the rational numbers!
0
- 1
1
- 2
- 1
1
2
- 3
- 1
1
3
- 4
- 3
- 2
- 1
1
2
1
1
1
1
2
2
1
1
3
3
1
1
2
3
4
4
3
b
1
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
b
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
So your intuition is ALL WRONG. A set and a proper subset can have the same number
of elements.
Gimblos: Now wait a minute, you have four rational numbers whose numerators
and denominators add up to 4, but eight where they add up to 5 and only two where
they add up to six. Well, I guess that’s right, all the others that add up to 6 aren’t in
lowest terms, like
- 4 - 3
- 2
. So, yes, you have matched up the positive
,
, and
2
3
4
integers and the rationals in a bijective way.
Gyrē (looking smug): Well, that’s what I said, there are the same number of positive
integers as rational numbers. Infinite sets are simply weird.
Gimblos:
… You are trying to
mess with my head!
You are like a magician
who does something over here so I don’t look at what you are doing over there so I think
you’ve done something impossible!
 You said there are the same number of positive integers as rational numbers.
 Let’s stipulate that you have proved that there is a bijection between the positive
integers and the rational numbers. (I just discovered that word “stipulate”. Isn’t it
great?)
 The positive integers and the rational numbers each form infinite sets.
 We count finite sets, not infinite sets.
 Two finite sets have the same number if there is a bijection between them.
 So you secretly extended the definition of “number of” to infinite sets by saying
two infinite sets have the same number of elements if there is a bijection between them.
 Then you tried to bamboozle me into thinking the set of positive integers and the
set of rational numbers have the same number of elements. But that is only because
you changed the meaning of the word “number”. You haven’t told me anything
astounding, you just tried to trick me. [Goes out and slams the door.]
Six Winged Seraph: Well, reader, who is right? Did Gyrē do something Astounding or
did she try and fail to trick Gimblon?
If you want to learn more:
Wikipedia article on cardinality
If you like Astounding Math Stories, you might like abstractmath.org.