Two Enumerations of the Rationals

integer

62947327

numerator

355

denominator

113

			Calkin-Wilf	Stern-Brocat	other system
integer	62947327	is	439 29	355 113	67107847
fraction	355 113	is	67107847	62947327	439 29

Here are two methods for enumerating the rational numbers. The Calkin–Wilf method starts with

1

corresponding to the binary number 1. If the binary number

x

corresponds to the fraction

a

b

, then the binary number corresponding to

a

a+b

is 0 appended to

x

, and the one corresponding to

a+b

b

is 1 appended to

x

. According to this scheme, the Calkin–Wilf enumeration begins with



1

;

1

2

,

2

1

;

1

3

,

3

2

,

2

3

,

3

1

;

1

4

,

4

3

,

3

5

,

5

2

,

2

5

,

5

3

,

3

4

,

4

1

;⋯

, where semicolons separate the rows of an array.

The Stern–Brocat method is more complicated, but it winds up reversing the Calkin–Wilf binary number except for the leading 1. The corresponding enumeration then reads



1

;

1

2

,

2

1

;

1

3

,

2

3

,

3

2

,

3

1

;

1

4

,

2

5

,

3

5

,

3

4

,

4

3

,

5

3

,

5

2

,

4

1

;⋯

.

Since either of the above arrays can be put into one-to-one correspondence with the set of natural numbers (positive integers) , the rationals must also comprise a set of cardinality

ℵ

0

.