Apostrophe Notation

The **apostrophe notation** is a extension to the standard notations used to denote numbers in other number systems. It is used to represent numbers in transdecimal bases where the symbols for transdecimal digits are not supported and are designed to only include symbols easily accessible with a keyboard. The transdecimal digits are encoded decimally unless specified.

Examples

* 1
′

0

[
n
]

{\displaystyle 1'0\_{[n]}}

 represents the base in any positional base

n

{\displaystyle n}

. (In pure text, this is written as 1'0 [n]).
* 1
′

2
′

34

[
60
]

{\displaystyle 1'2'34\_{[60]}}

 represents

1
×

60
2

2
×
60
34

{\displaystyle 1\times60^2+2\times60+34}

.
* 29.5

[
60
]

{\displaystyle 29.5\_{[60]}}

 represents

29

5
60

{\displaystyle 29+\frac{5}{60}}

, whereas

29.50

[
60
]

{\displaystyle 29.50\_{[60]}}

 represents

29

50
60

{\displaystyle 29+\frac{50}{60}}

.
	+ Leading zeroes are optional, and could be used when seen fit. For example,

	29.5

	[
	60
	]

	{\displaystyle 29.5\_{[60]}}

	is the same as

	29.05

	[
	60
	]

	{\displaystyle 29.05\_{[60]}}

	.
* An overline represents repeating decimals. For example,

1
7

=

8
′

34
′

17

¯

[
60
]

{\displaystyle \frac{1}{7}=0.\overline{8'34'17}\_{[60]}}

. In pure text, the overline is replaced by braces (curly brackets,

{

}

{\displaystyle \{\ \}}

)
* Scientific notation works the same as decimal numbers. For example,

1
′

30
′

0
′

0
′

0
′

0
′

0
′

0
′

0
′

0
′

0
′

0
′

0
′

0
′

0

[
60
]

=

1
′

30
′

0
⏞

13

[
60
]

=
1.30
×

1
′

0

14

[
60
]

=
1.30

E

14

[
60
]

{\displaystyle 1'30'0'0'0'0'0'0'0'0'0'0'0'0'0\_{[60]}=1'30'{\overbrace{0}^{13}}\_{[60]}=1.30\times{1'0^{14}}\_{[60]}=1.30\ \mathrm{E}14\_{[60]}}

.
* The radix

[
20
:
30
]

{\displaystyle [20:30]}

 represents a trigesimal-coded base 600, where the highest digit for the odd-position digit (1st, 3rd, 5th...) (from the right) is 29, whereas the highest digit for even-positions (2nd, 4th, 6th...) (from the right) is 19.
	+ For example,

	10
	′

	11
	′

	12
	′

	13

	[
	20
	:
	30
	]

	{\displaystyle 10'11'12'13\_{[20:30]}}

	 represents

	10
	×
	(
	30
	×
	20
	×
	30
	)
	11
	×
	(
	20
	×
	30
	)
	12
	×
	30
	13

	{\displaystyle 10\times(30\times20\times30)+11\times(20\times30)+12\times30+13}

	.
* The radix

[

20
¯

:
30
]

{\displaystyle [\overline{20}:30]}

 (or in pure text "{20:30]") represents a number base where the largest possible digit for each position is 19, except for the ones place which can go up to 29.
	+ For example,

	10
	′

	11
	′

	12
	′

	13

	[

	20
	¯

	:
	30
	]

	{\displaystyle 10'11'12'13\_{[\overline{20}:30]}}

	 represents

	10
	×
	(
	20
	×
	20
	×
	30
	)
	11
	×
	(
	20
	×
	30
	)
	12
	×
	30
	13

	{\displaystyle 10\times(20\times20\times30)+11\times(20\times30)+12\times30+13}

	.
* The radix

[

20

10

:

24

12

]

{\displaystyle [20\_{10}:24\_{12}]}

 represents a vigesimal-coded base 480, which the vigesimal digit is encoded decimally and the tetravigesimal digit encoded dozenally. In cases where bases are encoded in transdecimal bases, any proposal for symbols could be used, though the alphanumeric digits are preferred due to its accessibility. A small letter "c" could be added to the radix to denote the preference of "computerese", and similarly, "a" for argam numerals.
* The radix

B
[
n
]

{\displaystyle B[n]}

 represents the bijective base

n

{\displaystyle n}

.
* The radix

L
[
n
]

{\displaystyle L[n]}

 represents the balanced base

n

{\displaystyle n}

.
* The radix

S
[

n
1

,

n
2

,
b
]

{\displaystyle S[n\_1,n\_2,b]}

 or

S
[

n

1

n

2

b
]

{\displaystyle S[{}\_{n\_{1}}^{n\_{2}}b]}

 represents a "shifted base" b number system using integer digits from

n
1

{\displaystyle n\_1}

 to

n
2

{\displaystyle n\_2}

 inclusive. For example, S[10, 19] represents a number system that uses decimal place values but uses digits with values of ten through nineteen instead. S[0, 22, 2] represents a number system that uses dozenal place values but uses the even digits with values of zero through twenty-two instead.
	+ Similarly,

	[
	n
	,
	{

	d

	1

	,

	d

	2

	,
	⋯
	}
	]

	{\displaystyle [n, \{d\_{1},d\_{2},\cdots\}]}

	 could be used to represent base n using the set of digits

	{

	d

	1

	,

	d

	2

	,
	⋯
	}

	{\displaystyle \{d\_{1},d\_{2},\cdots\}}

	.
* The radix

m
[
n
]

{\displaystyle m[n]}

 represents that the positional base

n

{\displaystyle n}

 is encoded in base

m

{\displaystyle m}

. (Where

m

{\displaystyle m}

 is a base other than decimal)
* The radix

[
ω
]

{\displaystyle [\omega]}

 represents an "infinite" base

ω

{\displaystyle \omega}

, where

ω

{\displaystyle \omega}

 denotes the first transfinite ordinal.
	+ The radix of the canonical Chronary system is denoted with

	[
	ω
	:
	13
	:

	28
	7

	:

	24

	12

	:

	60

	10

	:

	60

	10

	]

	{\displaystyle [\omega:13:28\_7:24\_{12}:60\_{10}:60\_{10}]}

	, with the "infinite base" digit encoded in a base of choice (usually decimal).

Shorthand notations for commonly used number systems and other symbols (such as the "percent") still apply.

* The radixes

[
b
]
,
[
t
]
,
[
q
]
,
[
p
]
,
[
s
]
,
[
h
]
,
[
o
]
,
[
e
]
,
[
d
]
,
[
l
]
,
[
z
]

{\displaystyle [b], [t], [q], [p], [s], [h], [o], [e], [d], [l], [z]}

 represent the positional bases 2 through 12 (binary through dozenal) respectively, although the apoostrophe notation is intended for bases >10 only. The same applies for mixed radix bases, for example,

[
s
:
d
]

{\displaystyle [s:d]}

 could be used to represent decimal-coded sexagesimal.
* The Humphrey point ";" could also be used to denote numbers expressed in dozenal.