Chapter 18: lojbau mekso: Mathematical Expressions in Lojban

10. Non-decimal and compound bases

The following cmavo are discussed in this section:

ju'u    VUhU    to the base
dau PA  hex digit A ⇔ 10
fei PA  hex digit B ⇔ 11
gai PA  hex digit C ⇔ 12
jau PA  hex digit D ⇔ 13
rei PA  hex digit E ⇔ 14
vai PA  hex digit F ⇔ 15
pi'e    PA  compound base point

In normal contexts, Lojban assumes that all numbers are expressed in the decimal (base 10) system. However, other bases are possible, and may be appropriate in particular circumstances.

To specify a number in a particular base, the VUhU operator “ju'u” is suitable:

✥10.1    li pa no pa no ju'u re du li pa no
the-number 1010 base 2 equals the-number 10

Here, the final “pa no” is assumed to be base 10, as usual; so is the base specification. (The base may also be changed permanently by a metalinguistic specification; no standard way of doing so has as yet been worked out.)

Lojban has digits for representing bases up to 16, because 16 is a base often used in computer applications. In English, it is customary to use the letters A-F as the base 16 digits equivalent to the numbers ten through fifteen. In Lojban, this ambiguity is avoided:

✥10.2    li daufeigai ju'u paxa du li rezevobi
the-number ABC base 16 equals the-number 2748

✥10.3 li jaureivai ju'u paxa du li cimuxaze
the-number DEF base 16 equals the-number 3567

Note the pattern in the cmavo: the diphthongs “au”, “ei”, “ai” are used twice in the same order. The digits for A to D use consonants different from those used in the decimal digit cmavo; E and F unfortunately overlap 2 and 4 — there was simply not enough available cmavo space to make a full differentiation possible. The cmavo are also in alphabetical order.

The base point “pi” is used in non-decimal bases just as in base 10:

✥10.4    li vai pi bi ju'u paxa du li pamu pi mu
the-number F.8 base 16 equals the-number 15.5

Since “ju'u” is an operator of selma'o VUhU, it is grammatical to use any operand as the left argument. Semantically, however, it is undefined to use anything but a numeral string on the left. The reason to make “ju'u” an operator is to allow reference to a base which is not a constant.

There are some numerical values that require a “base” that varies from digit to digit. For example, times represented in hours, minutes, and seconds have, in effect, three “digits”: the first is base 24, the second and third are base 60. To express such numbers, the compound base separator “pi'e” is used:

✥10.5    ci pi'e rere pi'e vono

Each digit sequence separated by instances of “pi'e” is expressed in decimal notation, but the number as a whole is not decimal and can only be added and subtracted by special rules:

✥10.6    li ci pi'e rere pi'e vono su'i pi'e ci pi'e cici
    du li ci pi'e rexa pi'e paci
the-number 3:22:40 plus :3:33 equals the-number 3:26:13
3:22:40 + 0:3:33 ⇔ 3:26:13

Of course, only context tells you that the first part of the numbers in ✥10.5 and ✥10.6 is hours, the second minutes, and the third seconds.

The same mechanism using “pi'e” can be used to express numbers which have a base larger than 16. For example, base-20 Mayan mathematics might use digits from “no” to “paso”, each separated by “pi'e”:

✥10.7    li pa pi'e re pi'e ci ju'u reno du li vovoci
the-number 1;2;3 base 20 equals the-number 443

Carefully note the difference between:

✥10.8    pano ju'u reno
the-digit-10 base 20

which is equal to ten, and:

✥10.9    pa pi'e no ju'u reno
1;0 base 20

which is equal to twenty.

Both “pi” and “pi'e” can be used to express large-base decimal fractions:

✥10.10  li pa pi'e vo pi ze ju'u reno du li re vo pi ci mu
the-number 1;4.7 base 20 equals the-number 24.35

“pi'e” is also used where the base of each digit is vague, as in the numbering of the examples in this chapter:

✥10.11  dei jufra panopi'epapamoi
This-utterance is-a-sentence-type-of 10;11th-thing.
This is Sentence 10.11.