## 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
3:22:40```

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.```