## 18.10. 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:

Example 18.70.

 li panopano ju'u re du li pano The-number 1010 base 2 equals the-number 1 0.

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:

Example 18.71.

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

Example 18.72.

 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:

Example 18.73.

 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 for making 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:

Example 18.74.

 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:

Example 18.75.

 li ci pi'e rere pi'e vono su'i pi'e ci pi'e cici The-number 3 : 22 : 40 plus : 3 : 33
 du li ci pi'e rexa pi'e paci 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 Example 18.74 and Example 18.75 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 :

Example 18.76.

 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:

Example 18.77.

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

which is equal to ten, and:

Example 18.78.

 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 fractions:

Example 18.79.

 li pa pi'e vo pi ze ju'u reno The-number 1 ; 4 . 7 base 20
 du li revo pi cimu 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:

Example 18.80.

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