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.

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

pi'e

rexa

pi'e

paci

equals

the-number

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.