MEX paper, draft 1.0, part #1 of 5

[cowan (John Cowan)]

lojbau mekso: Mathematical Expressions in Lojban
Draft 1.0

1. Introduction

lojbau mekso is the part of the Lojban language that is tailored for
expressing statements of a mathematical character, or for adding
numerical information to non-mathematical statements.  Its formal design
goals include:

	representing all the different forms of expression used by
	mathematicians in their normal modes of writing;

	providing a vocabulary of commonly used mathematical terms
	which can readily be expanded to include newly coined words
	using the full resources of Lojban;

	permitting the formulation, both in writing and in speech,
	of unambiguous mathematical text.

The first goal requires that mekso not be constrained to a single
notation such as Polish notation or reverse Polish notation, but make
provision for all forms, with the most commonly used forms the most
easily used.

The second goal requires the provision of several conversion mechanisms,
so that the boundary between mekso and full Lojban can be crossed from
either side at many points.

The third goal is the most subtle.  Written mathematical expression is
culturally unambiguous, in the sense that mathematicians in all parts of
the world understand the same written texts to have the same meanings.
However, international mathematics does not prescribe unique forms.
For example, the expression

	3x + 2y

contains elided multiplication operators, but there are other possible
interpretations for the strings "3x" and "2y" than as mathematical
multiplication.  Therefore, the Lojban spoken form of this expression
must not elide the multiplication operators.

2.  Lojban Numbers

The general scheme of Lojban numbers will be briefly reviewed here.
There are cmavo for each of the 10 digits, and numbers greater than
9 are made by stringing together the cmavo.  Thus

	pa re ci
	one two three
	123 (one hundred and twenty three)

Therefore, there are no separate words for "ten", "hundred", etc.
A number can be given an explicit sign by the use of "ma'u" and "ni'u",
which are the plus and minus signs as distinct from the addition,
subtraction, and negation operators.  For example:

	ni'u pa
	minus-sign 1
	-1

Various numerical punctuation marks are likewise expressed by cmavo,
as illustrated in the following examples:

	ci pi pa vo pa mu
	three point one four one five
	3.1415

(In some cultures, a comma is used instead of a period in this function;
"pi" is the Lojban representation in either case.)

	pa fi'u ze
	one fraction seven
	1/7

	pi ra'e pa vo re bi mu ze
	point repeating one four two eight five seven
	0.142857142857...

	pa ki'o re ci vo ki'o mu xa ze
	one comma two three four comma five six seven
	1,234,567

(In some cultures, spaces are used in this function instead of commas;
"ki'o" is the Lojban representation in either case.)

There are also cmavo for a few special numbers:

	ci'i	infinity
	ka'o	imaginary i (the square root of -1)
	pai	pi (approx 3.14159...)
	te'o	exponential e (approx 2.71828...)
. 
Further details on numbers may be found in the selma'o catalogue under
selma'o PA.