Chapter 18: lojbau mekso: Mathematical Expressions in Lojban

4. Special numbers

The following cmavo are discussed in this section:

ci'i    PA  infinity
ka'o    PA  imaginary i, sqrt(-1)
pai PA  pi (approx 3.14159...)
te'o    PA  exponential e (approx 2.71828...)
fi'u    PA  golden ratio, phi,
        (1 + sqrt(5))/2 (approx. 1.61803...)

The last cmavo is the same as the fraction sign cmavo: a fraction sign with neither numerator nor denominator represents the golden ratio.

Numbers can have any of these digit, punctuation, and special-number cmavo of Sections 2, 3, and 4 in any combination:

✥4.1  ma'u ci'i
+¥

✥4.2  ci ka'o re
3i2 (a complex number equivalent to “3 + 2i”)

Note that “ka'o” is both a special number (meaning “i”) and a number punctuation mark (separating the real and the imaginary parts of a complex number).

✥4.3  ci'i no
infinity zero
À
ℵ0 (a transfinite cardinal)

The special numbers “pai” and “te'o” are mathematically important, which is why they are given their own cmavo:

✥4.4  pai
pi

✥4.5   te'o
e

However, many combinations are as yet undefined:

✥4.6  pa pi re pi ci
1.2.3

✥4.7   pa ni'u re
1 negative-sign 2

✥4.5 is not “1 minus 2”, which is represented by a different cmavo sequence altogether. It is a single number which has not been assigned a meaning. There are many such numbers which have no well-defined meaning; they may be used for experimental purposes or for future expansion of the Lojban number system.

It is possible, of course, that some of these “oddities” do have a meaningful use in some restricted area of mathematics. A mathematician appropriating these structures for specialized use needs to consider whether some other branch of mathematics would use the structure differently.

More information on numbers may be found in c18-§8 to c18-§12.