4. Special numbers

The following cmavo are discussed in this section:

     ci'i    PA  infinity
     ka'o    PA  imaginary i, sqrt(-1)
     pai     PA  π, pi (approx 3.14159...)
     te'o    PA  exponential e (approx 2.71828...)
     fi'u    PA  golden ratio, Φ, 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
       0 (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, π

4.5)   te'o
However, many combinations are as yet undefined:
4.6)   pa pi re pi ci

4.7)   pa ni'u re
       1 negative-sign 2
Example 4.7 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 Sections 8 to 12.