## 18.4.  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:

Example 18.15.

 ma'u ci'i +∞

Example 18.16.

 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).

Example 18.17.

 ci'i no
 infinity zero ℵ 0 (a transfinite cardinal)

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

Example 18.18.

 pai
 pi, π

Example 18.19.

 te'o
 e

However, many combinations are as yet undefined:

Example 18.20.

 pa pi re pi ci
 1.2.3

Example 18.21.

 pa ni'u re 1 negative-sign 2

Example 18.21 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 Section 18.8 to Section 18.12 .