This is an essay I started early in my toki pona learning days and have been refining since. I think this is its third revision, but it’s still a work in progress! This essay also references another essay—”toki tawa tuli sin”—to justify reviving the word “tuli”, but I’m still working on that essay.
tan jan Tapuni
Learning toki pona has been a fascinating experience. Its limited language makes translation a fun mental exercise that I often enjoy; instead of reaching for a word that I don’t remember, I’m learning to fit existing words together in new ways. However, like many new language learners, there are things about it I find frustrating. And like many new tokiponists, I’m frustrated with numbers.
I do think there’s an elegance to the one/two/many system of describing things, and I think in many situations it’s worth being imprecise to focus on the feeling or essence of a concept rather than a precise number. But in some cases precision is key, and here toki pona (and even nasin nanpa) leaves something to be desired. Aside from that lack, my brain loves numbers. She is very happy working with numbers, and numbers are one of the first things I pick up when learning a new language: while lists of vocabulary are challenging for me to memorize, math comes naturally and feels familiar regardless of language.
Here is my proposal for a new number system. At its core it’s based on how numbers are written in Japanese, and its goal is to make larger numbers less clunky to deal with. I call it “nanpa sijelo” due to its use of shape words to fill in the gaps for small numbers (see the Small Numbers section for details). Please note that this is not intended to replace the one/two/many system, only to add precision where it’s desired. You could say that it replaces nasin nanpa, or rather, extends that system. Let’s begin…
~ Small Numbers ~
nanpa sijelo requires unique words for all the numerals from one to five. toki pona already provides wan and tu, and nasin nanpa gives us luka, so we only need to assign words to three and four to proceed. This is where the name “nanpa sijelo” comes in; I believe leko is a natural fit for four, as it already represents squares and blocky objects with four sides. That just leaves us with finding a word for three.
Three is more tricky. Since toki pona lacks a word for triangle, I think the best way to fill this requirement is by reviving the word tuli (an old word for three), and expanding its semantic space to triangles, and giving it an appropriate glyph (an upside down triangle with a smaller triangle inside). Making the case for tuli is beyond the scope of this essay–see “toki tawa tuli sin” for more details on that–but as long as a word is assigned to three we can use nanpa sijelo.
In order to write any number from one to five, we just use its word on its own:
1 wan
2 tu
3 tuli
4 leko
5 luka
From that sound foundation, we can build all the rest. Let’s move on to…
~ Larger Numbers ~
nasin nanpa uses mute and ale for 20 and 100, respectively, and nanpa sijelo does the same. Like nasin nanpa, larger numbers are created via multiples of luka, mute, and ale. Unlike nasin nanpa, nanpa sijelo uses them as place values (similar to Japanese juu, hachi, sen, man for 10, 100, 1000, 10000). Thus, ten becomes “two fives” and fifteen “three fives.” Forty becomes “two twenties,” sixty “three twenties,” and eighty “four twenties.”
5: luka
6: luka wan
8: luka tuli
10: tu luka
15: tuli luka
20: mute
55: tu mute tuli luka
80: leko mute
99: leko mute tuli luka leko
100: ale
Note that there’s no need to write “wan luka” for five through nine; “luka” alone is sufficient. Also note that in nanpa sijelo, it’s customary to turn the glyph for mute on its side so it’s written with three horizontal strokes (similar to Japanese san). This is done to distinguish it from “tu wan” or “wan tu” to avoid confusion with nasin nanpa or three tally marks, and because the “flag” on one is often omitted in other languages. Three horizontal lines also looks like a stack of objects, which could conceivably contain twenty things.
Above 100, using ale as a place value is identical to nasin nanpa’s “multipicative ale” notation, but further simplified with the inclusion of three and four rather than having to repeat luka or mute several times.
234: tu ale mute tu luka leko
888: luka tuli ale leko mute luka tuli
1250: tu luka tu ale tu mute tu luka
2025: mute ale mute luka
6843: tuli mute luka tuli ale tu mute tuli
In nasin nanpa, numbers above 10,000 become less and less readable as ale is repeated multiple times. Instead of falling back on nasin nanpa here, let’s consider a new system.
~ Very Large Numbers ~
Using shapes for numeric values leaves us with an open question: What do we do about sike? Circles have many sides—an infinite number, in fact. But we already use “ale” for 100 and trying to introduce a number larger than that feels counterproductive. Instead, I use sike to represent scientific notation. Using our place value notation, we can write ” sike”, which means “add zeros to the end.” As such, sike is never used on its own (just as you’d never write “e” instead of “1e1”); it must have a number preceding it. However, if the number after sike is wan, that can be omitted.
10: wan sike
100: tu sike
300,000: luka sike tuli
63,000,000: luka wan sike tuli mute tuli
12,000,000,000,000 = 12e12: tu luka tu sike tu luka tu
6.022e23 (Avogadro’s Number) = 6,022e20: mute sike tuli mute ale mute tu
When using this notation, I strongly suggest multiplying the mantissa by a power of 10 in order to make it a whole number before writing it out using sike notation. The number before sike therefore denotes the least significant digit of the entire number. Numbers with many significant digits (e.g. 123,456,789) still can’t be expressed cleanly; like toni pona’s other number systems, it’s intentionally limited (just in different ways). It’s recommended that numbers be limited to four significant digits in nanpa sijelo.
Now we can describe arbitrarily large numbers with nanpa sijelo. What about small ones?
~ Fractions and Decimals ~
Numbers between one and zero are expressed using fractions. Since one is being broken into smaller pieces, “pakala” is used to denote the fractional part of a number, which comes after. A number after pakala represents one part after divided equally that many times:
1/2 (half): pakala tu
1/3 (third): pakala tuli
1/5 (fifth): pakala luka
1/8 (eighth): pakala luka tuli
Note that the numerator for these fractions is always one, and if the whole part of a number is zero, it may be omitted. If the numerator is not one, the denominator is followed by “sin” and then the numerator (i.e. the denominator is “repeated” multiple times). This is similar to Japanese fractions, which list the denominator first (e.g. san bunno ni = 2/3).
3/5: pakala luka sin tuli
9/10: pakala tu luka sin luka leko
11/16: pakala tuli luka wan sin tu luka wan
43/100: pakala ale sin tu mute tuli
Fractional parts can be combined with whole numbers to form mixed fractions, and numerators can exceed the denominator for improper fractions.
1 1/4: wan pakala leko
5/4: pakala leko sin
4 2/3: leko pakala tuli sin tu
13/5: pakala luka sin tu luka tuli
12 3/4: tu luka tu pakala leko sin tuli
Decimals are also expressed using sike notation after pakala. As with fractions, if the only significant digit is a single one (e.g. 0.1, 0.01, 0.0001), it may be omitted. Also note that like sike notation for large numbers, the number before sike denotes the least significant digit of the entire number, so you may need to multiply by a power of 10 in order to get a whole number mantissa.
0.0001: pakala leko sike
0.00000046: pakala luka tuli sike tu mute luka wan
6.626e-34 (Planck’s Constant) = 6,626e-31: pakala mute tu luka wan sike tuli mute luka wan ale mute luka wan
As with fractions, whole numbers and decimal parts can be combined:
1.25: wan pakala tu sike mute luka
12.34: tu luka tu pakala tu sike mute tu luka leko
It’s strongly recommended using sike only once in a number, either on one side of pakala or the other: if sike is already before pakala, any quantity after pakala must be insignificantly tiny by comparison.
~ Zero and Negative Numbers ~
nanpa sijelo uses ala for zero, though in many cases zero is simply omitted from numbers, such as with place values (i.e. 20 is “mute, not “mute ala luka ala”) or with fractions (i.e. 1/2 is “pakala tu”, not “ala pakala tu”). Unless you’re writing the number zero itself, ala is unnecessary. For numbers below zero, use weka before the number.
-1: weka wun
-4: weka leko
-6.5 weka luka wan pakala wun sike luka
~ Arithmetic ~
Numbers are used outside of arithmetic, but since arithmetic always uses numbers, it makes sense to consider them together. Arithmetic in nanpa sijelo uses infix notation, similar to most written languages. The toki pona words for the operators are as here:
+: namako
-: weka
*: kulupu
/: kipisi
^: sewi
=: li
A single variable in one or more equations can be expressed with nanpa. For example, consider this simple equation:
2x + 5 = 13: tu kulupu nanpa namako luka li tu luka tuli
Note that order of operations is unchanged. If the intent of the equation is to solve for the variable, nanpa can be replaced with seme. Presenting an equation with seme means “solve for <seme>.”
When there’s more than one variable, each variable should have a unique name using nanpa as its headnoun.
wh (Area of a square): nanpa Insa kulupu nanpa Sinpin
(pi)r^2: nanpa Sike kulupu nanpa Insa sewi ni
It’s worth noting that what to call variables is not well defined; it’s presumed that some common names (such as the variable for radius) will eventually be assigned by common convention. Also note that constants such as pi are written the same as variables: They are named numbers.
Functions or equations are to variables, except that they use the headnoun ilo.
Let f be the area of a rectangle: ilo Suli la nanpa Insa kulupu nanpa Sinpin
~ Spare Concerns ~
While this method describes a complete number system in my opinion, there are a few “edge cases” that are worth noting.
First, while large numbers can be written in decimal scientific notation, it’s confusing because sike is written twice:
6.022e23: mute tuli sike luka wan pakala tuli sike mute tu
Very small numbers written in decimal scientific notation are also ambiguous because weka could be negating either the exponent or the entire number:
6.626e-34: weka mute tu luka leko sike luka wan pakala tuli sike luka wan ale mute luka wan
Both of these are discouraged. See the Large Number and Small Number sections for the preferred method for representing these numbers. Likewise, using “ala sike” to mean one or “pakala ala” to mean .0 are both discouraged even though they’re technically correct:
1: ala sike (silly)
14.0: ala sike tu luka leko pakala ala pi tu leka (sillier)
…though an exception may be made if doing so is sufficiently amusing.
And that’s all! Happy numbering!