This is an essay I wrote early in my toki pona learning days and have been refining since. It’s a work in progress!
tan jan Tapuni
Learning toki pona has been a fascinating experience: I’ve been at it for a few days so far, and I’m already recognizing words and understanding simple sentences. 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, 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.
As I’m a new language learner, its not really my place to suggest changes, as I lack the breadth of knowledge necessary to know how a change will interact with specific phrases or sentence constructions. Instead, please read the following as describing a particular frustration. In crafting this system I tried to add precision while still staying true to the spirit of toki pona, or at least how that spirit feels to me today. My feelings may change in the future, and that’s okay.
Lastly, please note that this is not intended to replace one/two/many, only to add precision where it’s wanted, similar to the 1/2/5/20/100 system used today. In fact, this builds on that system. Let’s begin…
~ Small Numbers ~
Already, toki pona has this covered for one, two, and five. Three is equal to two + one, and four is equal to two + two. However, I argue that the small numbers up to five (the number you can count on one hand) are so key to thought that they should all be discrete concepts with value beyond that of simply counting objects.
Three in particular feels like a curious omission, as three is also the number of the triangle. Children’s blocks often come in three shapes: circle (cylinder), square (cube/rectangular prism), and triangle (wedge/ramp). Three represents stability and independence, such as the minimum number of legs a stool needs to stand alone, three objects stacked into a pyramid, or triangle shapes used for building. This shape also includes pyramids, and mathematically, three is the minimum number of points needed to describe a plane. Three is the number at which people become a group, where conversation is no longer simply one-on-one but a speaker must consider group as a whole.
Love triangles in particular are an interesting case study, I feel. To describe a love triangle as “two plus one” inherently suggests that a love triangle is composed of a couple plus an interloper, even if the three don’t know which is which yet. Or it suggests two suitors fighting for the attention of a single target. Either way the third wheel will be discovered and expelled and the couple can emerge. The concept of three rejects this framing and places everyone involved on equal footing, and suggests a future where all three could remain together as a unit.
While some of these “three” concepts drift into the semantic spaces of other words (independence could be considered part of wan, love triangle/conflict strays into utala, becoming a group is already the essence of mute), “three” encapsulates them all and specifically highlights the precise point at which less becomes more. For all those reasons and others I’m sure I have yet to discover, “three” deserves its own semantic space.
So how to call “three”? In my mind there are two leading options. First, three could be added to the semantic space of kule, as color is already “three shaped” (we have trichromatic vision) and the sitelen pona glyph for three already includes a triangle. However, kule encompasses the semantic space of all senses, of which color is only one. In particular, including “threeness” into kule could be ambiguous because we widely consider ourselves to have five primary senses.
Instead, I prefer reviving an old word: “tuli.” The word tuli already means “numeric three” but its semantic space would now include the entirety of “threeness” and “triangular.” I think an upside down empty triangle for its glyph in sitelen pona would work well. I also recognize that “kiki” already includes the concept of “triangle”, but since it describes a triangle as a “sharp shape” rather than a “three shape,” I feel it is a poor choice.
Four also includes concepts beyond the count of two plus two. Four represents balance, as the first even number beyond two. Four is the minimum number of points required to describe mathmatical space. Four points also describes two parallel lines, or squares. Fortunately toki pona already has “leko” for squares and the concept of “fourness” fits relatively cleanly into its existing semantic space, so I feel using “leko” for four is a natural fit.
Now that we have a sound foundation of small numbers (one through five), we can move on to…
~ Larger Numbers ~
We already have mute and ale for 20 and 100, respectively. For clarity, I like the idea of turning mute on its side when used for its numeric meaning: That way the number wan can be unambiguously written without its “flag” and still be understood, and three lines stacked scans more readily in large numbers than three lines side-by-side. Three horizontal lines won’t be confused with three tally marks. Three lines stacked also looks like a stack of objects, which could conceivably contain twenty things.
More importantly, we can now treat 5, 20, and 100 not only as numbers but as place values. Ten becomes “two fives” and fifteen “three fives.” Forty becomes “two twenties,” sixty “three twenties,” and eighty “four twenties.” Thus, we can write numbers similarly to other languages that use separate glyphs for place values, such as Japanese:
1: wan
2: tu
3: tuli
4: leko
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
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. Above 10,000, “multiplicative ale” becomes less and less readable as number gets larger. For that reason I prefer a different 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 to me. Instead, I like the idea of using sike to represent scientific notation. Using our place value notation, we can write ” sike”, which means “add zeros to the end.” As such, numeric 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. This method still doesn’t allow numbers like 123,456,789 to be expressed cleanly, but to me that feels in spirit with toka pona’s notion of only describing significant things. I feel like four significant digits is the most sike notation comfortably allows.
So that allows us to describe arbitrarily large quantities. What about small ones?
~ Fractions ~
Fractions are also a fundamental concept: In my opinion it’s important to be able to describe half or a third of something quickly and cleanly. For toki pona in particular, focusing on small fractions rather than decimal numbers feels more in spirit with the language anyways. To that end, I suggest “pakala” to mean “some part of one” (the whole is broken up into smaller parts). Like a decimal, the fractional part comes after pakala; in a sense, the number modifies pakala to describe which part.
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
To take more than one part, I use pi to create a fraction with a numerator and a denominator. Pi “modifies” the cut to mean “take (first number) pieces of (second number) cuts.” This also has the advantage of making a fraction look like a fraction.
3/5: pakala tuli pi luka
9/10: pakala luka leko pi tu luka
.37: pakala mute tuli luka tu pi ale
For very small numbers, sike after pakala “inverts” to describe a negative exponent.
.0001: pakala leko sike
.00000046: pakala tu mute luka wan pi luka tuli sike
6.626e-34 (Planck’s Constant) = 6,626e-31: pakala tuli mute luka wan ale mute luka wan pi mute tu luka wan sike
I highly recommended using sike only once in a number; on one side of pakala or the other. After all, if sike is already before pakala, any quantity after pakala would be insignificantly tiny by comparison.
Whole parts and fractional parts can be combined:
1 1/4: wan pakala leko
5/4: pakala luka pi leko
1.25: wan pakala mute luka pi ale
4 2/3: leko pakala tu pi tuli
12 3/4: tu luka tu pakala tuli pi leko
12.34: tu luka tu pakala mute tu luka leko pi ale
~ Arithmetic ~
This is technically a separate concern, but since mathematics is a very common use for numbers, it can’t hurt to define the operators here. They are as follows:
+: namako
-: weka
*: kulupu
/: kipisi
^: sewi
=: li
Variables are denoted by nanpa followed by a name, such as “nanpa A.”
~ Spare Concerns ~
I feel like this method describes a complete, concise numerical system which is still limited to describing quantities with at most four significant digits. Because of the use of shape glyphs for numbers, I call this system nanpa sijelo. I also see two “edge cases” in this system that are worth noting.
First, large numbers can be written in decimal scientific notation, but small numbers can’t, because pakala is already being used to “invert” sike. So while this is technically correct:
6.022e23: mute tuli sike luka wan pakala mute tu pi tu luka ale
…it’s discouraged because there’s no parity in using a decimal mantissa for very small 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 as noted above, an exception may be made for humor. And that’s all! Happy numbering!