Numbers

TLDR: Utopia uses zero-indexing, tau for the circle constant, and custom words for numbers that are regular, yet distinct.

Prerequisites: Counting and Writing

Since Utopia uses a distinct counting system and writing system, it’s only natural for it to have its own method of specifying numbers. We’ve already touched on a bit of the number system, in the essay on counting. But let’s make it a bit more concrete, now that we know a little more about the Utopian alphabet.

We know that Utopian numbers must start with a symbol that indicates that a number is beginning. There’s a symbol for simple numbers (⌊), which use base-6, and for positive (+) and negative (-) compressed numbers which use pseudo-base-36. It’s good for a language to have a correspondence between written and spoken forms, so let’s choose sounds for these symbols. Somewhat arbitrarily I’ll pick sm (voiced alveolar sibilant followed by voiced bilabial nasal) for simple numbers, s (voiceless alveolar sibilant) for positive compressed numbers, and sh (voiceless postalveolar fricative) for negative compressed numbers.

The first vowel letter in the alphabet is u, which most naturally corresponds to the close back rounded vowel (“oo” in “soon”), and the first consonant is n (“n” in “dune”). Thus the first (basic) number in our Utopian system is smun, which is basically pronounced like an “s” followed by the English word “moon.”

The smun’th Floor/Year/Century

Let’s say you walk in the front door to an apartment building. What floor are you on? Americans would say you’re on the first floor, while Brits would say you’re on the ground floor, and the first floor is up a flight of stairs. It’s almost as though British people start counting floors at 0, but alas, this is probably just a quirk of language.

But it shouldn’t be! When we count things, 0 is a totally valid number of things to count. How many zebras are in the room with you right now? Zero. This number comes before any other number. Thus the ordinal number that should come before any other ordinal is zero — i.e. we should count all things by saying zeroth instead of first (and first instead of second and so on).

This would alleviate all kinds of problems (after dealing with the chaos and confusion involved in switching), such as the numbering of years and centuries on the calendar. The year 1950 is “in the 20th century” because European monks literally didn’t know about the number zero until hundreds of years after the year numbering system was developed. Thus the year 50 took place “in the 1st century” and 150 was in “the 2nd century.” Ugh. Making matters worse, there’s no year zero for the same reason, making the year 100 “the 99th year in the 1st century”. The 21st century thus technically started on January 1, 2001, not in 2000.

If we started with 0, things would truly be blissful. The zeroth hour would be before 1-o-clock, and the first hour would be any time that starts 1:__. The year 1950 would be in the 19th century, and the millennium would have started in 2000.

Another way of thinking about this is that when using numbers as ordinals (a.k.a. for ranking/ordering) I am proposing that we talk about how many we’ve counted before the item in question, rather than how many we’ve counted after we count the item in question. English isn’t even consistent about this, as we usually say that two competitors with nobody before them are “tied for first-place,” rather than “tied for second place.”

Thus smun is more precisely the zeroth number, a.k.a. zero.

More Numbers

Let’s proceed with numbers by choosing consonants in order, and choosing vowels in order to maximize clarity. It would be no-good to have adjacent numbers sounding too similar! Here are the next five (simple) numbers:

1. smɛm — “sm” + (“em” in “them”)

2. smʌk — “smuck”

3. smit — “sm” + (“eet” in “beet”)

4. smap — “sm” + (“aa” in “saag”) + “p”

5. smof — “sm” + (“o” in “go” as pronounced with an Pakistani/Indian accent) + “f”

Compressed representations use the same rhyme with an onset of s (+) or ʃ (“sh”) (-) instead of sm for the first six numbers. After that I’ve selected rhymes for numbers based on a few simple rules based on their numerical properties:

1. Multiples of 6 take the diphthong oi (“oy” in “boy”), followed by a consonant based on the multiple¹:

   1. ±6 = _oim (e.g. “soim” for +6 or “shoim” for -6)

   2. ±12 = _oip (“soip”)

   3. ±18 = _oit (“soit”)

   4. ±24 = _oik (“soik”)

   5. ±30 = _oif (“soif”)

2. Prime numbers use the vowel ɛ (“e” in “met”) with (voiced) consonants chosen in order based on letters in the alphabet, eventually wrapping (after l/r) around to the plosives again (skipping the nasals):

   1. ±7 = _ɛx (“se” + “ch” in “loch”)

   2. ±11 = _ɛʒ (“se” + “ge” in “mirage”)

   3. ±13 = _ɛz (“sez”)

   4. ±17 = _ɛl (“sell”)

   5. ±19 = _ɛb (“seb”)

   6. ±23 = _ɛd (“said”)

   7. ±29 = _ɛg (“seg”)

   8. ±31 = _ɛv (“sev”)

3. If a number can be written as xⁿ (where n ≥ 2), use a w (voiced labial-velar approximant) followed by a vowel corresponding to the exponent and then a consonant corresponding to the base.

   1. ±8 = 2^3 = _wip (“sweep”)

   2. ±9 = 3^2 = _wʌt (“s” + “what”)

   3. ±16 = 2^4 = _wap (similar to “swap”)

   4. ±25 = 5^2 = _wʌf (“swuff”)

   5. ±27 = 3^3 = _wit (“sweet”)

   6. ±32 = 2^5 = _wop (“sw” + clipped “oh” + “p”)

4. All other numbers up to 35 can be written as the product of two different factors. We take the vowel from the smaller factor and the consonant from the larger factor.

   1. ±10 = 2×5 = _ʌf (“suff”)

   2. ±14 = 2×7 = _ʌx (“suh” + “ch” in “loch”)

   3. ±15 = 3×5 = _if (“seef”)

   4. ±20 = 4×5 = _af (similar to “soff”)

   5. ±21 = 3×7 = _ix (“see” + “ch” in “loch”)

   6. ±22 = 2×11 = _ʌʒ (“suh” + “ge” in “mirage”)

   7. ±26 = 2×13 = _ʌz (“suzz”)

   8. ±28 = 4×7 = _ax (“saw” + “ch” in “loch”)

   9. ±33 = 3×11 = _iʒ (“see” + “ge” in “mirage”; similar to “seige”)

   10. ±34 = 2×17 = _ʌl (“sull”)

   11. ±35 = 5×7 = _ox (similar to “loch” but with an “s” instead of “l”)

And then, of course, we can make larger numbers by combining digits, whether we’re using simple numbers or compressed numbers.

365 = ⌊5041 = smofunapɛm (“smoe-foon-apem”) = +5A = sofʌf (“so-fuff”)
186,282 = ⌊0324553 = smunitʌkapofofit (“smooneet-uh-cap-oaf-oaf-eet”) = +IQZ3 = soitʌzoxit (“soituzz-och-eet”)

While it takes some effort to memorize all the digits, as can be plainly seen, this system makes communicating numbers extremely efficient. This is especially true for scientific notation, which can be expressed by slipping in a rein (“rain”) or reiz (“raze”) syllable in the middle of a number to express a positive or negative exponent, respectively. Likewise, just inserting the “r” sound can mark the changeover into the fractional part of a number. No leading zero is needed for fractions less than 1, but the “r” changes to be pronounced as an “l.”

0.16666 ≈ ⌊0.1 = +0.1 = slɛm (“slem”)
3.14159 ≈ ⌊3.05033 = +3.53I = sitrofitoit (“seat-rofeet-oit”)
3 × 10⁸ = ⌊02521043454 = +CH1MY4 ≈ +5eA = sofreinʌf (“soaf-reinuff”)

For vowel sounds for xⁿ digits (e.g. wʌ or wi) immediately after an l/r, it’s acceptable change the “w” sound into an u (“oo” in “too”). Thus “slwʌt” (0.25) becomes “sluʌt” (“sloo-uht”).

Too Many

Of course, sometimes one wants to talk about quantities without specifying a specific number, even to an order-of-magnitude. There are two reasons for this, that blend together. The first is that it’s hard to count, and the speaker doesn’t want to bother. The second is that it’s impossible to say how many/much because the quantity is infinite. Both of these concepts use the same “s” vs “sh” distinction for positive vs negative quantities, and the diphthong “ai” or “aɪ” (like the English word “eye”).

• I can’t be bothered to count (>2 implied) = _ait (“sight”)

• I couldn’t count if I tried, but it’s possible in principle = _ail (“sigh” + “l”)

• Countably infinite (ℵ₀) = _ain (“sign”)

• Uncountably infinite (𝔠) = _aiθ (“scythe”)

Angles

I join now with the chorus of voices on my side of the internet that say, in unison: the circle constant is tau, not pi! Here’s a good, 5-minute Vi Hart rant from 2011.

Okay, so with that out of the way, let’s consider how we might specify tau in our Utopian framework. Well, we can always approximate it with a fraction:

τ ≈ 6.2831853 ≈ ⌊01.1411 = +6.A7 = soimrʌfɛx (“soimruffech”)

But this is imprecise. If I say “soimruffech” I’m giving a limited approximation, rather than specifying tau exactly. What we need is a number that is exactly tau. So we invent one! Let’s have “sk” be the prefix for mathematical constants, and have tau be the sixth constant, reflecting its proximity to the number six.

τ = skoim (“skoim”)

We can now easily talk about multiples of tau by tacking more digits on after the skoim. The most natural multiples of tau are actually fractions, which we can indicate via a standard insertion of the letter “l” between the skoim and the fractional digits.

π = 180° = τ×0.5 = τ×⌊0.30 = τ×(+0.I) = skoimloit (“skoim-loit”)
90° = τ×0.25 = τ×⌊0.13 = τ×(+0.9) = skoimluʌt (“skoim-loo-uht”)
60° = τ×⌊0.1 = τ×(+0.6) = skoimloim (“skoim-loim”)
45° = τ×⌊0.043 = τ×(+0.4I) = skoimlapoit (“skoim-la-poit”)
30° = τ×⌊0.03 = τ×(+0.3) = skoimlit (“skoim-leet”)
15° = τ×⌊0.013 = τ×(+0.1I) = skoimlɛmoit (“skoim-le-moit”)
10° = τ×⌊0.01 = τ×(+0.1) = skoimlɛm (“skoim-lem”)

Other Important Numbers

Since we’ve introduced “sk” to talk about mathematical constants, we can introduce some other important numbers. Just as we round tau off to 6, we give numbers to the constants that are near the numeric value.

1.618 = φ (Golden Ratio) = skʌk (“skuk”)
2.718 = e (Exponential Constant) = skit (“skeet”)

The mathematical constant associated with 1 is the square root of -1, often written with the letter i.

√-1 = i = skɛm (“skem”)

We can thus write imaginary numbers in the same way that we write multiples of tau.

3i = skemit (“skem-eat”)

In fact, since synonyms are nice, we have an alternative way of writing negative numbers by using the constant associated with zero.

-1 = skun (“skoon”)
-2 = skunʌk (“skoon-uck”)
-3 = skunit (“skoon-eat”)

Utopian Numbers

I believe Utopia has a far superior number system to our world. Thanks to clever design principles, numbers in Utopia are easy to say, easy to understand, and easy to remember. Utopians start by counting the zeroth item, making timekeeping more sane. In Utopia there is a clear linguistic distinction between a number being more-positive than usual and more extreme (positive or negative) than usual, rather than both concepts being lumped under an ambiguous adjective of “large.”

All angles in Utopia are measured in radians, but thanks to clever tricks of the language these are both easy to specify and make the relationship between radians and portions of a circle obvious.

1. Sometimes it’s particularly useful to reference the number 36 (normally sunɛm) with only one syllable. The way to do so would be soin.