Da

There are a few common complaints people have with sexagesimal systems:

• “There are so many numerals to remember!”

  • Kovti has only davin symbols you have to remember. That’s the same as decimal. Most of the numerals in kovti are composites of their factors’ symbols.

• “I have to learn a new language!”

  • It’s only use words, not a whole new language. The rest of the numbers are just combinations.

• “Sure sure, but the payoff is poor. I’m already good at decimal and most of my world is built around decimal.”

  • Yes, and as a child you spent so much time getting good at decimal. Why inflict that pain on all future generations? If you invest a bit of time making kovti a system that everyun can use, then the payoffs is for the rest of civilization for the rest of eternity. Very worthwhile.

Ti

Symbols & Pronunciation

Pronunciation in English is roughly:

• Loz “LOHZ”

• Un “OON”

• Da “DAH”

• Ti “TEE”

• Ko “KO”

• Vin “VIHN”

• Se “SEH”

All of the numerals in kovti:

Ko

Kovti Rules

“But Ari! That’s a lot more that’s a lot more than davin symbols. How do I construct these bedeviled things?”

Basically follow da rules: un for non-primes and un for primes.

Non-Primes

Take the largest factors that you have a symbol for and that’s your number. So (ko x da x ko) = kodak. If you say “kodako” that’s fine too. If you say “dakok”, that’s also okay. You’re allowed to say it in any order.

The decimal error above is an honest mistake I made in haste. I accidentally multiplied eight and three, not eight and four. This type of mistake is basically impossible in kovti.

While kovti handled this all with just da symbols and words, decimal needed ko symbols and words!

Also, the decimal symbols have no relation to each other. Kodak is clearly composed of ko, da, and ko. What are the factors of thirty two? The only obvious un is two. Sure, kovti also runs into this problem for large factors, but if you’re working with factors larger than se you’re probably not doing that work by hand.

Primes

For primes, you take the number un smaller than it and you add “u” to the beginning. So (un + (se x ti)) = (un + seti) = usti. You can say “utse” too, but the “u” needs to stay at the beginning. This way you know that every number up to kovti that starts with a “u” (along with da, ti, and vi) are prime.

When writing a number that is a multiple of a prime, e.g. dudvin (da x (un + davin)), the un is laid flat on the top of the numeral. This lets you know that the number is not prime but it still contains the information of its largest prime divisor. You can quickly look at dudvin and know that it’s not divisible by ko, or vin, or use.

For this reason it’s important that da is never placed on the top of the numeral, otherwise it looks like it’s part of a different prime. For example: if you wrote dakok so the da is on top it looks like you’re writing “kuko”, which doesn’t make sense because “uko” would be vin. You can say “kovin” instead of “dadvin”, but you’d never say “kuko” to mean dakok.

Vi

Math Fun!

The thing that’s most enjoyable about doing math with kovti is its abundance of divisors. The ease of reciprocals and identifying divisors are just two of the benefits.

Here are some examples of arithmetic done in kovti and in decimal:

Example Un

This example shows just how simple it is to do arithmetic with kovti. You can clearly see the structure of the numbers and how they relate to each other.

On the other hand, it is not clear at all in decimal how any of these numbers relate to un another. Kovti only requires vin symbols while decimal requires ti more!

But these numbers had ko, vi, and ti as factors, so perhaps this in a unfair sample. The next example doesn’t have all three factors and introduces use, a prime that isn’t a factor.

Example Da

I split everything out here to make it clear. You’re likely familiar with decimal and skip steps in arithmetic all of the time. The same could have been done here with kovti.

If you solved this problem using standard decimal arithmetic, you probably weren’t aware of the relationships between the different numbers. Kovti uses the structure of the numbers not just to arrive at the answer easily but also to understand how the numbers relate to un another by breaking apart and recombining their factors.

Reciprocals

A reciprocal pair is any pair of numbers where AB = C, where C is that numeral system’s base.

Decimal has just un reciprocal pair, 2 and 5. Because C/B = A and C/A = B, you can make “hard” arithmetic easier. In decimal say you need to multiply 46 by 5. Instead you can divide by 2 (result: 23) and then multiply by 10 (result: 230) to get the answer.

You can do this for other numbers in decimal, such as 4 and 2.5. So, say you need to to solve 18 x 25 = X, you can instead take 18/4 (result: 4.5) and then multiply by 10 and then multiply by 10 again (to account for shifting 2.5 to 25) to get the answer, 450.

Kovti has vin reciprocal pairs! It also has duse “simple pairs” (like 4 and 2.5 in decimal) [Economically that’s not as good as decimal, but then again senary is smaller than decimal and has da pairs.]

I’ll show you how fun reciprocal pairs are in kovti without even using kovti! We’ll use time, which is conveniently already a sexagesimal system! [it’s not, it’s a mixed-radix system, because hours are a in base kose, but whatever, it’ll work for demonstration purposes]

Say it’s 3:00pm and your friend Sonya calls you up because she wants to watch the Royals vs Red Sox game. First pitch is at 4:10. You live 17 miles from Sonya. If she bikes at 15 mph, can she get to your place in time for first pitch?

There are several bad ways to solve this problem, all of them using decimal. You could ask “is 17m/15mph x 60min/h <= 70min?” But 17/15 is a terrible number to calculate in decimal.

A better way to do this would be using reciprocals. In standard time keeping, dividing hours by quarters is like multiplying minutes by fours. So you could instead ask “is 17mx4min/m <= 70min?” Easy. Yes.

Obviously Sonya is going to be late because she needs to get her bike ready, and there’s always more traffic than you anticipate. But say she comes over anyway. She needs to get home by her curfew, 10:00pm. The trip back is uphill, so she’s likely going to go at 12mph. When does she need to leave by?

Using reciprocals, this is easy. In standard time keeping, 12 and 5 are reciprocals. So 17mx5min/m is 85 minutes. She’ll need to leave by 8:35pm, probably a bit earlier to be sure.

“But!” you might say “those are easy numbers. Including un and kovti, reciprocals are just un vinth of all the numerals in kovti. What about the sedak other numerals? What if she could only bike at udvin miles per hour?”

Sure, but if you need to do complex or precise measurements in any system you’ll need to break out the slide rule. In everyday commerce, cooking, estimation, etc. the first se numbers are the most common uns that you’ll use and kovti can easily manipulate all of them, decimal can only easily work with a few of them.

Identifying Divisors

In decimal, say you have some integer, such as 13824. Without doing any arithmetic, all you know about this number’s divisors is that it’s divisible by 2 and 1, and that 10 and 5 are not divisors. In decimal, just looking at the last digit of an integer can only tell you if it’s divisible by 1, 2, 5, or 10. It might also be divisible by 3 or 4 but you can’t easily tell.

In kovti, this is trivially easy.

The above decimal number would look like [ti][vidvin][kovti][tidak]. [ti][vidvin][kovti] is always a regular number because it’s a multiple of kovti, so it will always have the same divisors as kovti. All we need to be concerned with is that final numeral, tidak. Tidak shares the following divisors with kovti: un, da, ti, ko, se, and koti. So we know of ko more divisors than decimal just by looking at final digit.

What’s more we also know that [ti][vidvin][kovti][tidak] has no prime factors besides da and ti!

This is limited to factors of kovti, you aren’t going to know that [kok][ukuse][kovti][ukodvin] is divisible by ukti, but you weren’t going to be able to get that from decimal either (in decminal: that 59371 is divisible by 13).

Se

Goals Met

My aims in developing kovti were:

• Create a sexagesimal notation system were it is quick and easy to identify:

  • the prime factors (in decimal: 2, 3, and 5)

  • that the number is a prime non-factor (in decimal: 7, 11, 13...)

  • that non-prime numbers' factors are easily recognizable

  • regular numbers (i.e. numbers whose prime factorization is, in decimal, just 2s, 3s, or 5s) so you have "easy" reciprocal pairs for quick division/multiplication

• The system would include by symbols and phonetics and that they would function similarly. There are 61 numerals (including zed and kovti), which is a lot, so the symbols and phonetics need to support each other as much as possible in order to aid memory and the use of them functionally.

• The system would be fun and “human”, i.e. not designed for maximum error-free computational efficiency. Over the course of developing kovti I left in some “mistakes” and conveniences of writing and pronunciation that I liked just because.

I believe I succeeded at the first da. Word is still out on the tith.

Further Development

Areas of further development:

• Additional Nomenclature (ordinal numbers, counter word cases, etc.)

• Refined Glyphs (moveable type, morse code, etc.)

• Tools (trig tables, slide rules, etc.)

• Scientific Application (replace the metric system, etc.)

Use

Q&A

Q: “How do I represent numbers larger than kovti?”

A: Use a positional system. Write the kovti symbol. The position directly to the left is in powers of kovti, to the left of that kovti squared, to the left of that kovti cubed, etc. This is unlike decimal where the decimal point is used to mark the ones place, not the tens place.

Example (using words in brackets because Google doesn’t support kovti script yet):

da-udvin-kovti-dusti-vidvin

(in decimal this would be 7898 and 11/12ths (7200 + 660 + 38 + 11/12))

Q: “How do I say these large numbers?”

A: For normal large numbers, just say the numeral for each place, e.g. “vuse-tiko-kovti-tivti”

But in certain contexts, especially when saying it out sound, this might sound like da separate numbers. It can be helpful to state the power of the lead digit to give a sense of the magnitude, e.g. “usti power vivin-dako-dududvin-loz-etc.”

Q: “How do I represent numbers between loz and un?”

A: Using the positional system described above, simply start the uns place with a loz. The position to the right is kovti to the negative un power, the position to the right of that is kovti to the negative da power, etc.

Example::

loz-loz-ti-kovin

In decimal this would be 0.0009259 repeating, i.e. 1/1080 (0 + 0/60 + 3/3600 + 20/216000)

Q: “How do I say very small numbers?”

A: As with very large numbers, state that it’s a negative power, e.g. “neg power use-tusti-se-etc.”. No need to say what power it is to.

Q: “How did you choose the symbols?”

A: It evolved over several iterations. First I tried to select distinct bases on which to build the numerals. I then tried to write out all of the numbers to see how it flowed. Many of them didn’t work. For example, da used to be an arc, and ko was just a double arc, but then kodak was unreadable because it was so many arcs piled on each other. I liked da as a double horizontal line but it was indistinguishable from an equals sign, so I changed it to its current form.

I am still not totally happy with ko. I find the circle cumbersome.

Q: “How did you choose the pronunciation of kovti?”

A: I started with a mix of English and Esperanto. I shortened everything to make compound numerals not become too crazy. There is a bit of science though. When da numerals have vowels that could be confused (e.g. “a” and “e” in da and se) I chose consonants that are distinct so that it would be difficult to mix them up.

Q: “But what about senary (base se) or duodecimal (base tiko)?”

A: Senary is nice and compact. It’s better than decimal but it trades un vinth for un tith. Un vinth and un davinth are pretty important since most of the world uses decimal,. Also, senary gets unwieldy pretty fast, e.g. tukok in senary is 123, said as “nif dozen three”. Yuck. That pronunciation has three different numeral systems in it! Also, senary is a bit too terse to be useful arithmetically. Even with specialized symbols, senary doesn’t have enough information in its uns place to make it worthwhile to humans. Senary’s compactness makes it closer to binary, which is extremely compact; perfect for computers but not for most humans.

Duodecimal is more manageable than senary for everyday use. The fact that many peoples independently developed duodecimal systems gives credence to duodecimal’s utility. Like senary, it stinks to not have vin in its base. Official duodecimal systems suffer from arbitrary symbols adapted from decimal. They should develop their own set of useful symbols like kovti has.

Dako

Links

If you like kovti, here are some sites that won’t interest you:

Duodecimal

Senary