arithmeticreal numberssoft-question
I read the article "A Number System Invented by Inuit Schoolchildren Will Make Its Silicon Valley Debut".
The article claims that the system is strikingly visual when used for arithmetic.
My question, are there any computational advantages to such a representation?
Can it make doing numerical calculations more efficient, are the number bases used more beneficial in other contexts or use cases, are there advantages to teaching it, etc.?
You can write what you're after as
$$\lfloor{29\over2} \rfloor+\lfloor{29\over3} \rfloor+\lfloor{29\over5} \rfloor-\lfloor{29\over6} \rfloor-\lfloor{29\over10} \rfloor-\lfloor{29\over15} \rfloor+\lfloor{29\over30} \rfloor=14+9+5-4-2-1+0=21$$
In general, if you have a set of pairwise relatively prime (positive) integers $a_1,a_2,\ldots,a_n$ and want to know how many multiples of one of these there are less than some number $N$, the answer is
$$\lfloor{N-1\over a_1} \rfloor+\cdots+\lfloor{N-1\over a_n} \rfloor-\lfloor{N-1\over a_1a_2} \rfloor-\cdots-\lfloor{N-1\over a_{n-1}a_n} \rfloor+\cdots+(-1)^{n+1}\lfloor{N-1\over a_1\cdots a_n} \rfloor$$
by inclusion-exclusion, as JMac31 mentioned in comments. In general this gives you $2^n-1$ numbers to compute. I don't think you can do any better than that if you want to get the exact number.
On the other hand, if (as is the case in the OP's example), the $a_i$'s are distinct primes and $N$ is their product, then the answer is
$$N-1-(a_1-1)(a_2-1)\cdots(a_n-1)$$
This is mainly the multiplicative property of the Euler phi function.
Make a table with two columns, and enter the two numbers to be multiplied into the first row. Make the next row by halving the first number (discarding remainders) and doubling the second. Continue until there is nothing left to halve. Cross out all the rows where the left number is even. Add the remaining numbers in the second column. The result is the product of the first two numbers
Examples
Best Answer
It is difficult to answer a question like this objectively—any discussion regarding advantages or benefits of any representation of knowledge requires a nuanced discussion of what is considered an advantage and to whom they are advantages. Is it an advantage if the numeral system is easier to do "hand-calculations" with but is more computationally inefficient? Is it an advantage if it is more visually intuitive for representing numbers below 100 than the Arabic numerals, but less if not? (You get the point, I hope.)
Ultimately, the Kaktovik numerals appear to be a way to translate Iñupiaq word-conceptions of numbers into a numeral system that is much more intuitive in that language than the Arabic numerals. The stroke-based system, similar to the Roman numerals, makes developing a visual intuition for arithmetic operations on numbers smaller than 20, or close to 20, easier* than with their Arabic counterparts (*it will depend on what you find more intuitive, of course). I do not think that the numerals are more efficient for purely computation purposes, nor do I think there is any ground-shattering breakthrough in the conceptualization of numbers via the development of this system.
What I subjectively find very meaningful about the development of this numeral system, and the push to represent such numbers in Unicode, is more humanistic and concrete; it is allowing younger members of Iñupiat communities to learn arithmetic in a more efficient way by helping them describe numbers in a format that is more cohesive with their primary language, and allowing them to transfer those mathematical skills into computers (where everyone does most of their math nowadays) more efficiently. And it is also a beautiful (subjective!) mathematical construction in the sense that it is a number representation that appears to have been largely motivated, and developed by, schoolchildren and their needs.
So, should we all switch to using and/or teaching Kaktovik numerals instead of Arabic ones? Probably not. But, in my opinion, they're still a wonderfully creative and materially beneficial conceptualization of numbers!