AdvantagesAlthough a lot mathematicians figured out clear advantages of the duodecimal system and argued that we should switch to it because all numbers from 1 to 4 are a factor of 12, the unvigesimal system has other very useful advantages. Although less useful than duodecimal for dividing by whole numbers up to 4, it's more useful for dividing by whole numbers up to 18. Being 4-digit repeating also makes it easier to find the quotient and remainder of an arbitrarily large whole number on division by a number from 1 to 18. In fact, 19 and 23 are the only numbers up to 24 that don't offer a 4-digit repeating decimal notation on division of a whole number by but they're only 2 away from 21 so they're not that hard to do long division by. In addition to that, all fractions with a denominator up to 7 are 1-digit repeating and all fractions with a denominator up to 12 are 2-digit repeating. In addition to that, long division by 19 and 23 in unvigesimal is about just as fast as long division by 19 and 23 in binary for the same number of digits but unvigesimal is much faster for the same size number to be divided by.
Is it really worth making long division of larger numbers very hard just to avoid long division by 2 and 4 being only very minimally hard. Although duodecimal is better for adding fractions with a denominator up to 4 and giving the answer as a fraction, unvigesimal is better for adding fractions with a denominator up to 18 and leaving the answer as a 4-digit repeating decimal without converting back to fraction. If one desires to convert back to fraction anyway, there is a general strategy for convertinging any 4-digit repeating decimal back to fraction. Even if we did use doudecimal, it would still be useful to learn the method of adding fractions so that we could easily add fractions with a larger denominator than 4 as a fraction. Why should the people who want to use duodecimal get their say just because they're too stubborn to learn the method of adding fractions? Why should somebody be allowed to create their own problem then create somebody else's problem to undo their own problem? The real purpose of unvigesimal is for long division to get a quotient and remainder, not for adding fractions as a fraction.
The issue of struggling to memorize the single digit base 21 multiplication table can probably be removed by having students learn it on their own after training their memory. One way to train their memory might be by asking them a closed ended question they previously learned the answer to until they answer correctly and notifying them when they answer correctly and keeping on going with other questions. Hyperschooling might be a good way to train it that way. Learning the single digit multiplication table of base 21 after that training would remove the issue of struggling to memorize it. After having trained one's memory, already having memorized the single digit multiplication table, it would be so easy to mentally multiply or do long division in unvigesimal because one would have no trouble retaining the information they figured out mid computation and using it for later parts of the computation. If in addition to that, they also get hyperschooled to be great at noticing patterns in their past experience, they might figure out a faster multiplication and division algorithm on their own and use it mentally, as well as continuously building up faster multiplication short cuts with experience.
Another problem with unvigesimal is that it's hard to find a finger counting method for it. There's a solution to it. When addition and subtraction is defined it terms of unions of sets, we just have to hope they notice by themselves the short cut of counting up from a given number to add or counting down from it to subtract. When they're just taught directly the method of counting up or down from a given number, it's so easy for them to learn how to add and subtract without finger counting. They can learn later how to determine the number of objects in a set, and from that, it follows straight forward that you add to get the cardinality of a union and subtract to get the cardinality of a complement under the assumption that a set can't have more than one number of objects and a subset of a finite set is finite.
A whole number larger than 1 is prime iff no prime numbers whose square doesn't exceed it are a factor of it. One good use of unvigesimal is that the set of all positive integers that are not a multiple of 2, 3, or 7 is the set of all positive integers that are congruent to 1 or -1 mod 6 except those that are congruent to 7 or -7 mod 21 × 2. Even more, there's a head start for determining whether a number is prime by simultaneously seeing whether any prime number up to 17 is a factor of it. That can be done by determining what the number is mod 215 - 21. Those factors can be tested separately by computing other moduluses as follows:
- mod 21 to test 3, 7
- mod 212 - 1 to test 2, 5, 11
- mod 212 + 1 to test 13, 17
Even up to numbers as large as 174, about half the numbers that don't have any prime number up to 17 as a factor are prime making unvigesimal useful for finding prime numbers.
215 is incredibly close to 4,000,000. In fact, 215 = 4,084,101. For that reason, if the use of decimal gets replaced with the use of unvigesimal for almost everything in real life, the browser will list the number of views each YouTube video has in unvigesimal so, it's useful to create another article about YouTube videos with over 215 views with the same title except that 4,000,000 is replaced by the number 4,084,101 written in the yet to be decided unvigesimal digits.
- 26 is written as 31 in unvigesimal and so is easy to multiply by in unvigesimal, making it easy to convert from unvigesimal to binary or vice versa. Since in addition to that all numbers coprime to 21 are a power of 2 or a negative power of 2 modulo 21, that further simplifies the individual single digit multiplications that are part of the computation of a product of 2 larger numbers.
- 215 is incredibly close to a power of 2, making it easy to compute for any positive integer n, [nlog221] which is one less than the number of binary digits of 21n, where  is the floor function. In fact 215 is between 37 and 38. This is a mathematical coincidence since 215 is also incredibly close to 4,000,000.
- There are 7 days in a week and 7 is a factor of 21.
- 21 is odd making it easy to round for calculations in unvigesimal.
Conversion to and from binary
There are 2 ways to convert from unvigesimal to binary in your head, multiplication by 21 in binary and division by 2 in unvigesimal. Similarly, there are 2 ways to convert from binary to unvigesimal, multiplication by 2 in unvigesimal and division by 21 in binary. However, 2 is a really small number and so is a lot easier to multiply and divide by than 21 so conversion from unvigesimal to binary should be done as follows: divide by 2 and take the remainder as the last digit of the number in binary, divide by 2 again and take the remainder as the second last digit of the number in binary and keep going until you get a quotient of 0. Similarly conversion from binary to unvigesimal should be done as follows: start at 0 then double it in unvigesimal then add the 1st digit of the number in binary then double the result in unvigesimal then add the second digit of the number in binary and keep going until the last digit is added. Hyperschooling would actually be a very efficient method of teaching people how to do this in their head where a kid can't decide they can't be bothered to learn it and will be glad later that they were forced to make the effort to learn it.
For real numbers
A real number can be converted from binary to unvigesimal or vice versa by converting its integer part (floor function) and real part separately. To convert the real part from binary to unvigesimal, all you have to do is take the closed interval [0, 1] then if the 1st binary digit after the decimal is 0, divide both ends of the interval by 2 in unvigesimal to get a new interval and otherwise, add 1 to each end point then divide by 2, then do the same for the 2nd digit after the decimal; divide both end points by 2 in unvigesimal if that digit is 0 and add 1 then divide by 2 if that digit is 1 and keep on going. Each unvigesimal digit after the decimal can be computed once you narrow down the interval to one where both endpoints have the same unvigesimal digits up to that digit. If you know what the real part of a real number is up to a certain number of unvigesimal digits after the decimal point, all you have to do to convert it from unvigesimal to binary is take the closed interval with end points that number tapered off and the other number that can be gotten by incrementing the last digit by 1, then multiply by 2 and reduce mod 1 (take the real part of the result of multiplying by 2). If both endpoints increase, the 1st binary digit is 0 and if they both decrease, the 1st binary digit is 1, and if only the higher one increases, we don't know which it is. Given that both or neither of them increased during the operation on the interval, multiply both endpoints by 2 and reduce mod 1 again. If both endpoints increase, the 2nd digit is 0 and if they both decrease, the 2nd digit is 1, and if only the higher one increases, we don't know what it is. This is a strategy that guarentees that if the binary or unvigesimal digits of the real part are given correctly, all the digits in the other base after the conversion will also be given correctly and you lose information of what the exact real number was when you convert from one base to the other base in that way. Therefore, if you're given a certain number of digits of the real part of a real number and you convert to binary then convert back to unvigesimal using the methods described, you can't be guaranteed not to get back fewer unvigesimal digits than you started with.
Unvigesimal could be used for listing the price on a cash register but binary could be used for cash since it minimizes the number of coins in the cash. That is, there are only coins for monetary values that are worth a penny times a power of 2. It also creates evolutionary pressure for smartness if the cash register only says the price of the order in unvigesimal but does no converting to binary and does not have an option of punching in the amount of cash given to have it calculate the amount of change to give.
In addition to that, for each amount of money, there's only one way to have that amount of money in your pocket with at most one of any type of coin, so people could be taught in school how to calculate from the amount of money in their pocket expressed in binary and the price of their order expressed in unvigesimal, what the amount of money they should have left is in binary which tells them which coins they should have if they are to have at most one of each type of coin, and how to make themselves have those coins after the purchase by giving only those coins they shouldn't have after the purchase and getting back the coins they should have in the change that they don't already have, so that they will never suffer from a huge pileup of coins.