Having determined what could potentially be considered flaws in the current base system, I then decided to look at other mathematical systems to determine potential remedies. I must admit to my surprise that nearly every mathematical system discovered on our planet fits into a description of a base system. While base 10 is the most common, base 5, base 20, and base 12 systems are also commonplace. Some are purely additive in numbering, while some are both additive and subtractive.
First, Let’s take a look at each of the Mathematical systems as currently defined and named (later I will be redefining terms and renaming some of these to enable greater ease of understanding).
Base ‘X’ system: the one most commonly used in the world today, where ‘X’ can be any number. It declares that each position in any given number has a value of ‘d’*(‘X’^’N’) where “d” is the digit used, X is the base and N is the number of positions left of the decimal point -1 (making the first position left of decimal point X^0 which = 1s place)
Dual Base system: similar to the Base ‘X’ system, but uses two base numbers. The most common of these systems is a Bi-Quinary system (a 2 base and a 5 base) such as the Roman Numerals. In these systems the above formula for the Base ‘X’ system is revised to look like: ‘d’*(‘X’^’N’)*(‘Y’^’M’). At each position left of the ones place you would increase either ‘N’ or ‘M’ by one (alternating between the two for each new position)
Mixed Radix system: Seriously? You have to change the term for the decimal point into a “radial point”? Fine, we’ll just use “Radial point” from now on when we refer to “decimal point” (trust me, it actually makes things easier in the long run).
As I explained earlier the first position (a.k.a. inches) is still called the ones place and has the value of ‘d’=6… so (6) *1 = 6 inches. The second position (feet) is called the twelves position (from the base of the first position)… and so on and so on, with each new position multiplying the previous position’s radial base to the previous positions positional value to arrive at the new position’s value. Lastly, in a Mixed Radix system, if there is a finite number of positions the final (first position written) will have the ‘base’ of ∞ to show that there can be no additional positions beyond it, and that there is no max numeral value in that position.
Positional systems: So far we have primarily talked about Positional Systems. In a Positional system the position of a symbol (I.e. a number) as well as its form, both have a determination on its value, usually in conjunction with the base.
Order-Positional systems: Now, in an Order-Positional system, the actual position of the symbol has only a minor influence on the overall value. Roman Numerals are an example of an Order-Positional system. I = 1, II=2, III=3, IV=4, V=5, VI=6, etc. In Roman Numerals the position of each “I” doesn’t matter, the “I” means one… it only matters if it comes before or after the “V” (5) or any other larger valued numeral. When a lower value numeral (such as “I”) comes before a higher value numeral (such as “V” or “X”) it means “less than” (I.e. IIX = 2 less than 10… or 8). When a lower value numeral comes after a higher value it means “with ### more” (I.e. XVI = 10 with 5 more with 1 more… or 16). Combine these and you get something akin to: XXIV = “10 and 10 with 1 less than 5 more”… or 24.
Non-Positional system: However in a Non-Positional system the actual position of the symbol has absolutely no influence on the overall value. Nowadays this type of system is commonly seen as a “thinking puzzle” where you are given three or more images or symbols, which are placed in mathematical formulae, you are then given a value for the answer to these formulae and requested to determine the specific value of each given symbol or another given formulae. In these image oriented numbers, each number is only separated by a mathematical function, and each number can contain any combination of the images. The total value of the images is the number value, and order or placement of these images in the number does not matter, only the separation of the numbers by the aforementioned mathematical functions. An example is shown below:
Base Equivalent Values: most common in Order-Positional system, a Base Equivalent Value means that the value of the numeral is always equal to the total base value of that numeral. I.e. in Roman Numerals I = 1 (base value 2^0*5^0) V= 5 (2^0 * 5^1) X = 10 (2^1*5^1). Each numeral is unchanging, and has a specific value of it’s total base value.
Base Limiting Values: Nearly all of the systems I have come across are Base Limited Systems… or more precisely, the numerical value of a position can not be equal to or greater than the base value of that position. For example, in a base 10 system the highest numeral in any given position would be a 9(the numerals are 0,1,2,3,4,5,6,7,8,9… with 9 being the highest).
Unlimited Values: Any system in which the positioned numeral value can exceed that position’s base value. While not commonly seen or viewed as mathematical systems there are a number of Unlimited Value methods. Earlier I showed an example of a Mixed Radix System which is commonly known as the Imperial Measurement System. This system is a system of Unlimited Values, in that while 12 inches = 1 foot, we can declare an object’s measurement to be 15 inches… greater than the base of that position would normally allow. We can also say 6 feet and 3 inches instead of 2 yards and 3 inches. Monetary Systems are another type of Unlimited Value System, as we can physically hold 13 $5 bills. Just because 2 $5 bills is equal to one $10 bill does not mean that as soon as we gain that second $5 bill the two merge and become a $10 bill.
Declared Values: Normally only given for numerals in a Non-Positional system (Declared Numeral Values), but can also be given to the base values of a position (Declared Positional Values), similar in methodology to a Mixed Radix System.The difference between a Mixed Radix System and a Declared Positional Value system is three fold. First, a Declared Positional Value System is always considered to be an Unlimited Values system; second, a Declared Positional Value System can have positional values that cannot be naturally formed by solely increasing the positional value preceding it; thirdly, Declared Positional Value systems have no base.
One example of the second difference in a Declared Positional Value can be seen in the American Monetary System. When talking about change (cents) we have Penny (1), Nickle (5), Dime(10), and Quarter(25). In the case of cents you can never have an amount of Dimes equal precisely 1 Quarter. If you take this to Dollar Values, we have $1, $2, $5, $10, $20, $50, etc. And here you can not have an amount of $2s to equal precisely one $5, or any amount of $20s equal precisely one $50. As money is a physical medium for counting it is often considered to be Non-positional(Declared Numeral Values), though it *can* be written in the specific amounts of each coin and bill value, thus equating it to a Declared Positional Value.
The reason a Declared Positional Value system has no base is that it’s position’s values can not be written in whole integers as a base system. (seriously, try writing the above example as a base system). Because of this, these systems should be written differently than a base system… but lo and behold, I could find no evidence that it ever is written differently. In fact, most countries simply write their money amounts using their base system, regardless of the actual amount of individual values. It’s not until you look at business ledgers that you might even come across each bill value being counted separately. I believe it is due to this fact that Declared Positional Value systems are not considered as a legitimate system amongst mathematical methodologies.
Unfortunately, even after looking at each of these base systems, none of them truly corrects the flaws of our base system (as talked about in Article 1: Problems with the Current Base Mathematics System). In fact, other than the Declared Positional Value system, none of the ‘base systems’ could correct the flaws present inherent in trying to use a negative bases. This outcome fully reinforced my belief that our Current Base Mathematics System is WRONG. To see how I finally found the simple way to correct these flaws please continue to: Article 4: Correcting How our Base System works. If, however, you want to see a new non-base system for mathematics feel privileged to continue to: Article 3: A more intuitive numerical system
Article 3: Correcting How our Base System works
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