1. Numbers (numerals: Duodecimal System)

1-4-5. Duodecimal is another system of numerals in which the base is twelve. Compared to 10, 12 has more factors so it can be divided more easily, besides it can be counted by fingers: excluding thumb, we have four fingers, each is divided into 3 parts or consists of 3 finger bones; so we can count up to 12. We can use the thumb moving over other fingers to count the amount.

We need to define two symbols for ten and eleven. Either A and B, or T and E is used to represent ten and eleven accordingly. Furthermore, a rotated 2 (ᘔ) and a reversed 3 (Ɛ) are proposed by Sir Isaac Pitman as symbols for ten and eleven respectively.

Duodecimal is also called dozenal system. That’s why we have a pack of dozen or half dozen products especially drinks. Now we can see why 13 is considered as an unlucky number. The 13^th product, let’s say can of drink, can ruin the divisibility of 12, moreover 13 is a prime number (it doesn’t have any factors but 1 and itself) which will be discussed later. So imagine you wanted to share 12 cans of drink between six friends, and now you had 13 cans, what would you do? Then the mythologies have amplified the unlucky power of 13.

That’s also why the words eleven and twelve are different from teen numbers. They stemmed from the Deutsche words elf and zwölf respectively.

In order to compare different numeral systems, we need to know the fractions and index form (that’s why I try to introduce them briefly), therefore, all different bases will be discussed elaborately after the history of numerals will have been discussed up to Hindu-Arabic numerals that we use these days. We will try to develop all arithmetic operations based on different bases, then we can compare their efficiency and drawbacks.

1. Numbers (numerals: Roman numerals)

1-4-4. I've deliberately avoided to define the base-system since index form is the pre-requisite and I had planned to explore mathematics step by step and to build the new concepts upon the pre-requisites. I’d planned that after developing the index form, I could get back to numbers and discuss base-system with a new perspective. I, however, found it almost impossible to go through Roman numeral and compare in to Egyptian numerals without a precise definition of the base. So I have to rely on the knowledge of my readers about exponent and index form which I’m certain of, then I can move forward. Chronologically speaking, Egyptians and Romans were aware of the base, so we just study it using Hindu-Arabic numerals which we presently use.

In a nutshell, since it’s going to be discussed thoroughly, the base-system is grouping numbers according to the progressive powers of the base. For instance Egyptian numerals follow the base system because they group objects based on 10, then 10 groups of 10s which is a hundred; 10 groups of 100s which is a thousand and so on. Tally mark numerals, on the other side, doesn't have such property because the grouping is limited to 5. We could say that it’d follow the base system if they grouped 5 bundles of 5; then 5 bundles of 25 and so on; but the base doesn't develop progressively.

Romans developed their numerals based on Tally marks. However, they used five and ten to group numbers. The other difference was using alphabetic letters to represent numbers and they developed it to 1000 in the first place. Here are the letters Romans used to write their numbers.

Symbol	Value
I	1
V	5
X	10
L	50
C	100
D	500
M	1000

In order to simplify the symbols, Romans found out that they can define new numbers by changing the position of digits which can be considered one of the first attempts to come up with the place-value or positional notation. VI represents 6 while IV represents 4. LX is 60 but XL is 40.

So 1974 in Roman numeral is written MCMLXXIV.

The last thing needs to be mentioned about Roman numeral is writing 5000 and larger quantities. They drew a bar on top of a letter to represent 1000 units of the value. So x̄ means 10000.

In conclusion, though working with Roman numerals, to some extent, is easier than Egyptian numerals or Tally marks, calculation with it is still a hardship which normally arises through multiplication and division. There’s still a need to ameliorate the numerals.