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.