Roman

The roman numeral system is familiar to us from clock faces and tv / film date notices. A curious system where straight angular figures, which can easily be carved with a straight knife, are used there are no rounds or curves.
     The real reason as to why TV and Movie dates are persistently numbered in this curious way, has been recently revealed as to prevent the general public from realizing that the film or programme is a few years old. But now we've got to 2000 - MM it's obvious to everyone, and the only way for them to continue the ruse would be to use a bar code on the screen instead.

     The Romans used a different symbol for each order of magnitude, there was no zero at all and the numbers get increasingly complex and long.
 

     The obvious disadvantages with this system is that it only deals with integers between 1 and a few thousand. And extremely long and cumbersome numbers often occur.
For example :-

Not to mention that 1 million would have to be represented by 100 M's, they obviously had no concept, or need to use numbers to represent numbers as large as this.

The Romans didn't actually use L, C, D and M as we do today, but non aphabetical symbols which we have approximated to these letters.

Fractions
Everyday romans
spelled out fractions, e.g.

Scientific Romans
Used sexagesimal fractions like the sumerians but using Ancient Greek symbols instead.
Pretty much the same system we use today for angles and time...

The Ancient Greeks used the alphabet to represent their number systems. If we represent this system using our modern alphabet, we get the following :-

The problem with this is that you rapidly run out of letters, and the large number of basic letters mean that you would have to spend a long time learning and getting a feel for the number system.

Because you don't give powers to the symbols, then you could put them in any order as you are just adding them together.

If you had a large amount of symbols, eg 26 letters upper case then 26 letters lower case, then this would be extremely wasteful in characters.
large numbers with say over seven digits, would still be seven digits long
26 characters would only reach 899, and another 26 would only double it.

The ancient people of Sumer, were very forward thinking and advanced, they are thought to be the first people to have come up with the now familiar system that we use today of ordered positions in the string. Also known as powers.

These systems use a small number of symbols and the position in the string indicate the magnitude. The positions are very important and a place holder symbol is necessary ( zero 0 ) for us, but a shell shape for the Sumerians.

The difference between our system and the Sumerians one is that we use base ten ( decimal ) and they used base sixty ( sexagesimal ). So influential was their system, that we still use it today.
Time in minutes and seconds is counted in Sumerian Sexagecimal, and also angles in degrees ( which are also divided into degrees, minutes and seconds ).
Long before the greek mathematician Pythagarus, the Sumerians were doing complicated mathematics with angles. It's possible that Pythagarus may have only re discovered, or re translated old Sumerian work.

Today when we're using a sexagecimal system, instead of having sixty symbols for the various digit figures, we use a shorthand decimal with special place markings denoted with apostrophes (') and speech marks (").

Eg

As there are 60 angular seconds in an angular minute, and 60 angular minutes in a degree, the full circle of 360 degrees elegantly matches the total number of minutes (360) in 1 degree.  Which means with this system that we are accurately dividing a full circle into 360 x 360 = 129600ths
360 degrees minus 1 second is 359º59'59"

The choice of 360 to describe a full circle is an excellent choice, and far better than a metric number like 1000 or 100, because 360 can be divided exactly by many different factors and is thus far more useful than 1000.

The first few prime factors of 360 are 7,11,13 which means that most of the very useful numbers up to ten can easily divide a circle with exact degrees.