Formation of Modern Mathmatics
Before Western society was introduced to the “Arabic” — technically, the number system originated in India — numeral system, it used the Roman system which uses six symbols to represent a base 10 numeric system (repeats every 10 digits) I,V,X,L,C, and M which referred to the numbers 1,5,10,50,100,and 1000 respectively, and the placement of the symbols determined their value. If a symbol with a smaller value came before a larger, it was subtracted from the larger; if it cam after it was added, so IV meant 4 while VI meant 6.
However, these numbers quickly become cumbersome and are difficult to add: to add the numbers IVXLCMM and CMLXXXII, for example, one would first cancel out any numbers that are subtracted, then grouping like letters, then simplify from the highest to the lowest. Finally after a few minutes of effort it should end up with MMCMXXVI. Obviously, this is hard to do and takes a lot of time to add simple numbers (1944 and 982) and this process becomes even more complex when multiplying.
The second problem with Roman Numerals is that there are no fractions or decimals. Fractions can not be easily written out as 1⁄4 instead they would have to be written out in words. These factors combine and create bulky and inefficient system that actually hampered the spread and development of mathematical concepts such as pi.
In contrast the Hindu system, which was developed from around 2BC to 5AD in India, had three important innovations that would make it extremely useful. The first is that there was a 0. While this concept is not unique in the world, it did make it possible to do many new things. The second is the placement system: unlike the Roman system where numbers are not grouped, the Hindu system introduced the idea of columns and place values. Thus, by combining the 0 with the concept of base 10, we see the creation of a system where each column is 10 to a power (the 1’s column is 10^0, 10’s column is 10^1, 100’s column is 10^2 etc). To see the importance of this one can imagine a tower of champagne glasses, when the top is filed it over flows and fills the four below it, which in turn fill the 16 below them. Similarly when one column has a value of more than 9 the extra spills over into the next column making simple vertical addition possible. The third innovation that made the Hindu system unique is the decimal place. Unlike the roman system where you would have IV and twenty-two out of twenty-five the Hindu system allows a person to write 4.88. This idea is fundamental for calculating precise numbers, and the creation of many mathematical constants like pi could only be done with a decimal place.
However, even though the mathematical benefits of the Hindu system are obvious, it did not just spread across the world immediately. The numbers were first introduced in the 7th century AD, but it wasn’t until the 12th century AD that the entire system spread to Western Civilization.
The first wave was the numbers themselves which allowed people to write much more succinctly and easily. The Hindu’s had several different number systems over the period where the number system was developed, but the one we are most familiar with, and was absorbed by the Arabs, is the Nagari number system which is very similar to ours. In the system however there are a few modifications from its original form over the years; for example, originally the 1 resembled a 9, the number 4 was represented by an 8, a 5 was represented by a 4, the 6, 7 and 8 were quite unlike any of our current numbers, and the 9 was backwards. The numbers first spread to the Arabs around the 7th century — the earliest and most complete records come from the writings of a man named al-Biruni from the 11th century.
However, the mathematical concepts didn’t follow until the 12th century. A 9th century Arab named Al-Khwarizmi is credited with introducing the concepts of the mathematical possibilities of the numbers through his treatise which was translated into Latin in the early 12th century. After this the numbers quickly spread throughout Europe and the rest is history.