Monday, September 28, 2015

Around the Bases

I was bothered by a prompt concerning our number system. Why is this base 10, with 9 distinct symbols, system of counting the one in practice today? After some initial research, I found many other systems that seem just good. What could make one system better than another? How would the nature of mathematics have changed given a different system in place? Let's go through some of the history and make sense of how and why we count to ten.

The Babylonians 3100 BC: A base 60 system? seems absurd right? 
The Babylonian numeral system is credited as being the first positional numeral system. This means the symbol, and its position determined the value. A milestone in human history. We still see the effect this new development on time. 60 seconds in a minute, 60 minutes in an hour, however not exactly the same. Since arithmetic for large number, and powers of numbers in this system were not easy, it is no wonder it didn't survive. I look at this and see that calculations would be similar to that of Z[60].

The Egyptian Numerals 3000 BC: aha, a base 10 system.
Value1101001,00010,000100,000million, or


The difference here being that position didnt matter. However, surprisingly, by 1740BCE, the egyptians had developed a symbol for zero. They thought of this as a baseline and eventually thought of numerals as either positive and negative. Also, they even had symbols for fractions,  a huge contribution to how our base 10 numeral system works today.

The Maya numerals: 

A base twenty system in which 5 dots make a bar, and numbers after 19 were written vertically as powers of twenty like so:

So, their place holders were of vertical orientation. The use of bars and dots made arithmetic very straight forward. The Maya numeral system is most famous for its appearance in the Long Count Calendar. Our numeral system today doesn't reflect much of genius system the Maya used, however their use of zero rivaled that of their predecessors. 

The Roman Numerals 1000BC: Probably the most well known historic system of counting since some small applications using the roman numerals still exist today.
Surprisingly, this system did not have a 0. So the idea of negative numbers did not exist. Further, instead of writing out 4 I's, like IIII, the Romans used a sort of subtraction like IV, which we would say 5 minus 1 today. Aside from still using the Roman Numerals for basic applications like numbering lists, or telling time, the system was outclassed by another for it's ease of arithmetic and basic operations.

In a sense, the most widely used numbering system today, the Arabic numeral system, was not invented overnight. The Babylonians told us position matters. 61 is certainly different than 16 in our system. The Egyptians had a base ten for ease of application into multiplication and division, ours is also a base ten. The Maya had a base twenty, but knew that higher mathematics required very large numbers, something the Arabic systems also handles well. And the Roman Numerals, well... they are a special breed, much like the Chinese counting rods. These are used as special, or ceremonial techniques. This may be because they look nice, and can be detailed to show such special meaning.

It's was Fibonacci's famous book, Liber Abaci, that changed the way people described quantitative elements. As Fibonacci traveled the Mediterranean, he fell in love with a number system popular with Hindu and Arabic merchants.  The Hindu Arabic numeral system was born and Fibonacci set out to make it known to the world. In his book, he provided detailed instructions on how to convert from another system to the HA system, along with providing details on how to use this system in everyday applications like weight, money, and other quantities. He followed up with a second section on business, which helped economies grow. 

The role of our number system today: One thing I notice different about our modern day system is the decimal system. Of course, with a lot of work, any of the above systems could make numbers as big as they like,  but our system can handle values as small as we like. The role of our number system is especially important in application of very small numbers; like in chemistry or physics. In my opinion, the system we have today opened the door for the exploration into the hard sciences which in turn lead to a more advanced, technological civilization. However, scientist and researchers are still coming up with better number systems as the development of computer science field progresses. Traditionally, a binary system, 0s and 1s, is used in computer programming. However, some claim other systems, like that with base 16, can provide better programming and data storage. Who knows, maybe with the next 25 years we will be counting to VI (16) instead of 10.

1 comment:

  1. What divides the hindu arabic numerals is the use of 10 distinct symbols for the digits, combined with the positional value. Good on the extension to decimals, which led to the discovery of the continuum - quite an advance. We have imagined unimaginable numbers. Content: you could add a bit about our system, or tie in a little history (Indians, Persians or Fibonacci) to make this an exemplar.
    Other Cs +