Tag Archives: decimal

Early Chinese Mathematics

Math is something that is found all throughout history.  It was used for may different reasons, in many different cultures.  What I find interesting is how these different cultures learned some of the same ideas without even having knowledge of the others’ work. These works could be anything from counting systems to Pascal’s triangle.  It can also include how one culture passed its knowledge on to another. This makes you wonder how some ideas that were known in western civilization could also be found in Asia.  As I was looking into this I found some very interesting facts about mathematics in China. Some small examples of math found in China begin with something called oracle bone scripts: scripts carved into animal bones or turtle shells. These scripts contain some of the oldest records in China.  This, like the clay from babylonian times, had many different uses including math.  Chinese culture also had something called the six arts: Rites, Music, Archery, Charioteerring, Calligraphy, and Mathematics.  Men who excelled in these arts were known as perfect gentlemen.

In China, like in India, one can find the use of a base ten numeral system.  This is quite different from the Babylonians, which makes it seem like there must have been some conduit of knowledge between India and China.  In China, around 200 BCE, they used something called “rod numerals.”  Rod numeral counting is very similar to what we use today.  This counting system consisted of digits that ranged from one to nine, as well as 9 more digits to represent the first nine multiples of 10.  The numbers one through nine were represented by rods going vertically, while the numbers of the power of 10 were horizontal.  This means that every other digit was horizontal while its neighbor was vertical.  For example 215 would be represented like this ||—|||||.  If one wanted to use a zero you would have to use an empty space.  The empty space is also something that can been seen in the Babylonian counting system.  As with the Babylonians, a symbol was eventually used for zero.  Interestingly enough, before there was a symbol for zero, counting rods included negative numbers. A number being positive or negative depended on its color: black or red.  This idea of having negative numbers didn’t come about in another culture until around 620 CE in India.  It seems quite apparent that several ideas that originated in China could possibly have been passed on to a neighboring country. 

Rod numerals. Image: Gisling, via Wikimedia Commons.

Rod numerals. Image: Gisling, via Wikimedia Commons.

The use of counting rods as a counting system brought about another very interesting mathematical concept, the idea of a decimal system.  China first used decimal fractions in the 1st century BCE.  Fractions were used like they are today, with one number on top of another.  For example, today if you used a faction for one half, it would be written like this: 1/2.  Using rod numbers you can do the same thing like this: | / ||.  Not only could this be represented as a fraction but it could also be written as a decimal.  To do this one would simply write the number out and insert a special character to show where the whole number started.  For example, if you wanted to say 3.1213, you would write it as a whole number like this: |||—||—|||.  To show where the left side of the decimal starts, you would mark it with a symbol under the number to the left of the decimal point, in this case under the first 3.  To me the use of rod numbers is so similar to how we use our numbers today that even the arithmetic that was used can be done easily by someone in our culture.  Addition is done almost the same except they would work from left to right.  Multiplication and division were used as well.   The use of base ten as well as using rod numerals made complicated equations much easier to attain, such as the use of polynomials and even Pascals triangle.

The Yanghui Triangle. Image: Public domain, via Wikimedia Commons.

The triangle known as “Pascal’s” in the west, in a Chinese manuscript from 1303 CE. Image: Public domain, via Wikimedia Commons.

Centuries before Pascal, the Chinese knew about Pascal’s triangle.  Shen Kuo, a polymathic Chinese scientist was known to have used Pascal’s triangle in the 12th century CE.  It appears that knowledge of Pascal’s triangle begins even before this. The first finding of Pascal’s triangle was in ancient India around 200 BCE.  We can see that this idea was sprouting around and found evidence in different cultures, from Persia to China and to Europe.  This again makes one wonder how this knowledge base was passed around from one culture to another.  Lacking historic details, it is hard to see if this idea of Pascal’s triangles was thought up individually or if this concept was somehow passed from one culture to another.

It seems that in all cultures there is a need for counting, which in turn brings about the need for math.  The cultural implications can mean that you are a “perfect gentlemen” by having mathematical knowledge, or it could lead a greater knowledge that can be passed on to other cultures.  In China, we see that many ideas of numbers and mathematics were thought up on their own without having other culture’s ideas intervening.  We can also see that the knowledge that was passed on was able to thrive and turn into something even more intriguing.  It is apparent that we can always learn and teach others to help our knowledge grow.

Source:

Boyer, Carl B., and Uta C. Merzbach. A History of Mathematics. 3rd ed. Hoboken, NJ: Jon Wiley and Sons, 2010. Print.

http://en.wikipedia.org/wiki/Decimal#History

http://en.wikipedia.org/wiki/Rod_calculus

http://nrich.maths.org/5961

http://en.wikipedia.org/wiki/Pascal’s_triangle

We Need more Fingers

2010-07-20 Black windup alarm clock face

Image: Sun Ladder, via Wikimedia Commons.

Our base ten number system is so ingrained in us that it is difficult to imagine using anything else.  With our ten fingers to count on, it makes sense that we have ten symbols to represent numbers.  Despite this, other cultures have used extremely different number systems.  The Ancient Mesopotamians used a sexigesimal, or base 60, number system.  The Mayans used a vigesimal, or base 20 system.  Roman numerals are used to number the Rocky movies despite them being almost completely useless.  Most computer techies are familiar with binary and hexadecimal.  Many early peoples even used a system with only 5 symbols (Boyer 3).  Our current number system may be intuitive but it may not be the best one around.  What if we had an extra finger on each hand?  We would be using a much more useful number system.  We should move away from the decimal numbers we currently use, and switch to a base twelve, or dozenal, number system.

There are several reasons to seek more mainstream use of base 12.  The factors of a number, or the numbers that divide into it evenly, determine a lot about the number.  Twelve can be broken down into more factors than ten can be.  Ten is divisible by only 2 and 5, whereas 12 is divisible by 2, 3, 4, and 6.  This gives 12 an advantage over 10.  The additional factors make it easier to think of many fractions, such as fourths and sixths, since they will now have only have a single significant digit after the point.  This is particularly effective when dividing 1 into thirds because it will not leave us with an infinite series of 3s like it does in decimal.  The simple tricks that help us do arithmetic, such as the fact that in base ten if a number ends in an even number the whole number is even and thus is divisible by 2, depend on the factors that make up our number base.  Since 12 has more factors, similar tricks can be used for more numbers.  In base twelve, if a number ends in 0, 4, or 8 the entire number is divisible by 4, if it ends in a 0, 3, 6, or 9 then it is divisible by 3.  We will not even miss out on the trick for evens that base ten has since twelve is also divisible by 2.  The trick we now currently use for 11, where you alternate adding and subtracting the digits of a number and see if the resulting number is divisible by 11, will work for 13 when we make the switch, because now 13 will be the number that is one larger than our base.  Don’t be concerned about 11, because we will have a new trick for 11 in dozenal.  The trick we currently use for 9, where we just add up the digits of a number and then check if the sum is divisible by 9, will work with 11 once we change to dozenal.  It is easy to check the divisibility of far more numbers in dozenal than it is to check in decimal.

For a dozenal system, we would have to make some changes to the actual symbols we use.  We have 10 symbols to use for the numbers 0-9 and a base twelve number system would need 2 more symbols to represent ten and eleven.  There are many different sets of symbols we can use to fill the two new places.  Some number sets use *, and # to represent ten and eleven in order to correspond to the symbols on most phone number pads.  Others use X, and a backwards 3, and some use a backwards and upside down 2 and 3.  Some sets completely replace all the symbols we use for 0-9, along with adding two new symbols.  We could use any group of numbers that would help us acclimate to a base twelve system.

Some things that we already do everyday would assist in our transfer to a base twelve system.  We already have specific terminology for 12 and several of its powers.  We use the word dozen to refer to twelve, a gross for a dozen dozens, or twelve sets of twelve, and a great gross for a dozen gross, or twelve gross.  When looking at a clock, we already deal with twelves to determine the time.  Figuring out what the time is 5 hours after 10 pm is basically the same thing as adding 5 to 10 in base twelve.  As Professor James Monroe notes, thinking of egg cartons makes thinking of dozenal numbers easy.  If 1 egg carton holds 12 eggs, and 1 case holds twelve cartons, a number like 426 in base 12 can simply be thought of as 4 cases, 2 cartons, and 6 loose eggs.  Twelves seem almost as prominent in daily life as the number ten.

Despite the familiarity we already have with base twelve, switching to dozenal will still be incredibly difficult.  Because we would have more digits, kids would have to memorize larger multiplication tables.  Luckily, the tables will not be anywhere near as large as they would be if we still used Cuneiform.  However, the real difficulty in switching has little to do with what number base we want to use.  The trouble will be in converting all the numbers on everything that we use.  Every road sign, price tag, page number, and countless other places have numbers that will need to be converted to base twelve.  This will be more difficult than changing from using imperial units to metric units, and America still has not completely converted to metric.  Here in the U.S., some things are measured with metric units, but we still measure distances in miles and a sack of potatoes at the grocery store is measured in pounds.  Also, to add more difficulty, in metric we would have to come up with new prefixes that are based on powers of 12 instead of powers of 10.  Changing to dozenal numbers is such a monumental task we may not be able to accomplish it.

In spite of the difficulties, I believe we should do it.  The conversion will not happen overnight.  We must look further down the road.  Perhaps, start by teaching people how to do arithmetic in dozenal in addition to teaching them the usual decimal system that we use.  Then when people are comfortable with it, we could move on to using base twelve alongside decimal.  Eventually, our ancestors will be able to move on to a better number system than what we currently have.  Like the Kwisatz Haderach from Frank Herbert’s Dune, we must endure temporary struggles in order to achieve the Golden Path.

Source

Boyer, Carl B., and Uta C. Merzbach. A History of Mathematics. 3rd ed. Hoboken, NJ: Jon Wiley and Sons, 2010. Print.

Additional Material

http://www.dozenal.org/drupal/

http://www.dozenalsociety.org.uk/

http://youtu.be/U6xJfP7-HCc