I will return the topics on Monday - with brief comments & suggestions.  Do check out the annotated bibliography assignment.  

Quick Answers §3.1

Things get named however they get named.  There are lots of misnomers throughout history, and almost never is there an attempt to rename things.  

I do not know of another purging of Chinese history like emperor Ti.  Ti was from Ch'in.  He seemed to want to erase the past to solidify the new ways.    He only saved things that he particular needed urgently.  

Both positional and verbal systems were used for numeration in China at the same time in the same places.  The verbal system is still used today, and is very similar to the way we write number words, except with characters instead of letters.  In the verbal system, there is a different symbol (word) for each of the digits.  Here are the symbols.  I'm guessing the verbal was more for common use and the positional was more for computation.  

We have not seen mention of a zero before this.  It's most needed when a place value system is used (otherwise it's not really important), and Babylonians didn't have one (I don't know how they compare in time - but remember when cultures are so separate in space, time is less important).  I think the Chinese rod numerals were placed on counting boards with actual places for digits (kinda like this).  Thus empty places could be noticed.  I'm guessing they dealt with the decimal point by context.  Nine chapters is notable for dealing with fractions in a modern way.  I do not know how they denoted them, but they worked with them as we now do.

Babylonians had a sexigesimal system.  It is the only decimal-like system about as old as the Chinese.  

We will see the Indian and Arabic history of our Hindu-Arabic numerals.  I do believe them to be aptly named.  

We are less familiar with Chinese mathematics because it isn't a direct precursor to our culture, and because there's not as extensive records of it.  Also, much of their accomplishment does not influence western mathematics directly because of the geographcal separation.  We see similar things, but they mostly don't lead to things you know directly because no one in the west knew about them.  

Nine chapters had rules for adding and subtracting integers, but I've not read evidence for multiplying or dividing.  Even adding/subtracting so early is very significant.  Europe will struggle with this for over 1000 years after.  

Said before, "fl.'' in a person's dates stands for "flourished" they were around and doing something notable around that time.  

We do not know who wrote Nine chapters.  Detailed information is quite difficult.  We know it written between 100 BCE and 100 CE.   The work on the

The Greeks seemed to get into their mind for something to be sure it needed to be done with geometry.  This was both a boon and a hindrance.  The Chinese, Indians and Islamics will succeed in different ways because of not having this limitation.  

The Great Wall separates china from beyond, not within China.  

Definitely chinese work on the pythagorean theorem was completely independent of Greek work.  

I do not believe the Chinese used variables.  I would say almost definitely not as far back as the Nine chapters.  

We have no archaelogical evidence of the Hsia dynasty, only later references to it.  It may have been created as part of the mythology, or it may have existed.  We do have artifacts from the Shang and all later dynasties.  This is why Shang is deemed historical and Hsia is uncertain.