390 Quick Answers 5 February 

I have many done.  I will probably get project topics processed before I go home today.  Some may need to find a new topic for one reason or another (taken, or not meeting the intent).



Lecture Reactions

Curious question - why would we use an approximation for π?  For any practical purpose having an infinite irrational decimal is not useful, and for that matter it is not known.  Furthermore, just having π in the answer is not useful either.  We are nowhere near proving π is irrational, and the best approximations so far are the two by Archimedes.  They are better ones in today's reading.  For all approximations, presume they are done by finding circumference of polygons with a large number of sides. 

I don't think Caesar took credit for the calendar, he just made it legal. 

Sometimes we see references to where they got their work from.  Often not. 

Pythagoreans surely also used proofs by contradiction.  

Yes, numbers can be in more than one figural category. 

We've got a long way to go, but yes Hipparchus and Ptolemy are taking early steps to trigonometry.

Why isn't Heron's formula more often used?  1. It's harder to justify (surely), 2. it involves square roots.  It's more _useful_, but not really easier. 

The problem with Hypatia and the earlier Pandrosion is that we lack information.  Here's a different way to view both Hypatia and Pandrosion … neither of them made progress for women in mathematics.  Nothing changed because of them.  I'm willing to wait and give significant credit where we know it is due.  I'm not convinced either of these ancient examples are a big deal, nor that there aren't others doing more at the same time.  We just don't have evidence.  Good question - is there anything noteworthy about what either Hypatia or Pandrosion did that is not connected to them being women?  I would say … not much at best. 

Not all ancient mathematics features circles, we can find plenty that do not, and you easily could just look through our work, but … circles _are_ important because they are one of the most basic (that and lines) figures.  They are also subtle and interesting, and so worthy of our attention. 

When Intercalaris was used, the year was 355 days without it.  It intended to be used every 2 or 3 years, but it was neglected.  It ended with the year of (last) confusion when the Julian calendar began in 45 BCE. 

We did use results similar to the exterior angle theorem on the sphere.  The exterior angle theorem (either the equality or inequality version) is not true on the sphere. 

Spherical geometry is a non-Euclidean geometry.  It is worth remembering that, but it is also more different from Euclidean than hyperbolic is (for those who know, for those who don't - we'll get there).  Spherical geometry is more motivated by studying the earth and astronomy (celestial sphere), than studying the logic of alternatives with Euclidean.

Mathematicians used Greek, both Greek numerals and letters. 



Reading Reactions

Good question about Chinese numeration systems.  I think it's fair to think of them as number-words and numerals.  Rod numerals played the role of numerals, and there were also words (which were also symbols, only because all Chinese words are symbols).  We'll see some of both today. 

This is our first view of a base ten numeration system.  Alternating places helps with a lack of zero, but not completely.  Jeff obliquely hints that we don’t have evidence that our system can be traced to China.   It is a big step and a big deal.  Much of their algebra is by successive approximation.  Because of this, it mattered more if an answer was good enough than if it was correct.  Therefore they had little concern about termination or not.  If you only work with decimals and not fractions, there is no noticeable difference between irrational numbers and non-terminating rational numbers.  The Chinese were definitely working with something equivalent to decimals as you know them.  

It seems now to be growing common for new US presidents to spend their early time undoing all their predecessors did.  Emperor Shih Huang Ti took that to an extreme.  Maybe they would do that today if they had the power.  I'm sure some would. 

Chinese work with negative is impressive at this time, and it is curious their colour choices are switched from our current choices. 

Coins are much older than paper money. 

There's a small mention of zero in this section.  I will say more about it next time with India and some recent (2017) developments in history that are clearly not in the book. 

I wasn't going to talk about this, so I will say a little becaues it was asked:  Zhau Shuang in a comment on the 9 Chapters finds the legs of a right triangle if the hypotenuse and difference of the sides, d, is known.  From that Zhau Shuang suggests solving x^2 + dx = (c^2 - d^2)/2.  This produces a, see:  the LHS would be a^2-(b-a)a = ab, and the RHS is (c^2-b^2-a^2+2ab)/2 =ab.  Once you know a you can find b.  Solving the quadratic they did by successive approximation, as we will discuss.  It is related to synthetic division, yes. 

I believe anything you know just as "the remainder theorem" is not "the Chinese remainder theorem" which is a was of solving equations given remainders.  We will talk about it, it's not expressed the way that we will. 

The circular wall town is probably fiction for math problems. 

There is more new mathematics being made today than ever before in human history.  It is one of the big reasons we stop at 1950, not because there is not enough, but because there is way too much.  Just because you don't see it, doesn't mean it's not there. 

Old, but hinted at this time:  “would most mathematicians have been related to someone in the government or royal family, because I’d imagine it takes a lot of expensive schooling.”  Definitely “elites” in some way.  We will see exceptions, but this is worth admitting throughout our exploration.