390 Quick Answers 21 February


"Why didn't anyone"?

Quick answers are short today - not because of you, but because last lecture was so focused and the reading was rather thin.  

Diversity Summit next Tuesday.  I will ask about it on Friday.  Come prepared to talk.

Is it two weeks for your annotated bibliography?  Are there questions?   

For that matter, is it three weeks for the exam?  Are there topics?  As we get closer, please remember that I am happy to look at outlines.  Let's look at the syllabus about this now …

Be wary about date shifts from chapter to chapter or even section to section.  The book is _not_ strictly chronological, but ties region to region together.  Section 4.3 Ends in 1450.  Section 4.4 ranges 450 to 1215.  Section 5.1 ranges 900 to 1300.  Then 5.2 starts in around 650.  None of those numbers are important.  Being aware that we are jumping around when making your exam plans is important.  And, keeping somewhat aware of what is contemporary is valuable.  As I said, a timeline is a significant and important project for this course.   


Lecture Reactions

The Islamic prohibition of depiction of sentient beings is out of respect, something like not stealing their souls. 

We read through Khayyami's work for the last step.  The first step reads quite the same as I presented it, and as the cubic step.  The middle step would read quite differently, as he would not use our notation.  It is very important that algebra is still all verbal.  There is no notation like we use today - variables, operations, equals, none of it. 

Here's a review of the entire almond work:  The artisans set the constraint of the almond problem for their aesthetic properties.  They set the goal of wanting the original sides congruent.  al-Khayyami gives either a ratio of two sides, i.e. the tangent of the angle, or just the angle itself.  The tangent of the angle was x/10 where al-Khayyami picked the second length to be 10 (it was a ratio so you can pick one length - you might’ve picked 1).  All the geometry led to x satisfying x^3 + 200x = 20x^2 + 2000, which motivated Khayyami to solve cubics.   The solution to this cubic is approximately 15.4369, hence the angle is, rather interestingly, quite close to 1 radian, Not close enough that it could be exactly.  I think from my computations the angle is 0.99597 radians to at least those 5 decimal places. 

It's worth thinking about "what is difficult?"  The longest part of the argument was nothing more sophisticated than repeated use of similar triangles.  But, there is extensive detail to keep track of - all in rather small contained figure. 

Similar triangles and congruent triangles are all in Euclid and know to those before him. 

I believe that al-Khayyami was the first to solve cubic equations exactly (remember that the Chinese have a method for approximating any polynomial equation).   Because they were still working so geometrically, even asking the question for quartic takes a long time.  That being said, it wasn't viewed geometrically in India and China, when they did make progress on such questions. 
 
We have lots and lots more stories, but that was probably our biggest day, i.e. one big problem. 


Reading Reactions


Yes, you see that understanding of mathematics in the West has dropped to very low levels.  Don't forget what the dark ages were like in Europe. 

Leonardo’s notation is more than just mixed numbers, it is successive fractions.  We will see it in context.  It is similar to something one would use for 3 days 5 hours 12 minutes and 30 seconds.  One number that has a mix of different size subunits.  His notation isn’t inaccurate, just wasn’t widely used.

The trivium (grammar, rhetoric and logic) were not branches of mathematics, but the quadrivium (arithmetic, geometry, music, astronomy) were.  Although Iogic is today. 

Definitely the most significant consequence of Leonardo of Pisa was introducing and promoting Hindu-Arabic numerals in the west.  Liber Abaci is merely poorly translated to mean “Book of the Abacus”.  

The first known irrational number was probably the golden ratio in the time of the Pythagoreans, long before this.  There’s no much “first” for today’s reading.  It was also long known that √2 is irrational. 

Yes, Roman use is the origin of our foot and inch.  There were common units only within the Roman empire, but when the lands were conquered and repopulated, the new cultures didn't have common units.