390 Quick Answers 19 February


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?   

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.  We're about to jump quite far back in time for Friday.  


Lecture Reactions

Our numerals began in India and then evolved as they passed through the Islamic empires.

How long does this take al-Khayyami?  I have no records, so I'm only guessing … weeks to months? 

I haven't thought about this before, but it's easy to see - an almond is a kite with right angles joining the two non-congruent adjacent pairs. 

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. 

Ok, I will say a tiny bit about geometric means since it's a valuable geometric step to know, and was often used. 

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).  

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. 

Each variable in x^3 + ax^2 + b^2x = c is a length, aside from c which is a volume. 

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.  No ruler questions the value of physicians.  That much should obviously be true.  The others require insight to appreciate. 

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”.  



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.  

Yes, Roman use is the origin of our foot and inch. 



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.