390 Quick Answers February 14

Lecture Reactions

The Jain large numbers are part of their mythology.  Telling these stories has a happy side benefit that people are exposed to large numbers on a regular basis. 

I am optimistic that we'll get the Bakhshali manuscript dating sorted in my lifetime.  Remember, 2017 is _very_ recent.  There is surely room for progress here.  There is no question that zero is actually in the manuscript.  It is all over it (as I showed).  The only question is how old it is.  It could be recent enough that this is commonplace, or old enough that this is revolutionary. 

I've got a lot to say about Islamic mathematics and not much to clarify from India … so on we go.

Reading Reactions

The single most important thing to remember about the rise of Islamic culture is being open to ideas from other cultures.  Islamic mathematics combines input from the Grecoromans to the west, the Indian and Chinese to the east, and even some from ancient Babylonian in the same region. 

We will talk about how and why we read numerals backwards.  Don't believe otherwise, we do. 

I enjoy that “algebra” which sounds fancy to many students has such a simple meaning.  Please share it with students learning it for the first time.  Algebra at this level (precalculus algebra) is mostly not something that is proven, but a language to speak and a way to express and find answers.  This is consistent with the fact that in HS algebra you didn't spend much time proving.  Furthermore, remember this is still all verbal algebra, no symbols, no equations. 

Neither ibn Turk nor any of the Islamic mathematicians were considering complex numbers.  They merely said the solution was not possible.  We will say more.

ibn Labban shows a curiosity of early use of base ten numerals, early on the whole number part was in base ten, but the fraction part was in base 60.  This is not so different from using degrees, minutes, seconds, e.g. 75° 24' 34"

Oh no, it’s time for the silly bible π story.  I had so happily banished this from my mind.  There is _one_ value for π in the bible.  It is 3.  That is the value that the Indiana state legislature attempted to make law.  Yes, that is true.  What isn’t true is that there is a much more accurate value encoded in the bible.  That is a fanciful tale and please see that and neither believe that or any similarly outlandish conspiracy that you read ever in your life.  Please never mention this again - to me, or to anyone else.  And please don't spread any other outlandish conspiracies. 

Negatives are … complicated at this stage.  Many times when seeking an answer it is about seeking a length, but negatives could show up part way toward a solution. 

Reminder:  trisecting angles in general is impossible by compass and straightedge (not proven until , so people occasionally search for other methods.

Let's talk a bit about the names for trigonometry functions.  Ah, more tradition that doesn't get changed.  Now that you know the peculiar history behind the term "sine", please do know that there are excellent reasons for the co-functions ("of the complement") and tangent (to the circle) and secant (to the circle). If you want to give a good meaningful name for sine, call it half-chord. 

Abu Kamil’s pentagon isn’t so difficult.  Put x on all the sides of the pentagon.  Remember the square sides are 10.  You’ll get some right triangles where the side lengths are all known in terms of x.  The MCRTT can be used to find x.  This is very likely his method, which is noteworthy for being similar to what you would do.  The angles are clearly not congruent - one is a right angle, and the others clearly are not. 

Surely memorisation in history, long throughout history, was much more relied upon than it is now. 

al-Sijzi’s derivation of (a+b)^3 would be similar to what one might do now with Algeblocks.  I will show here. 

We will discuss the solution to the cubic, and Khayyami in depth on Friday.  These are geometric solutions to equations.  Upcoming schedule is a bit odd, but it all fits together, I promise.  Ok, here’s what’s happening … al-Khayyami is getting pushed to Friday, and some of Friday is getting pushed to Monday for when you will read about what’s happening in Europe all this time.  Spoiler alert - it’s not much.  So, we’ll wrap up Islamic mathematics next Monday.