390 Quick Answers 16 February

Hint for your project:  finding and analysing original sources (usually with help from a secondary source) is a great thing to do. 

Unusual Announcement:  There won’t be as many different topics in today’s lecture, as we discuss only al-Khayyami from two different perspectives.  As a consequence you may earn full credit for Friday’s reactions with only 3 lecture reactions (at most 1 from quick answers, as usual).  This will be the grading scale for this one time only:

If you have 8 reactions (5+3) I will score it as 10.
If you have 7 reactions (5+2) I will score it as 8.
If you have 6 reactions (5+1) I will score it as 6.  

Thank you for your participation always.

General hint:  If you think "oh, I know an easier way that is very simple" that usually means that you don't understand.  Be careful.  At least say "I don't understand" instead of "it looks simple to me" or "they just need to do this simple thing". 

Lecture Reactions

Someone asked about Islamic naming conventions, this is mentioned from the beginning of 4.1.2, but I'm putting it up here, since it is relevant to all.  al, i.e. al-Khwarismi is "from" or their location.  ibn is son of, and abu is father of. 

Although our language reads left to right, we didn’t change how numerals were written.  Very interesting question if in Arabic, 4321 is pronounced one twenty three hundred four thousand.  I honestly don’t know.  I would love to know more.  Our numerals are Hindu-Arabic.  They originated in India and then traveled into the Islamic culture, where they were refined.    I asked Ahmad about this, and he said the number is pronounced left to right, with the last two places inverted, as one might say "four and twenty".  He also says that in Arabic they don't use Arabic numerals (this may be changing) but something they call Indian numerals.  I also don't know if the sequence of reading has changed in history. 

You do recognise (we hope) that some quadratics have complex solutions.  This was not a consideration in the early days of algebra.  We'll notice when that starts.  Good general rule - if we haven't said that something important has started, it hasn't yet.  Catch to that - you need to remember what we've said (see negatives). 

Law of sines is a common research topic, but not this class.  The planar law of sines was known in India about 300 years before the spherical law of sines.  Sines don't cancel in fractions.  The rule of 4 quantities is a spherical result that does something similar to what similar triangles do on the plane.  It is only about spherical geometry and it is not the same as anything from planar geometry. 

Is the qibla problem an example of research mathematics?  No.  But, it's an example of applied mathematics, definitely (and not a silly made up story problem as an excuse to do mathematics).  Your angle to Mecca depends on where _you_ are, and so cannot be referenced by star positions.  The work we did is the way to solve this problem.  The basic ideas are unavoidable.  Remember - mathematicians make the table.  Others merely _use_ the table.

It won't be part of our al-Khayyami story, but he also tried to prove the fifth postulate by looking at quadrilaterals.   al-Haytham's examples were opposites, hence one was hyperbolic and the other spherical. 


Reading Reactions

Philosophy includes the nature of knowing, what it means for something to be true, it is somehow even more basic than mathematics. 

Fractions don't really have a base, base is important for decimals. 

al-Samawal is doing full polynomial long division, exactly like you do, just not writing the powers of x.  Synthetic division is more specialised.  (Varahamihira had worked with negatives fluently earlier.) 

Remember that astronomy was a part of mathematics.  Trigonometry was largely developed for use in astronomy.   This is true for planar and even moreso for all spherical trigonometry. 

al-Kashi's method (it was he, not Ulugh Beg) for finding roots is what we discussed with the Chinese.  Once the angle sum formula is known, double and triple identities are mere computations.  Angle sum was known long before al-Kashi.  "Calculator" = one who calculates.  805,306,368 = 2^28*3, so start with a hexagon (or 60° angle) and then bisect 26 times. It'd tedious, but not very interesting to do.

I think the comment about al Mu'taman and plagiarism is from a modern point of view, not in the time.  However, when text is duplicated word for word, there is no doubt, that is true then and now.