390 Quick Answers 17 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.

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.  Relevant here:  you may not ever refer to someone just by "ibn" "abu" or "al".  That makes no sense. 

We're still well before the time when people get "credit" for ideas.  Same goes for plagiarism.  On the other hand, we do this now, and it is worth the effort to recognise those who contributed.

Lecture Reactions

Although Suzuki and I agree to call this time Islamic mathematics.  There are plenty of sources that call it Arabic mathematics.  I deeply respect both options. 

We are definitely seeing last time and this time how geometric early algebra was. 

The work for al-Khwarizmi and ibn Turn on solving quadratics is a different case than what we saw earlier from the Babylonians.  These two would have solved the other case in the way the Babylonians do.  It's not that one is more complete, but that they have different geometries.  When working geometrically without negatives, the cases all look different.  This work is much more tangible and physical than working with "x3" today. 

The value of decimals comes in the first three letters.  Using base ten for both the whole number and the fractional parts.  Before this it was done with sexigesimals in the fractional places.  Fractions don't really have a base, base is important for decimals.  And by "fractional" I mean the parts that are between 0 and 1. 

Postulates by their very nature as assumed and not proven.  The first four of Euclid's postulates are natural to assume.  The fifth is far from natural. 

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.   We have talked rather extensively about spherical geometry, and it is the geometry of the sphere.  We will say more about hyperbolic later, for those who aren't quite clear what it is. The main reason I'm not saying much more now is that _they_ weren't quite sure what it is yet. 




Reading Reactions

Please remember that I am discussing this reading on Friday.  I am not neglecting or ignoring it. 

"The idea of an approximate answer is very interesting because as I mentioned before they needed exact measurements when they are building structures."  We discuss this extensively in my elementary geometry course.  It is worth saying here - exact measurements are never possible.  Thankfully they are also never necessary. 

The counting board of al-Samaw'al was probably more like a place value "frame" than an abacus.  It is a big deal that he is moving beyond geometric algebra.  It seems so mundane to us, but x^4 and higher powers are a significant change of perspective.  We'll see why this is so when we look at Khayyami's cubic work today.  Throughout all of this, algebra remains verbal.  There were no variables.  We will see when they start.  In general it's a good rule to assume if we haven't said something about it, that is hasn't begun.  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.) 


I won't say much about this on Friday, but … we have seen the arithmetic triangle before, in China (Yang Hui) and India (Varahamihira).  I'm … surprised that you're surprised this time. 


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.