Quick answers §4.1

Some particularly good questions this time.  Please try to be thoughtful and keep it up.  I do appreciate it.

Sebokht felt that *others* were too focused on the Greeks, not him.

It is not true that "most of the Arab's [sic] contributions to mathematics comes from Al Khwarizimi", as you will see in section 2 especially.  There's plenty to come.  There's a reason we're spending four days on islamic mathematics - they do a vast amount and take great steps to making mathematics what it is today.  

The House of Wisdom (a physical building which was a combination of a library and a museum and a place for processing knowledge)  was destroyed in 1258 when Hulagu sacked Baghdad.

Both Suzuki and I would recommend against making too much of the numerology of pi in the bible.  Although the Indiana State Legislature did try to pass a law declaring pi to be 3 based on the reference in Kings I.

There's plenty of geometry in Islamic mathematics - probably more than any other culture except the Greeks - there's just the rest of mathematics, too.

Arabic numerals are mostly just a refinement of the Indian numerals with some additions we will discuss.  

There is significant Islamic work on astronomy.  We will see discussion of it to come.  

Someone asked if in arabic the lower places are pronounced first.  I do not know about this, but I imagine it would still be so if it were so.  It is an interesting question, and I would guess yes, but this is complete speculation.  Modern Arabic is still written and read right to left.  The sequence of our numerals does come from this source.  And they do current still use numerals in the same order as we do, as they have.  

Variables are still a long way off, and Diophantus' pseudo-variables hadn't been transmitted into Islamic mathematics yet.  For reference, as you can see in our nifty new timeline, Diophantus was working in the 3rd century and al-Khwarizmi in the 9th century.  

The Islamic calendar is a lunar calendar (hence Ramadan isn't at a fixed time in our calendar).  I think there are 33 Islamic years every 32 of our years.  This is approximately correct - I didn't do the research to make it exactly correct.  

In discussions of al-jabr and al-muqabala, I only find them referring to addition and subtraction.  Clearly they divided and multiplied, too, but I haven't seen any explicit reference to that.  Maybe those methods didn't have names.  

Unlike other cultures before them, the Islamic empire spanned several historically disparate cultures, and intentionally went on quests for knowledge and information.   This probably has much to do with their great success.  

I think we see repeatedly in our explorations of history one non-mathematical theme - tolerance of other cultures leads to intellectual progress.  

I don't know anything about this (and we didn't discuss it in my Islamic history class), but exploring seems to suggest that "Islam" and "Muslim" do come from a common linguistic root.  

I'm skipping the estate problem in lieu of much more interesting work of al-Khwarizmi



Class notes


al-Khwarizmi - religious and philosophical study of mathematics "all numbers come from one" influenced by Indian, Greek, Babylonian, Chinese sources
    on Indian numbers
        add, subtract, multiply, divide, square roots, practical applications of ease of representing large numbers.
        rules of negatives:  
start with
(10 + 1)(10 + 2) = 10x10 + 10x1 + 10x2 + 1x2
generalise to
(10+x)(10+y) = 100 + 10x + 10y + xy
specify back to
(10 - 1)(10 - 1) = 100 - 10 -10 + (-1)(-1)
what must that be?  requires (-1)(-1) to equal 81.  

[Abu'l-Hassan al-Uqilidisi (Euclid copyist) - The book of Chapters on Hindu Arithmetic - 952
    oldest extant arabic arithemtic.  New numerals will succeed because of ease of computation.  Presents written form of arithmetic (rather than using dust boards).  Includes long multiplication.  Refers to decimal fractions - first time out of China  Precursor to decmial point.  .  As an example, halves 19 five times.  Describes how to increase a number by one tenth.  But only divides by two and ten.  ]But does use same base for positive and negative powers, not base sixty for negative powers, as had been common.  


 
    al-jabr waal-muqabala
        restoring & balancing
        first algebraic proofs
        still verbal algebra - symbolic is still a ways away.
        analysed quadratics - considering cases based on which terms were +/-/0.  Recognised (more than Diophantus) two solutions.  solutions included only positives, but some irrationals.  negatives will be mostly not studied in islamic mathematics.  (geometric algebra from greeks, rule of three and number facility from india, verbal relates to bablyonian roots [and arguments] - and lack of symbols indicates no connection with diophantus)
        not groundbreaking, but thorough, practical, and well-timed  
        
    example:
        "What must be the square which, when increased by ten of its own roots, amounts to thirty- nine?  The solution is this:  You halve the number of roots, which in the present instance yields five.  This you multiply by itself; the product is twenty-five.  Add this to thirty-ninel the sum is sixty-four.  Now take the root of this which is eight, and subtract from it half the number of the roots, which is five;  the remainders is three.  This is the root of the square which you sought for."
    his reasoning we saw in babylonian examples.  
    
    see x^2 + c = bx
    
Ibn Turk

    see other version of x^2 + c = bx

Abu Kamil (850-930)
    general statements of rules rather than merely presenting specific problems
includes Indian numerals with words.  includes abbreviated words for exponents - similar to diophantus, which still hadn't been read in arabic.  
    pentagon - equilateral but not equiangular - has a right angle.  notable answer with square roots of square roots - moved to strictly computation and away from geometry.   maybe (x/10-x)^2 - (10-x/x)^2=2.