I've been very disheartened to hear people, both here and elsewhere, regard the Islamic world as if it were a barbarous place devoid of any redeeming value. To go some ways toward showing just how false that view is I thought I might start writing a series of diaries on the many great advances that the world owes to these "cave dwelling barbarians."
It is fitting in a way to start the series with mathematics because math is not an end unto itself but rather a means by which to accomplish so much else. It is the base of science and economics which means it is the foundation on which US society was built.
So how has Islam contributed to mathematics? First let me say that I am not literally here talking of Islam the religion. Rather I am speaking of the region and cultures which either preceeded and led up to, or postdated and developed within the area where Islam grew to power. This essentially means the arabs and persians in the time period shortly before Muhammed to current day. For example the earliest mathematical texts yet found came from Egypt, Mesopotamia, and India but at ~1300BC these too far predate Islam for us to tie them directly to the culture (as opposed to religion) of Islam. For Islam's contributions we have to look much later.
Roman Numerals
The Roman notation of numbers (I, II, III, IV, V, VI...) was used in Europe commonly into the 1400s. This system originally developed from simple tally sticks. The system worked (clumsily) for basic addition but was difficult for anything more complicated. The reasons for this are varied.
In the case of fractions a new symbol was used for each fractional value. This meant that only a very few commonly used fractions were ever codified. For instance 1/12 had a symbol but you can bet that 1/137th didn't.
Subtraction, multiplication, and division with Roman numerals is so complicated that it is virtually always done with the use of a special abacus.
The Romans and later Europeans had the concept of "zero" but did not treat it as a number in and of itself (with a few very rare exceptions). The Roman systems of numbers had no symbol for zero and so a number of very important mathematical ideas remained unassailable, not the least of which is negative numbers (if you have no zero you don't think of what's on the other side).
Arabic Numerals
Arabic numerals are not strictly speaking arabic. They originate in India around 300BC. They are referred to as "Arabic" because it was the work of arab mathematicians that really spread the use of these symbols and made them pretty universal from the Indian subcontinent through the Middle East eventually into Europe. Furthermore the arabs made a significant advancement of making zero a number unto itself in the 800s AD.
While they would look strange to us these symbols morphed over time to become the number system we all grew up with (1, 2, 3, 4...).
This system of numbers was much more capable than the Roman version. It was more compact in general (compare 58 to LVIII or 1997 to MCMXCVII). The Roman system requires very careful reading since some numbers are subtractive and some additive based on position (IV is 4 but VI is 6), by comparison the Arabic numbers are all "additive" or all subtractive (if used in conjunction with a negative sign). Addition and subtraction are simple, enough so that any school age child is expected to routinely do them. Multiplication and division are only a little more complicated.
Very large numbers and very small numbers can be a bit cumbersome (but not nearly so cumbersome as in Roman Numerals) but this was eventually relieved with the use of Exponential or Scientific notation (i.e. 450,000,000,000 is written 4.5x10^11 or 4.5E11). Interestingly enough I can't seem to find any info on where/when scientific notation originated. I'm assuming it is European but may be wrong.
Ultimately it boils down to the Arabic system being open (in the sense of continuously expandable) and the Roman system being closed (as in it had finite discrete states and to go outside of that required the creation of new symbols). This is somewhat analagous to the difference between a lettered language and logographic systems, like Kanji, where every symbol is a single word. In the former system it is easy to continuously expand the vocabulary in the latter each new term requires a totally new symbol and the language quickly becomes unwieldy and potentially incomprehensible.
It is no coincidence that calculus developed first among those using the arabic system and only really started developing in Europe after the adotion of same. The impact of this contribution to mathematics can't be overstated. Without calculus most of science is simply impossible.
Beyond the numbers
Okay so the Arabic number system was a big improvement, one which the arabs made significant contributions to if not actually originating themselves. What else have they got?
Oh boy.
Our very word "algebra" is derived from Al-Jabr wa-al-Muqabilah, a treatise on mathematics by Muhammad ibn Mūsā al-Kwārizmī from whom we also get the word "algorithm." Al-Kwarizmi, a persian, is considered "the father of modern algebra" and algebra of course is one of the most fundamental building blocks of mathematics.
Al-Biruni by age 27 had written nine books and a host of shorter works. Much of this work had to do with the application of geometry to both astronomy and geography. Today a crater on the moon is named in his honor.
The poet Omar Khayyam created a geometic solution to cubic equations (ax^3 + bx^2 + cx +d =0), a feat that is regarded as "one of the most original discoveries in Islamic mathematics" (no small feat clearly). He also contributed to the synthesis of mathematical approaches (combining trigonometry and approximation theory to provide solutions to other algebraic problems).
The greatest contribution to the Ptolemaic planetary system (our understanding of the motion of the planets including our own) until the coming of Copernicus was made by the Persian Nasir al-Din al-Tusi due to his work on spherical trigonometry.
Ghiyath al-Kashi was considered the inventor of decimal fractions (1.16 as opposed to 29/25ths) because his contribution to the field was so enormous (he did not actually invent them). He furthermore computed pi to the 16th decimal place. What's more he developed an algorithm for calculating nth roots and put together tables of sines accurate to 8 decimal places (for use in astronomy).
The hindu-arabic number system was originally desgined for "dust boards," basically primitive chalk boards. Abu'l-Hasan al-Uqlidisi modified the system to make it usable for the much more convenient (and potentially permanent) method of pen and paper. While this sounds minor consider the awesome potential of being able to write a mathematical work and include directly in it the very mathematical symbols to which you refer. Quite simply without this development the very idea of math text is impossible. Each problem would be described in words while the reader would have to write out the equation in numbers on their personal dust board.
The list of somewhat less titanic but still very important Islamic mathematicians and the fields they contributed to is long; Thabit ibn Qurra (geometry), Abu Kamil (algebra, irrational numbers), Al-Batanni (trigonometry), Sinan ibn Thabit (geometry), Ibrahim ibn Sinan (calculus), Abul Wáfa (invented the tangent function, trigonometry), Abu Bakr al-Karaji (numerical analysis, major advancements to algebra), Al-Haytham (number theory), Abu Nasr Mansur (discovered sine law, trigonometry), Abu Sahl al-Kuhi (geometry), Al-Baghdadi (arabic decimal system), Al-Samawal (algebra), Sharaf al-Din al-Tusi (founded algebraic geometry), Al-Farisi (number theory), Ulugh Beg (trigonometry in his spare time from ruling the Timurid Empire).
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