Post 44: Science & Islam: Mathematics

Most people will probably know that the mathematicians of the Islamic world have played a big part in mathematics as we know it – terms like “algebra” and “algorithm” is a bit of a give away, and the invention of zero is a common bit of trivia. Not as many people may be familiar with the characters and setting behind these inventions or the full extent of their mathematical achievements. These weren’t just abstract creations either, they often had practical uses for both everyday life and for other academic arts. This post is a bit of a deviation from a book (Science & Islam by Ethan Masood) I reviewed in a recent post – I’ve given a bit of a general run through of the general history there if you want to check it out (please do, views are always nice to have), but I’ve tried to keep this post relatively self contained.

Why?

Why did they study mathematics? Well, much like anyone else it was a mixture of the practical and the academic. Some of these scientists and philosophers would investigate more and more complicated work for its own sake, but there were also more down to earth reasons. Al-Khwarizmi’s work on algebra was explained as a way to speed up the complicated process of Islamic inheritance, and work on trigonometry was needed for new techniques in navigation and astronomy (okay, maybe that last one is a bit less down to earth). The work was, as with more scientific work, generally funded by a patron or ruler.

Al-Khwarizmi

The most famous mathematician of the Islamic world, Al-Khwarizmi was probably born somewhere in central Asia around the 780s – we don’t really know. That’s not exactly a great start, but our knowledge of him gets better once he ends up in Baghdad – to join the great collection of scholars known as the House of Wisdom that was being gathered by the Caliph Al-Mamun. Part of their scholarly work would have been the translation of previous works from India or Greece into Arabic, so they could easily be distributed and discussed. Through these translations, Al-Kwarizmi would have become familiar with the likes of Pythagoras, Euclid and Brahmagupta and it would give him a great base to build off of.

Number one in his contributions to mathematics must be the adoption of so-called Indian Numerals, ie. the numbers 0 to 9. His translation of works on this spurred the Islamic world and the west to adopt this much simpler and more compact numbering system, compared to the old method of Roman Numerals. Obviously a lot of credit has to go to the Indians here, but our man was not slack in spotting and adopting a more useful convention. Incidently, his book on the subject Algoritmi de numero Indorum is credited as the derivation of the term Algorithm – used here to describe his methodical process for doing arithmetic.

Number two is potentially just as big – his work on Algebra. In fact his book Al-Kitab al-mukhtasar fi hisab al-jabr wa-l-muqabala dividing the process into two parts completion – al-jabr – and balancing – al-muqabala. By taking an equating with some unknown value (known as shay or ‘the thing’ – the notation still needed a bit of work) and “completing” it by shifting any negative values to the other side, and then “balancing” it by removing anything on both sides of the equation. For example,
x^2 – 2x + 7 = 4
x^2 + 7 = 2x + 4 (complete)
x^2 + 3 = 2x (balance)
Then use a set of standard answers that he had derived to read off the result to this. Al-Khwarizmi had a set of six that covered pretty much anything going. Working with generic prototypes like these was a big step forward in mathematics and that only cements the importance of Islam and Al-K in the subject.

That wasn’t all for him either – he also contributed towards trigonometry, astronomy and geography but I’ve been told my posts are long enough as it is, so I’ll move on …

Notation

This work was brilliant but it was still very awkward to read. The new Indian numerals were compact, but ‘shay’ was the only really step towards modern algebraic symbolism – everything else was written out in full. That would change with al-Marrakushi in the 13th century and al-Qalasadi in the 15th. They weren’t the first to use generic symbols – Brahmagupta had done the same in India hundreds of years earlier – but it had fallen by the wayside. These symbols:
wa meaning “and” for +
illa meaning “less” for –
fi meaning “times” for ×
ala meaning “over” for ÷
j from jadah meaning “root”
sh from shay meaning “thing” (x, the unknown)
m from mal for x2
k form kab for x3
l from yadilu for =
allowed a compact and generic form of problems to be stated. Think of how much terrible mathematics would have been in school if we were writing “three times the cube of shay plus seventeen times shay equals four minus two times the square of shay” all the time.

Al-Haitham

Here’s someone who will definitely come up again in my posts on science and Islam. Perhaps better known for his work in Astronomy or in Optics, Al-Haitham also contributed towards the early stages of integral calculus. He seems like an interesting character – born in Basra, he moved to Cairo to work for Al-Hakim in Cairo. He claimed that he had a method to regulate the flooding of the Nile, but this quickly proved impossible so he did what any sensible person would do and feigned insanity until Al-Hakim finally died. After that, free from the threat of an angry Caliph, he released brilliant papers on a huge variety of topics.

For Mathematics, his great contribution was to demonstrate the link between the generic algebraic formulation that was becoming so useful and geometry. His motivation was a particular problem (now known as Alhazen’s Problem) that was relevant to optics. The problem involves terms with a power of four and looked fiendishly difficult as algebra, but Al-Haitham used sections of a cone to find a geometric solution to the problem. Centuries later this kind of technique would go on to be developed into the geometry of Descartes and even later into integral calculus.

He also contributed various ideas towards number theory and attempted a proof of Euclid’s Parallel Postulate. This postulate stated that if a line segment intersects two straight lines forming two interior angles on the same side that sum to less than two right angles, then the two lines, if extended indefinitely, meet on that side on which the angles sum to less than two right angles. For this he attempted a proof by contradiction – could he come up with a new shape that would disprove this statement? In the end he couldn’t, but his ideas hinted at future investigations of non-Euclidean geometries.

Irrational numbers

All this would have been so much more difficult if we were stuck with rational numbers as answers – ones that we can express simply as a fraction of whole number a divided whole number b. They had come up in Indian and Greek work previously, but generally were treated with a little bit of embarrassment until ibn Aslam in the 9th century decided to treat them as an acceptable solution and coefficient in equations. His writings on algebra would go on to be particularly useful for the Italian mathematician Fibonacci. It would also be used by Al-Karaki in the 10th century – a mathematician who development proof by induction and investigated exponents (ie. the idea that a power of two multiplied by a power of three becomes a power of five).

Omar Khayyam

Another of the big names – Khayyam was a poet, philosopher and scientist of the 12th century. His major contribution to mathematics was to find general cubic solutions for Algebraic expressions – basically the next step on from Al-Khwarizmi. He did this using some very clever geometry but, despite attempts, was unable to find an arithmetic solution (as would Sharaf al-Din al-Ṭusī). He also attempted a proof of the Parallel Postulate. Or sort of anyway. His version was more of a derivation from and critique of previous work. It seems to have been a popular past-time, the astronomer Nasir al-Din al-Tusi would do similar.

Al-Battani and Trigonometry

An astronomer from the 9th century in what is now Turkey, Al-Battani added much to the field of trigonometry. He found the famous expression tan is sin divided by cos as well as some of the more complicated ones that people tend to just read off a datasheet if they need them (sec^2 = 1 + tan^2 ?). He and other trigonometrists would go on to use their results for astronomical and navigational measurements. These results would also become important for the Western astronomers like Copernicus or Brahe who would go on to build a more accurate model of our solar system. More on that in a post coming soon …

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