As the Islamic empire flourished, many Muslim intellectuals began to contemplate and eventually advance an indispensable academic field that is now taken for granted: mathematics.

The caliph Harun al-Rashid first sparked an interest in mathematics when his reign began in 786 A.D. by encouraging the knowledgeable mathematicians and scientists of the time to relinquish the then-forgotten works of famous Greek scholars such as Euclid, Archimedes, and Ptolemy and begin studying and translating them. As the Greeks had originally laid down the foundation for modern mathematics with their work in geometry, the advent of newly-translated Greek works undoubtedly encouraged scholars to explore the field even further.

Perhaps the most notable Muslim scholar was the mathematician Muhammad ibn Musa **al-Khwarizmi**, who is credited for the discovery of algebra. Khwarizmi began his career at the House of Wisdom in Baghdad by studying the newly-invented decimal system and idea of a set of numerals that have become the nine Arabic numerals we know today. He also read the works of Euclid extensively, using the Greek’s discoveries in geometry as a basis for much of his work.

From this research, Khwarizmi began to develop a form of mathematics that grouped rational numbers, irrational numbers, geometric magnitudes, and other mathematical figures into one discipline that followed a specific set of rules. He was the first to understand and discuss the principle of exponents, roots, and how they are equal to numbers (although our modern-day notation was not invented until the 16^{th} century^{1}).

Khwarizmi wrote and published his findings in one of the most famous books in all of mathematics: Al-Kitab Al-Mukhtasar fi Hisab al-Jabr wa’l Mugabala, or in English, The Book on Calculations by Completion and Balancing. It is from the title of Khwarizmi’s book that the name for this new form of mathematics, algebra (or al-jabr), was given. This “gave mathematics a whole new development path so much broader in concept to that which had existed before, and provided a vehicle for future development of the subject.”^{2}

In addition to algebra, Khwarizmi also lent his name to what are now known as algorithms, penning another book, Liber Algorismi de Numero Indorum (“Khwarizmi’s Book on Indian Numerals”), on the subject. He helped discover the tangent function by drawing detailed trigonometric tables on the sine function, and also found differentiation by developing the calculus of two errors. Modern mathematics as we know it today would not exist without the invaluable contributions of Khwarizmi.

However, Khwarizmi is not the only Muslim mathematician to whom we are indebted. Other notable Muslim scholars include:

**Al-Karaji**, who not only founded a school of algebra but also freed algebra from the confines of geometry, laid down many of its most important rules, and began to develop numerical analysis with decimals;

**Ibrahim ibn Sina**, known as Avicenna in the West, who advanced the field of geometry by refining Archimedes’s original integration method and studied optics to learn more about how conic sections could be applied to find algebraic solutions; and

**Al-Baghdadi** or Ibn Tahir, whose book, al Takmila fi’l Hisab, clarified the arithmetic systems used by scholars at the time and helped disprove many false claims made by Western mathematicians about algebraic theories during the Renaissance.

Although mathematics may seem like a bore to us now, the advances that Muslim mathematicians made during the Islamic empire by debunking the work of intellectuals that came both before and after their lifetimes are invaluable to modern-day society. So the next time you have to open up your math textbook and stare at page after page of problems, you can take comfort in the fact that there were truly intelligent and gifted Muslims before you who made learning how to solve all of those problems immensely worthwhile.