Islamic mathematics flourished between the 7th-16th centuries under Islamic empires spanning the Middle East and North Africa. Key developments included establishing algebra as its own discipline through the works of al-Khwarizmi, who wrote the first book systematically dealing with algebraic equations up to the second degree. Omar Khayyam expanded on this by providing both geometric and algebraic solutions to cubic equations. Later mathematicians like al-Karaji, al-Tusi, and al-Qalasadi further advanced algebraic notation and concepts like the binomial theorem and trigonometry. Their works had significant influence on mathematics in other cultures.

1. ISLAMIC
MATHEMATICS
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2. INTRODUCTION TO ISLAMIC MATHEMATICS
•
• Islamic mathematics refers to the mathematics developed in the Islamic world between 622-1600, Islamic science
and mathematics flourished under the Islamic Empire, established across the Middle East. Central Asia, I North
Africa, Sicily, the Iberian Peninsula, and in parts of France and India in the 8th century. The center of Islamic
mathematics was located in Persia, but expanded to the west and east over time.
• Most scientists in this period were Muslims and Arabic was the dominant language. Arabic was used as the chosen
written language of most scholars throughout the Islamic world at the time-contributions were made by people of
different ethnic groups (Arabs, Persians, Berbers, Moors, Turks) and sometimes different religions (Muslims,
Christians, Jews, etc

3. ISLAM & MATHEMATICS
The Islamic law of inheritance served as an drive behind the development of algebra (Arabic:
al-jabr) by Muhammad ibn Mūsā al-Khwarizmi and other medieval Islamic mathematicians.
Al-Khwarizmi’s Hisab al-jabr w’al-muqabala had a chapter formulating the rules of
inheritance as linear equations (& his knowledge of quadratic equations was unnecessary).
Later mathematicians dedicated to the Islamic law of inheritance included Al-Hassar, who
developed the modern symbol for fractions in the 12th century, and Abu al Hasan ibn Ali al-
Qalasādi, who developed an algebraic notation which affected the rise towards the
introduction of algebraic symbols in the 15th century.

4. ISLAMIC SCHOLAR AND THEIR CONTRIBUTION
Al-Hajjaj ibn Yusuf ibn Matar (786 833)
• translated Euclid’s Elements into Arabic
Muḥammad ibn Mūsā al-Khwarizmi (c. 780 c. 850)
• Persian mathematician, astronomer, astrologer and geographer; worked
most of his life as a scholar in Baghdad; Algebra was the first book on linear and quadratic equations; introduced the decimal positional number system to the Western world in the 12th century;
revised and updated Ptolemy’s Geography as well as writing several works on astronomy and astrology.
Al-’Abbas ibn Sa’id al-Jawhari (c. 800- c. 860)
• mathematician who worked @ House of Wisdom (Baghdad); most important work: Commentary on Euclid’s Elements (contained 50 additional propositions and an attempted proof of the
parallel postulate)
‘Abd al-Hamid ibn Turk (830)
• wrote a work on algebra (only 1 chapter of quadratic equations survived)
Ya’qub ibn Ishaq al-Kindi (c. 801 – 873)
• contributions to mathematics include many works on arithmetic and geometry.

5. Banu Mūsă (c. 800 – 873)
• three brothers in Baghdad; most famous mathematical treatise: The Book of the Measurement of Plane and
Spherical Figures; The eldest, Ja’far Muhammad (c. 800) specialized in geometry and astronomy; Ahmad (c. 805)
specialized in mechanics and wrote On mechanics; The youngest, alHasan (c. 810) specialized in geometry and wrote
The elongated circular figure.
Ikhwan al-Safa’ (first half of 10th century)
• group wrote series 50+ letters on science, philosophy and theology. The first letter is on arithmetic and number
theory, the second letter on geometry.
Labana of Cordoba (Spain, ca. 10th century)
• Islamic female mathematicians & secretary of the Umayyad Caliph al-Hakem II; could solve the most complex
geometrical and algebraic problems known in her time.
Al-Hassar (ca.1100s)
• Developed the modern mathematical notation for fractions and the digits he uses for the ghubar numerals also
cloesly resembles modern Western Arabic numerals.
Ibn al-Yasamin (ca. 1100s)
• first to develop a mathematical notation for algebra
Abu al-Hasan ibn Ali al-Qalasādi (1412-1482)
• Last major medieval Arab mathematician; Pioneer of symbolic algebra

6. MUHAMMAD IBN MŪSĀ AL-KHWARIZMI
Muhammad ibn Musa al-Khwarizmi, better known simply as al-Khwarizmi, was a highly influential Muslim
mathematician who made several notable contributions to mathematics and other subject areas. He was
born in the Middle Ages around 780 CE in Khwarizmi, which was part of Persia at the time. He lived until
approximately 850 CE.

7. ALGEBRA
The term algebra is derived from the Arabic term al-jabr in the title of
Al-Khwarizmi’s Al-jabr wa’l muqabalah. He originally used the term al
jabr to describe the method of “reduction” and “balancing”, referring to
the transposition of subtracted terms to the other side of an equation. Before
the fall of Islamic civilization, the Arabs used a fully abstract algebra, where
the numbers were spelled out in words (four). They later replaced the words
with Arabic numerals (4), but the Arabs never developed a symbolic algebra
until the work of Ibn al-Banna al Marrakushi (13th cent) & Abu al-Hasan ibn Ali
al-Qalasādi (15th cent)
There were 4 stages in the development of Algebra :
• Geometric Stage where the concepts of algebra are largely geometric
• Static equation-solving stage : find #s satisfying certain relationships
• Dynamic function: where motion is an primary idea
• Abstract Stage: where mathematical structure plays a essential role

8. STATIC EQUATION-SOLVING ALGEBRA
Muhammad ibn Mūsā al-Khwarizmi (c. 780-850) was a staff member of the
“House of Wisdom” in Baghdad (established by Al-Mamun). Al-Khwarizmi, wrote
more than half a dozen mathematical and astronomical works. One of al
Khwarizmi’s most famous books is entitled Al-jabr wa’l muqabalah (The
Compendious Book on Calculation by Completion and Balancing), about solving
polynomials up to the second degree. The book also introduced the fundamental
method of “reduction” and “balancing”, referring to the cancellation of like terms
on opposite sides of the equation. This is the operation which Al-Khwarizmi
originally described as al-jabr.
6 Chapters of Al-Jabr
• Squares equal its roots (ax2 bx)
• Squares equal a number → (ax² = c)
• Roots equal a number → (bx = c)
• Squares & roots equal a number→ (ax² + bx = c)
• Squares & numbers equal roots →→ (ax² + c = bx)
• Roots & numbers equal squares → (bx + c = ax2)
Arabic Mathematicians were the first to treat irrational #’s as algebraic objects

9. MUHAMMAD AL-KARAJI
Abū Bakr Muḥammad ibn al Ḥasan al-Karajī was a 10th-century Persian
mathematician and engineer who flourished at Baghdad. He was born in
Karaj, a city near Tehran
Born: 953 AD, Karaj, Iran
Died: 1029, Iran

10. WORKS/CONTRIBUTION
• Wonderful on calculation , glorious algebra , sufficient On calculation
• One of the first major contribution al-Karaji made to the development of the discipline of
mathematics was in his work on algebra in al-Fakhri. Al-Karaji managed to completely free algebra
from geometrical operations.

11. Now, the other contribution al-Karaji made is in the development of binomial
theorem.
Al-Karaji dealt with the problem of expanding binomials that are being raised by an
exponent greater than 2.
The binomial (a + b)² is relatively easy to expand (to get a² + 2ab + b²), but as we go
higher up, the work will get more and more tedious, say, for example, to expand the
binomial (a + b)⁶.

12. Al-Karaji tried to deal with this problem by constructing a structure similar to the famous Pascal’s
Triangle – in fact, we may assert that Karaji’s table is a pioneer to the Pascal’s Triangle! – in order to
figure out the binomial coefficient as we go higher up the power of a binomial

13. Al-Karaji was among the first mathematicians who employed the method of mathematical induction as
proof to his theorem

14. GEOMETRIC ALGEBRA
Omar Khayyám (c. 1050-1123) wrote a book on Algebra that went beyond Al-Jabr. Omar
Khayyám gave both arithmetic & geometric solutions for quadratic equations, but only gave
geometric solutions for general cubic equations (he thought that arithmetic solutions were
impossible). His method of solving cubic equations by using intersecting conics had been used
by Menaechmus, Archimedes, and Alhazen.However, Omar was abut to generalize the method
using only positive roots and didn’t go past the 3rd degree. He also saw a strong relationship
between Geometry and Algebra

15. OMAR KHAYYAM CONTRIBUTION
Omar Khayyam’s most important contribution to mathematics was his work involving cubic
equations. A cubic equation is an equation whose highest degree variable is three, for
instance x³ + 3x² - 2x + 5 = 0.

16. MUHAMMAD IBN MUHAMMAD
IBN AL-HASAN AL-TŪSĪ,
Muhammad ibn Muhammad ibn al-Hasan al-Tūsī, better known as Nasir al-Din
al-Tusi, was a Persian polymath, architect, philosopher, physician, scientist,
and theologian. Nasir al-Din al-Tusi was a well published author, writing on
subjects of math, engineering, prose, and mysticism.

17. One of al-Tusi’s most important mathematical contributions was the creation
of trigonometry as a mathematical discipline in its own right rather than as
just a tool for astronomical applications.

18. Al-Tusi gave the first extant exposition of the whole system of plane and
spherical trigonometry.
Another mathematical contribution was al-Tusi’s manuscript, dated 1265,
concerning the calculation of nn-th roots of an integer.