Contributions of Muslim Mathematicians

From the 9th to the 15th century, Muslim mathematicians in Baghdad, Nishapur, Maragha and Samarkand collected Greek and Indian mathematics, added new methods of their own, and passed it all on to Europe. Words like algebra, algorithm and the name sine (from Arabic jayb, through Latin sinus) are part of this story.

1. Al-Khwarizmi (c. 780–850 CE)

Full name: Muhammad ibn Musa al-Khwarizmi · Worked in: the House of Wisdom (Bayt al-Hikma), Baghdad

• Called the father of algebra. His book Al-Kitab al-Mukhtasar fi Hisab al-Jabr wal-Muqabala (about 820 CE) gave the subject its name: algebra comes from al-jabr.
• Showed, step by step, how to solve linear and quadratic equations, including the method we now call completing the square.
• His book on Indian numbers spread the digits 0–9, zero and place value to Europe.
• The word algorithm (a step-by-step method) comes from the Latin form of his name, Algoritmi.

2. Thabit ibn Qurra (826–901 CE)

Worked in: Harran (Turkey) and Baghdad

• Found a rule for making amicable numbers: pairs where the factors of each number add up to the other. 220 and 284: the factors of 220 (1, 2, 4, 5, 10, 11, 20, 22, 44, 55, 110) add up to 284, and the factors of 284 (1, 2, 4, 71, 142) add up to 220.
• Translated and improved the geometry books of Euclid and Archimedes.

3. Al-Battani (c. 858–929 CE)

Known in Europe as: Albategnius · Worked in: Raqqa (Syria)

• Used sines and tangents in place of the Greek "chords". This made trigonometry calculations much simpler.
• Measured the length of the solar year very accurately.

4. Abu al-Wafa al-Buzjani (940–998 CE)

Worked in: Baghdad

• Worked with all six trigonometric ratios (sin, cos, tan, cot, sec, cosec) and made very accurate sine and tangent tables.
• Proved the sine rule for spherical triangles (triangles drawn on a ball), used in astronomy.
• Wrote a book of practical arithmetic for clerks and business people.

5. Al-Karaji (c. 953–1029 CE)

Worked in: Baghdad

• Freed algebra from geometry and worked with numbers, powers and polynomials directly.
• Used the triangle of binomial coefficients (later called Pascal's triangle) to expand expressions such as (a + b)ⁿ.
• Used an early form of proof by mathematical induction.

6. Omar Khayyam (1048–1131 CE)

Worked in: Nishapur and Isfahan (Iran)

• Classified cubic equations (with x³) and solved them geometrically, using curves called conic sections.
• Led the team that made the Jalali calendar (1079 CE), more accurate than the calendar used in Europe at that time.
• Also famous as a poet.

7. Nasir al-Din al-Tusi (1201–1274 CE)

Worked in: Maragha (Iran)

• In his Treatise on the Quadrilateral, he made trigonometry a separate subject, not just a tool for astronomy.
• Stated the sine rule for ordinary flat triangles: a ÷ sin A = b ÷ sin B = c ÷ sin C.

8. Jamshid al-Kashi (c. 1380–1429 CE)

Worked in: Samarkand (Uzbekistan)

• Calculated π correct to 16 decimal places in 1424 CE, a world record that stood for about 170 years.
• Explained decimal fractions clearly in Miftah al-Hisab (The Key of Arithmetic, 1427 CE).
• In France, the cosine rule is called "Al-Kashi's theorem".

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