Mathematics in the medieval Islamic world facts for kids
During a special time called the Islamic Golden Age, especially in the 800s and 900s, mathematics grew a lot. This growth was built on ideas from Greek mathematics (like Euclid and Archimedes) and Indian mathematics (like Aryabhata and Brahmagupta). Big steps forward included using decimal fractions in our number system, creating the study of algebra, and making progress in geometry and trigonometry.
The medieval Islamic world saw huge changes in math. A key person was Muhammad ibn Musa al-Khwārizmī. In the 800s, he made algebra a separate field of study. Al-Khwārizmī's new way of thinking about math, different from older methods, set the stage for how algebra works today. Other mathematicians like al-Karaji built on his ideas. These new math methods were very useful and helped spread Arabic math to the West. This greatly helped Western math grow.
Arabic math ideas spread widely during the Islamic Golden Age. This happened because al-Khwārizmī's methods were so practical. Not only trade and politics, but also cultural exchanges helped spread these ideas. Events like the Crusades and the translation of books played a part. The Islamic Golden Age, from the 700s to the 1300s, was a time of amazing discoveries in many sciences. Scholars from Europe came to learn from this knowledge. Trade routes and cultural meetings were important for bringing Arabic math to the West. Translating Arabic math books, along with Greek and Roman ones, from the 1300s to the 1600s, shaped the Renaissance.
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How Arabic Math Began and Spread
Arabic mathematics, especially algebra, grew a lot during the medieval period. Muhammad ibn Musa al-Khwārizmī (around 780–850 AD) made a big difference in Baghdad between 813 and 833 AD. He used the word "algebra" in his book, "Kitab al-jabr wa al-muqabala." This made algebra a new, separate subject. He saw his work as "a short book on Calculation by (the rules of) Completion and Reduction." It was meant to be easy and useful for everyday math. People later realized his work was not just theory. It was also practical for solving problems in business and land measurement.
Al-Khwārizmī's method was new because it didn't come from older math traditions. He created new words for algebra. He also showed how numbers are important in daily life. He wanted to find a simpler way to do math, which later became algebra. His early algebra focused on simple equations and basic math with two or three terms. This way of solving equations influenced math for a long time.
Al-Khwārizmī made a huge step in algebra by proving how to solve quadratic equations. These are equations like ax2 + bx = c. This discovery created a clear way to solve these equations. It became a basic part of algebra as it grew in the Western world. Al-Khwārizmī's method, called "completing the square," gave a practical solution. It also introduced a more general way to think about math problems. His important book, "Al-Kitab al-Mukhtasar fi Hisab al-Jabr wal-Muqabala," was translated into Latin in the 1100s. This translation was key to bringing algebra to Europe. It greatly influenced mathematicians during the Renaissance and helped shape modern math.
Arabic math spread to the West for several reasons. Al-Khwārizmī's methods were very useful and could be used for many things. They helped turn number or shape problems into equations that could be solved easily. His work and that of others like al-Karaji led to new ideas in number theory and how to solve equations.
Al-Khwārizmī's algebra was a new field with its own history. It led to algebra becoming more about numbers. His followers built on his work. They changed it to solve new problems and made it more about abstract math.
Arabic math, especially Al-Khwārizmī's work, was vital in shaping math. It spread to the West because it was practical. Also, his followers expanded math ideas. And these ideas were translated and used in the West. This spread was a complex mix of money, power, and culture. It had a huge impact on Western math.
The Islamic Golden Age (700s to 1300s) was a time of great progress in many areas, including mathematics. Scholars in the Islamic world made big contributions to math, astronomy, medicine, and other sciences. Because of this, European scholars in the Middle Ages wanted to learn from them. Trade routes like the Silk Road helped move goods, ideas, and knowledge between East and West. Cities like Baghdad, Cairo, and Cordoba became learning centers. Scholars from different cultures came to these cities. So, math knowledge from the Islamic world reached Europe in many ways. The Crusades also connected Europeans with the Islamic world. Even though the Crusades were mostly about war, there was also cultural exchange. European scholars who traveled to the Islamic world found Arabic math books. From the 1300s to the 1600s, translating Arabic math books, along with Greek and Roman ones, was very important for the Renaissance. People like Fibonacci, who studied in North Africa and the Middle East, helped bring Arabic numbers and math ideas to Europe.
Key Math Ideas from the Islamic Golden Age
Algebra: Solving Equations
The study of algebra grew a lot during the Islamic golden age. The word "algebra" comes from an Arabic word meaning "completion" or "reunion of broken parts." Muhammad ibn Musa al-Khwarizmi, a Persian scholar in Baghdad, is known as the founder of algebra. He is often called the "father of algebra," along with the Greek mathematician Diophantus. In his book The Compendious Book on Calculation by Completion and Balancing, Al-Khwarizmi showed how to solve equations with positive answers. These included first-degree (linear) and second-degree (quadratic) equations. He introduced a method called "reduction." Unlike Diophantus, he also gave general ways to solve the equations he worked with.
Al-Khwarizmi's algebra was "rhetorical." This means equations were written out in full sentences. This was different from Diophantus's work, which used some symbols. The move to using only symbols in algebra came later. This can be seen in the work of Ibn al-Banna' al-Marrakushi and Abū al-Ḥasan ibn ʿAlī al-Qalaṣādī.
J. J. O'Connor and Edmund F. Robertson said about Al-Khwarizmi's work:
"One of the most important steps in Arabic mathematics began with al-Khwarizmi. This was the start of algebra. It's important to understand how big this new idea was. It was a huge change from Greek math, which was mostly about shapes. Algebra was a way to connect different types of numbers and shapes. It gave math a whole new path to grow. It also allowed math to be used on itself in a new way."
Many other mathematicians during this time built on Al-Khwarizmi's algebra. Abu Kamil Shuja' wrote an algebra book with pictures and proofs using geometry. He also found all possible answers to some of his problems. Abu al-Jud, Omar Khayyam, and Sharaf al-Dīn al-Tūsī found many ways to solve cubic equations. Omar Khayyam found a general way to solve a cubic equation using geometry.
Cubic Equations: More Complex Problems
Omar Khayyam (around 1038–1131 in Iran) wrote the Treatise on Demonstration of Problems of Algebra. This book had a clear way to solve cubic or third-order equations. This went beyond what al-Khwārizmī had done. Khayyám found the answers to these equations by finding where two conic sections (like circles or parabolas) crossed. The Greeks had used this method, but they didn't make it work for all equations with positive answers.
Sharaf al-Dīn al-Ṭūsī (died 1213/4 in Tus, Iran) found a new way to study cubic equations. He looked for the highest point a cubic equation could reach. For example, to solve x3 + a = b x, he found the highest point of the curve y = b x - x3. He then saw if the height of the curve at that point was less than, equal to, or greater than a. This told him if the equation had no solutions, one solution, or two solutions. We don't know exactly how he found his formulas for these highest points.
Induction: Proving Patterns
The idea of mathematical induction helps prove that a pattern works for all numbers. Early signs of this idea are in Euclid's proof that there are endless prime numbers (around 300 BCE). The first clear statement of induction was by Pascal in 1665.
Before Pascal, al-Karaji (around 1000 AD) used a form of induction for arithmetic sequences. al-Samaw'al continued this. He used it for special cases of the binomial theorem and properties of Pascal's triangle.
Irrational Numbers: Beyond Simple Fractions
The Greeks found irrational numbers, like the square root of 2, which cannot be written as a simple fraction. They saw these as "magnitudes" (lengths) rather than "numbers." So, they only dealt with irrationals using geometry. Islamic mathematicians like Abū Kāmil Shujāʿ ibn Aslam and Ibn Tahir al-Baghdadi slowly removed this difference. They allowed irrational numbers to be part of equations. They worked with irrationals as real math objects.
In the 1100s, Latin translations of Al-Khwarizmi's book on Indian numerals brought the decimal number system to the West. His book, Compendious Book on Calculation by Completion and Balancing, was the first to show a clear way to solve linear and quadratic equations. In Europe during the Renaissance, he was thought to be the first to invent algebra. However, we now know his work was based on older Indian or Greek ideas.
Spherical Trigonometry: Math for Globes
Spherical trigonometry deals with triangles on the surface of a sphere, like the Earth. The rule for sines in spherical triangles was found in the 900s. It's linked to Abu-Mahmud Khojandi, Nasir al-Din al-Tusi, and Abu Nasr Mansur. Abu al-Wafa' Buzjani also helped. Ibn Muʿādh al-Jayyānī's book in the 1000s introduced the general law of sines. The law of sines for flat triangles was described in the 1200s by Nasīr al-Dīn al-Tūsī.
Negative Numbers: Below Zero
In the 800s, Islamic mathematicians knew about negative numbers from Indian mathematicians. But they used them carefully. Al-Khwarizmi did not use negative numbers. But within 50 years, Abu Kamil showed how to use rules of signs for multiplying numbers. Al-Karaji wrote that "negative quantities must be counted as terms." In the 900s, Abū al-Wafā' al-Būzjānī thought of debts as negative numbers in his book for scribes and businessmen.
By the 1100s, al-Karaji's followers stated the general rules for signs. They used them to solve complex math problems. As al-Samaw'al wrote:
The product of a negative number by a positive number is negative. The product of a negative number by a negative number is positive. If we subtract a negative number from a larger negative number, the answer is their negative difference. The difference stays positive if we subtract a negative number from a smaller negative number. If we subtract a negative number from a positive number, the answer is their positive sum.
Double False Position: A Clever Way to Solve Problems
Between the 800s and 900s, the Egyptian mathematician Abu Kamil wrote a book about using "double false position." This method is also known as the Book of the Two Errors. The oldest book still existing on this method from the Middle East is by Qusta ibn Luqa (900s). He was an Arab mathematician from Baalbek, Lebanon. He proved this method using geometry. In the Islamic Golden Age, double false position was called hisāb al-khaṭāʾayn ("reckoning by two errors"). It was used for hundreds of years to solve real-world problems. These included business questions and dividing inheritances. It was also used for fun math puzzles. People often memorized the steps using rhymes or diagrams.
How Arabic Math Influenced the World
Medieval Arab-Islamic mathematics had a huge impact on science and math worldwide. This knowledge reached the Western world through Spain and Sicily during a time of many translations. In the 1200s, King Alfonso X of Castile started the Toledo School of Translators in Spain. Here, scholars translated many science and philosophy books from Arabic into Latin. These translations included Islamic work on trigonometry. This helped European mathematicians and astronomers in their studies. European scholars like Gerard of Cremona (1114–1187) were key in translating and sharing these works. This made them available to many more people. Cremona is said to have translated 90 Arabic books into Latin. European mathematicians built on the ideas of Islamic scholars. They further developed practical trigonometry for navigation, map-making, and finding positions by stars. This helped push forward the Age of Exploration and the Scientific Revolution.
Al-Battānī was an Islamic mathematician who greatly helped trigonometry grow. He "created new trigonometric functions, made a table of cotangents, and wrote some formulas for spherical trigonometry." These discoveries, along with his very accurate astronomy work, greatly improved calculations and tools for astronomy.
Al-Khayyām (1048–1131) was a Persian mathematician, astronomer, and poet. He is known for his work on algebra and geometry. He especially studied how to solve cubic equations. He was "the first in history to create a geometry-based theory for equations up to the third degree." He greatly influenced Descartes, a French mathematician often seen as the founder of analytical geometry. Many problems in Descartes's "La Géométrie" (Geometry) have roots in al-Khayyām's work.
Abū Kāmil (around 850–930), also known as "The Egyptian Calculator," studied algebra after al-Khwārizmī. His Book of Algebra was "mostly a comment on and expansion of al-Khwārizmī's work." Because of this and its own value, the book was very popular in the Muslim world. It has 69 problems, which is more than al-Khwārizmī's 40 problems. Abū Kāmil's Algebra was very important in shaping Western mathematics. It especially influenced the Italian mathematician Leonardo of Pisa, known as Fibonacci. In his Liber Abaci (1202), Fibonacci used about 29 problems from Abū Kāmil's Book of Algebra with few changes.
Other Important Math Figures
- 'Abd al-Hamīd ibn Turk (around 830) (worked on quadratics)
- Sind ibn Ali (died after 864)
- Thabit ibn Qurra (826–901)
- Al-Battānī (before 858 – 929)
- Abū Kāmil (c. 850 – c. 930)
- Abu'l-Hasan al-Uqlidisi (around 952) (worked on arithmetic)
- 'Abd al-'Aziz al-Qabisi (died 967)
- Abū Sahl al-Qūhī (c. 940–1000) (worked on centers of gravity)
- Ibn al-Haytham (c. 965–1040)
- Abū al-Rayḥān al-Bīrūnī (973–1048) (worked on trigonometry)
- Al-Khayyām (1048–1131)
- Ibn Maḍāʾ (c. 1116–1196)
- Ismail al-Jazari (1136–1206)
- Jamshīd al-Kāshī (c. 1380–1429) (worked on decimals and estimating pi)
Images for kids
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Engraving of Abū Sahl al-Qūhī's perfect compass to draw conic sections
See also
- Arabic numerals
- Indian influence on Islamic mathematics in medieval Islam
- History of calculus
- History of geometry
- Science in the medieval Islamic world
- Timeline of science and engineering in the Muslim world
