A Soviet postage stamp, issued in 1983 celebrating Al Khwarizmi’s 1200th birthday.
Introduction
Al-Khawarizmi was a polymath born in 780 CE, who produced influential works in mathematics, astronomy, geography, and cartography. His major contribution to the world of mathematics was his book ‘Al-Jabr’ (the Compendious Book on Calculation by Completion and Balancing). He was of Persian descent; his name means ‘from Khwarazm’, a region that was part of Greater Iran and is now part of Turkmenistan and Uzbekistan. He was a member of the House of Wisdom in Baghdad which was believed to be a major Abbasid-era public academy and intellectual center in Baghdad.
This book presented the first systematic solution of linear and quadratic equations. His demonstration of solving quadratic equations, by completing the square along with its geometric justification, was one of his achievements in algebra.
He is called the father of algebra, introducing the concepts of balancing and reducing, I.e. working around both sides of equality to solve or simplify it, by transposing and canceling terms across equality. Algorithm term today is named after him. His systematic approach to mathematics led to the philosophy of modern algorithms.
Al-Khwarizmi’s textbook on Indian arithmetic was translated into Latin in the 12th century, after almost 400 years. This translation codified the various Indian Numerals and introduced the decimal-based positional number system to the Western world.
The translation of Al-Jabr by Robert of Chester in 1145, was used until the 16th century as the principal mathematical textbook of European universities.
Al-Khwarizmi authored a major work on geography, “Zijes,” which gives calculations of movements of planets and stars, latitudes, and longitudes for 2402 localities as a basis for a world map.
In trigonometry he produced accurate sine and cosine tables and the first table of tangents.
Contributions
Al-Khwarizmi’s work on the Hindu-Arabic numeral system is the basis of the modern number system. Translation of Al-Khwarizmi’s work in this field in the 12th century named Algoritmi de numero Indorum (Al-Khwarizmi on the Hindu art of reckoning), brought the modern number system to the Western world, and also the term “algorithm” was introduced.
Some of his work was based on Persian and Babylonian astronomy, Indian numbers, and Greek mathematics. He systematized and corrected Prolemy’s data for Africa and the Middle East. His book Kitab surat al-ard (“The Image of the Earth”; translated as Geography), presents the coordinates of places based on those in the Geography of Ptolemy, but with improved values for the Mediterranean Sea, Asia, and Africa.
He also wrote about mechanical devices like the astrolabe and sundial. One of his projects was to determine the circumference of the Earth and make the world map, for Al-Ma’mun the Abbasid caliph, and for this, he oversaw 70 geographers.
The main impact of his work happened in the 12th century when his work was translated into Latin and spread across Europe.
Algebra
On the Left is the original Arabic print manuscript of the Book of Algebra, i.e. Al Jabr by Al Khwarizmi, and on the right is the book Algebra by Al-Khwarizmi by Fredrick Rosen, in English.
The book was written in 820 CE and Al-Khwarizimi was encouraged for it by Caliph Al Ma’mun. It was a popular work on calculation having examples and applications useful to a range of problems in trade, surveying, and legal inheritance.
In 1145 it was translated into Latin by Robert of Chester. In 1831, F. Rozen translated the original book into English.
An extensive account of solving polynomials up to a second degree, fundamentals of reduction and balancing i.e. transposing terms across sides and later on simplification by canceling is the main core of this book. Back in those days, modern symbols were not developed, so Al-Khwarizmi used plain language to describe the solution. In looks something like what we call pseudo-code in programming.
An Example of Al-Khwarizmi’s approach to polynomial equations
From F. Rozen’s book, an example is
If some one says: “You divide ten into two parts: multiply the one by itself; it will be equal to the other taken eighty-one times.” Computation: You say, ten less a thing, multiplied by itself, is a hundred plus a square less twenty things, and this is equal to eighty-one things. Separate the twenty things from a hundred and a square, and add them to eighty-one. It will then be a hundred plus a square, which is equal to a hundred and one roots. Halve the roots; the moiety is fifty and a half. Multiply this by itself, it is two thousand five hundred and fifty and a quarter. Subtract from this one hundred; the remainder is two thousand four hundred and fifty and a quarter. Extract the root from this; it is forty-nine and a half. Subtract this from the moiety of the roots, which is fifty and a half. There remains one, and this is one of the two parts.
In modern mathematics it means
Let the roots of the equation be x = p and x = q. Then 
So a root is given by
Six Standard Forms
His method was to use transposition (reduction and balancing) to get an equation to one of Six standard forms and then use a particular approach for solving it.
- squares equal roots (ax2 = bx)
- squares equal number (ax2 = c)
- roots equal number (bx = c)
- squares and roots equal number (ax2 + bx = c)
- squares and number equal roots (ax2 + c = bx)
- roots and number equal squares (bx + c = ax2)
In the above case, we reduced the equation into a form as ‘squares and number equal roots (ax2 + c = bx)’
Mathematics beyond Geometry
Many historians and mathematicians like Solomon Gandz, Victor J. Katz, John J. O’Connor, Edmund F. Robertson, and Florian Cajori have appreciated Al-Khwarizmi’s work in algebra. They define his work as the first to teach algebra in an elementary form and the first true algebraic text which is still extant.
Before this, mathematics was just geometry. But when algebra came into form, for the first time the geometrical aspects of mathematics, rational numbers, irrational numbers, etc were treated as algebraic objects. It was so important that mathematics became applicable to things it was not previously.
Arithmetic
Al-Khwarizmi’s work on the subject of arithmetic died in its original form, but its Latin translations survived. His books Kitāb al-ḥisāb al-hindī (‘Book of Indian Computation) and Kitab al-jam’ wa’l-tafriq al-ḥisāb al-hindī (‘Addition and Subtraction in Indian Arithmetic’), were based on algorithms on decimal numbers. Many of his original work is lost, but a mere fraction was preserved in terms of Latin translations. His work led to the introduction of Arabic numerals, which were based on the Hindu-Arabic number system, to the Western world. Algorithm is derived from Al-Khwarizmi’s name and is defined as the technique of performing arithmetic using Hindu-Arabic numerals.
His work gradually replaced the abacus-based methods used in Europe by first utilizing a takht, i.e. dust boards to perform arithmetic and later on using pen and paper.
Many of his work is preserved in many Western Institutions till date.
Astronomy
Al-Khwarizmi’s Zīj as-Sindhind is a work comprising approximately 37 chapters on calendrical and astronomical calculations. It has 116 tables of calendrical, astronomical, and astrological data and a table of sine values. He utilized Indian astronomical methods.
It has tables for movements of the sun, the moon, and the five planets (known at that time). This was a turning point in Islamic astronomy, as before this all the astronomical work was primarily a research approach and translation of already discovered facts.
Unfortunately, the original Arabic version was lost, but the Latin translation is still preserved. The preserved work consists of four surviving manuscripts which are preserved in libraries like Bodleian Library (Oxford), Bibliothèque Mazarine (Paris), Biblioteca Nacional (Madrid), and Bibliothèque Publique (Chartres).
Trigonometry
Zīj as-Sindhind contains tables for trigonometric functions of sine and cosine and the first table for tangents. There is also a related work of spherical trigonometry attributed to him.
Geography
Kitāb Ṣūrat al-Arḍ (“Book of the Description of the Earth”), by Al-Khwarizmi, also known as his Geography was finished in 833. It is a major rework of Ptolemy’s second-century Geography. There is one preserved copy of it, and a Latin translation is also included. Like his other books, these too are preserved in European Libraries.
The book describes various latitudes and longitudes in order of weather zones. Paul Gallez a 21st-century cartographer notes that the system allowed deduction for many other latitudes and longitudes, but the condition of the document makes it practically unreadable. Both the Arabic and Latin versions of this book missed the world map, but Hubert Daunicht was able to reconstruct the missing map.
This was a reconstruction and the original map was a refinement and correction of Ptolemy’s world map, with accurate focus on Middle East and Africa. He “depicted the Atlantic and Indian Oceans as open bodies of water, not land-locked seas as Ptolemy had done and also corrected some overestimations of distance.
Ptolemy’s world map
Jewish Calendar
Risāla fi istikhrāj ta’rīkh al-yahūd (“Extraction of the Jewish Era”) is a work by Al Khwarizmi in which he described the Metonic cycle, rules for defining the day of the week the first day of the month Tishrei shall fall, calculates the distance between Jewish calendar and the Seleucid era, and rules for determining the mean longitude of the sun and the moon using Hebrew calendar.
Other works
Kitāb al-Taʾrīkh by Al Khwarizmi is described as a book recording historical events in chronological order. No direct manuscript survived, but a copy was in circulation in the 11th century. Several Arabic manuscripts in Berlin, Istanbul, Tashkent, Cairo, and Paris contain further material that surely or with some probability comes from al-Khwārizmī. The Istanbul manuscript contains a paper on sundials.
Some other papers were on spherical astronomy, leading to the determination of the direction of Mecca.
