Muhammad al-Khwarizmi: A Life in Five Chapters

Chapter 1 · The House of Wisdom

Baghdad, translation, and a library to run

c. 780 – c. 820 · Khwarazm · Baghdad

Muhammad ibn Musa al-Khwarizmi was born around 780. His name points to Khwarazm, the region south of the Aral Sea, though whether he or his family came from there is not certain, and little else about his origins is recorded.

He worked in Baghdad under the caliph al-Ma'mun, at the institution later called the House of Wisdom, where he had charge of the library. The Abbasid court was running a translation programme on an extraordinary scale: Greek, Persian, Syriac and Sanskrit works were being rendered into Arabic, and scholars had access to Euclid, Ptolemy, Indian astronomy and Persian tables in one place.

This matters for what he did next. He was not working in one tradition. He had Greek geometry, which was rigorous and proof-based but clumsy with numbers; and he had Indian arithmetic, which had a positional system with a zero and was superb for calculation but arrived without demonstrations. He took both seriously and would not choose between them.

That combination — Indian calculating methods, justified by Greek geometrical reasoning, written out in plain Arabic prose — is the whole of his method. Neither tradition on its own would have produced it.

He dedicated his algebra to al-Ma'mun, who had encouraged it.

Why this matters

Al-Khwarizmi's work was possible because Baghdad had assembled Greek and Indian mathematics in one library — a case of translation producing original science.

You have the library and two traditions in it. What would you ask him?

Ask al-Khwarizmi

• “What did you have on the shelves at the House of Wisdom?”
• “What did the Indian methods have that the Greek ones did not?”
• “Why not simply choose one tradition and work in it?”
• “What did al-Ma'mun actually want from you?”
• “Who else was working in that library alongside you?”

Chapter 2 · Restoring and Balancing

Two operations, and the words that became algebra

c. 820 · Baghdad

The book is *al-Kitab al-mukhtasar fi hisab al-jabr wa'l-muqabala* — the Compendious Book on Calculation by Restoring and Balancing — written around 820.

The two words in the title are the two operations. *Al-jabr*, restoring, means taking a subtracted quantity and moving it to the other side so that it becomes added: what a modern student does when they move a term across the equals sign. *Al-muqabala*, balancing, means cancelling identical quantities appearing on both sides.

Between them these two moves reduce any equation of the kinds he treats to one of a small number of standard forms. That is the essential idea of algebra: not solving a particular problem but transforming any problem into a shape you already know how to solve.

The difficulty was the notation, which did not exist. He had no letters for unknowns, no symbols for operations, no equals sign. Everything is written out in words: a quantity, its root, its square, the number. A modern one-line equation becomes a paragraph.

And he recognised no negative numbers. A negative quantity was not a number; it was nothing at all. That single restriction is why he needed *six* separate types of equation instead of one general form — because he could not allow a coefficient or a constant to be negative and so had to enumerate every arrangement of signs separately.

Why this matters

The idea of transforming any problem into a standard form you already know how to solve is what makes algebra a general method rather than a collection of tricks.

You have two operations and no symbols to write them with. What is your question?

Ask al-Khwarizmi

• “What do al-jabr and al-muqabala actually do to an equation?”
• “Why did you need six types of equation instead of one?”
• “What is it like to write mathematics with no symbols at all?”
• “Could you not accept a negative quantity even as a step?”
• “How did you decide the six cases were all of them?”

Chapter 3 · The Square With Rectangles Along Its Sides

A diagram still drawn twelve hundred years later

c. 820 · Baghdad

For each of his six cases al-Khwarizmi gives a procedure — and then a geometrical justification, which is the part that matters.

Take his standard example: a square and ten roots equal thirty-nine. In modern notation, a quantity squared plus ten times the quantity equals thirty-nine.

He draws it. A square whose side is the unknown. Along two of its sides he lays rectangles, each five units wide, together accounting for the ten roots. The figure now has area thirty-nine, but it is not a square — there is a piece missing at the corner, a small square five by five, area twenty-five.

Supply it. The whole figure becomes a true square of area thirty-nine plus twenty-five, which is sixty-four. Its side is therefore eight. And that side is the unknown plus five. So the unknown is three.

That is completing the square, and the diagram is the one taught today. It is worth noticing what it achieves: it does not merely tell you a rule works, it shows you *why* it works, in a picture you can check with a ruler. Al-Khwarizmi's stated principle is that a procedure which works should be shown to be right, not merely reported to be reliable.

He finds only the positive root. The negative one, being no number, is invisible to him — half of every solution, lost.

Why this matters

Completing the square is one of the few school methods that comes with its own visual proof, and the proof is essentially unchanged since 820.

You have the square, the rectangles and the missing corner. What would you ask?

Ask al-Khwarizmi

• “Walk me through the square and the two rectangles.”
• “Why give a proof at all when the rule already works?”
• “What happened to the second solution?”
• “Why 'a square and ten roots equal thirty-nine' as the example?”
• “Could the same picture be made to work for a cube?”

Chapter 4 · Bringing the Zero West

A second book, and the route by which digits reached Europe

c. 825 · Baghdad · Toledo

Around 825 al-Khwarizmi wrote a separate treatise on calculating with the Indian numerals — the positional system with nine digits and a zero. The Arabic original is lost; it survives in Latin translation, and its opening words, *Dixit Algorizmi* — al-Khwarizmi said — are how the word entered European languages.

He did not invent these numerals and never claimed to; he credits their Indian origin. What he did was set out plainly how to compute with them: how to write numbers, how to add, subtract, multiply, divide and halve, and how the zero holds an empty column so that position keeps its meaning.

The advantage over the alternatives is enormous. Roman numerals are workable for recording a number and nearly useless for calculating with one; the abacus is fast but leaves no record of the working. A positional system with a zero lets you do the arithmetic on paper, in a way that can be checked afterwards by somebody else.

That last point was his real preoccupation. His stated aim is a method an ordinary person can follow alone, without the author present, step by step, with no case omitted.

Which is what an algorithm is. The Latin form of his name became the word, and every computer program written since is a descendant of the idea that a procedure can be written out completely enough for someone else to execute it exactly.

Why this matters

The concept of a procedure written out completely enough for a stranger to execute without its author is the foundation of computing, and it carries al-Khwarizmi's name.

You have the digits, the zero and the word that came from your name. What would you ask?

Ask al-Khwarizmi

• “What can you do with a zero that you cannot do without one?”
• “Why was writing the working down so important to you?”
• “What would you make of your name becoming the word 'algorithm'?”
• “Why credit the Indian origin when you could have claimed it?”
• “Who were you actually writing these methods for?”

Chapter 5 · Tables, Coordinates and the Shape of the Earth

The other half of the working life

c. 830 – c. 850 · Baghdad

The algebra is what he is remembered for; it was not most of his work.

He produced astronomical tables — the *Zij al-Sindhind* — combining Indian, Persian and Ptolemaic material, giving positions of the Sun, Moon and the five known planets, together with tables of sines and tangents. These were used, corrected and re-used across the Islamic world and later in Europe. He wrote on the astrolabe and on the sundial, and on determining the direction of Mecca from any given place — a problem in spherical geometry with a daily practical use for millions of people.

He also revised Ptolemy's *Geography*, producing the *Kitab Surat al-Ard*, the Book of the Description of the Earth, with corrected latitudes and longitudes for well over two thousand places. He worked, according to one account, as part of a team commissioned by al-Ma'mun to measure the length of a degree of latitude, and hence the circumference of the Earth, by survey on the plain of Sinjar.

His stated view of what mathematics is for sits behind all of it: inheritance, trade, surveying, legacies, the digging of canals. The algebra book itself is half worked problems about dividing estates under Islamic inheritance law.

He died around 850, with the methods already moving west.

Why this matters

Al-Khwarizmi's mathematics was written for inheritance, trade and surveying — a reminder that abstraction and practical use were not opposed for him.

You have the tables, the coordinates and the estates. What is your question?

Ask al-Khwarizmi

• “How do you find the direction of Mecca from an unfamiliar city?”
• “How was a degree of latitude measured on the ground?”
• “Why is half your algebra book about dividing inheritances?”
• “What was wrong with Ptolemy's map of the world?”
• “Which of your books would you expect to last longest?”

What al-Khwarizmi changed

Two English words come from al-Khwarizmi — algebra from the title of his book and algorithm from the Latin form of his name — and one diagram: the square with rectangles along two sides, which is still how completing the square is taught. His larger legacy is the principle that a method should be written out plainly enough for a stranger to execute it without its author present, which is what an algorithm is and what every computer program depends on.

A debate that continues

How much of the Indian material he transmitted versus developed is not fully settled, and his own origins — whether he or only his family came from Khwarazm — are uncertain.

Keep exploring — ask al-Khwarizmi

• “What would you have done with a symbol for the unknown?”
• “Is a rule without a proof worth teaching?”
• “Which of your six cases gave you the most trouble?”

Related lives

• Euclid — Who Made Proof The Standard
• Ibn al-Haytham — Who Tested What the Ancients Wrote
• Brahmagupta — Who Gave Nothing Its Rules
• Leonardo Fibonacci — Who Brought Europe The Nine Figures

Related themes

Solving quadratic equations · Number systems · Algorithms