Algebra in Baghdad

Historical evidence

An equation stated in words

Around 825, Muḥammad ibn Mūsā al-Khwārizmī wrote an Arabic book on calculation by restoration and reduction in Baghdad. The cited studies discuss rhetorical algebra: types of equations are expressed in words rather than in the compact symbolism used here.

The example now written x² + 10x = 39 is accompanied in historical teaching by a geometric demonstration. This page identifies its symbols and animation stages as modern translations.^{[ALGEBRA][ALGEBRA-39]}

Historical evidence

Squares, roots and quantities

A square, roots and a known quantity name kinds of magnitude in the rhetorical procedure. For positive magnitudes, an area arrangement makes the transformation visible.

The Atlas writes x, b and c so readers can vary the example. That general symbolic notation is a modern layer and should not be mistaken for the wording of the Arabic text.^[ALGEBRA]

Reconstruction

Complete the square

Begin with x² and split bx into two equal strips. Their L-shaped arrangement is missing a corner of area (b/2)². Add that corner to both sides and the left side becomes a square with side x + b/2.

Use the stage buttons or Back and Next, then vary positive b and c. With b = 10 and c = 39, the completed square has area 64 and positive side 8, so x = 3.^[ALGEBRA-39]

Derivation

The modern identity

Taking the positive geometric length gives x = √(c + (b/2)²) − b/2. Substitution into the original equation checks the result independently.

Modern symbolic algebra also returns −√(c + (b/2)²) − b/2. The live result reports this second root outside the historical positive-area diagram rather than drawing a negative length.^[ALGEBRA-39]

Open question

Scope of the reconstruction

The geometric reconstruction depends on positive lengths and areas. Its clarity comes with that boundary.

The title “algebra in Baghdad” locates this example in a scholarly setting. It does not reduce the varied mathematics of the Islamic world to one author, city or line of descent.^{[ALGEBRA][ALGEBRA-39]}

A classroom area model demonstrates the positive case shown. It does not by itself represent modern signed coefficients, complex roots or every equation type treated in later algebra.

Historical evidence

Writing, translation and reception

The word algebra has a history tied to the Arabic title, while the mathematical practices around equations have many earlier and later settings. Describing al-Khwārizmī as the sole inventor or father of algebra would overstate the evidence.

The Atlas transmission view names documented translations into Latin and keeps translation distinct from the broader history of influence and reception.^[ALGEBRA]

Open question

Why the area argument matters

Completing the square is more than a symbolic trick: the added term is exactly the corner required to turn an L-shaped area into a square. The diagram exposes why the same quantity must be added to both sides.

Continue to Geometry for area and metric reasoning, or to equations and transmission in the historical Atlas.^[ALGEBRA-39]