Algebra Tiles Explorations of al-Khwārizmī’s Equation Types: al-Khwārizmī’s Compendium on Calculating by Completion and Reduction

Muhammad ibn Musa al-Khwārizmī (780–850 CE) was the first mathematician to write an algebra book: Al-Kitab almukhtasar fi hisab al-jabr wa’l-muqabala (Compendium on Calculating by Completion and Reduction¹). In this treatise, al-Khwārizmī first emphasized the notion of number via his statements [Rosen 1831, 5]:

When I considered what people generally want in calculating, I found that it always is a number.

I also observed that every number is composed of units, and that any number may be divided into units.

Moreover, I found that every number, which may be expressed from one to ten, surpasses the preceding by one unit: afterwards the ten is doubled or tripled, just as before the units were: thus arise twenty, thirty, etc., until a hundred; then the hundred is doubled and tripled in the same manner as the units and the tens, up to a thousand; then the thousand can be thus repeated at any complex number; and so forth to the utmost limit of numeration.

He then proposed a classification of numbers into three types [Rosen 1831, 5–6]:

I observed that the numbers which are required in calculating by Completion and Reduction are of three kinds, namely, roots, squares, and simple numbers relative to neither root nor square.

A root is any quantity which is to be multiplied by itself, consisting of units, or numbers ascending, or fractions descending.

A square is the whole amount of the root multiplied by itself.

A simple number is any number which may be pronounced without reference to root or square.

Here, the phrase “in calculating by Completion and Reduction” is a reference to the practice of equation-solving. This, together with al-Khwarizi’s descriptions of the kinds of numbers he had observed in that practice, suggests a natural correspondence between his categories and the three terms we associate with quadratic equations today: “roots” with the linear term; “squares” with the quadratic term; and “simple numbers” with the constant term. 

We also notice the quantitative aspect of al-Khwārizmī’s roots, squares, and simple numbers categories, which allows any two of the three to be used to generate a simple equation [Rosen 1831, 8]:  

A number belonging to one of these three classes may be equal to a number of another class ; you may say, for instance, “squares are equal to roots," or “squares are equal to numbers,” or “roots are equal to numbers."

We further see the emergence of a new kind of quantity, which al-Khwārizmi used to describe three “compound species” equations [Rosen 1831, 8]:

I found that these three kinds; namely, roots, squares, and numbers, may be combined together, and thus three compound species arise; that is, “squares and roots equal to numbers;” “squares and numbers equal to roots;” “roots and numbers equal to squares.”

al-Khwārizmī’s treatise is thus generally considered a handbook for solving linear and quadratic equations. His text classified these, however, as six different equation types. The following table outlines this classification, along with the examples of each type that al-Khwārizmī solved in the first part of his text. Note that the word “dirhem” that appears in the example for Type 4 refers to a type of coin that was historically used in Baghdad; it was often used by  al-Khwārizmī as the units for the simple number in an equation.

Notes

1. Al-jabr = completion; al-muqabala = reduction.