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.

Since a quantitative approach is prevalent in al-Khwārizmī’s text, in order to set up the historically based portion of the activity, I first introduced algebra tiles to my students in a manner which shows how the basic quantities involved in today’s quadratic equations can be represented as products. For that purpose, each quantity is derived using a multiplication mat by placing the two dimension tiles (one on top and the other on left) respectively:²

Note that negative quantities are included in the algebra tile representations. In the Al-Kitab almukhtasar fi hisab al-jabr wa’l-muqabala, however, al-Khwārizmī used only positive integers and fractions as the coefficients and solutions of equations. The system of numbers used by al-Khwārizmī was thus different from the one that is used in present-day classrooms. Interestingly, however, al-Khwārizmī did provide rules for multiplying negatives [Rosen 1831, 25–26]:

If the instance be, “ten and thing to be multiplied by thing less ten, then you say, thing multiplied by ten is ten things positive; and thing by thing is a square positive; and minus ten by ten is a hundred dirhems negative; and minus ten by thing is ten things negative. You say, therefore, a square minus a hundred dirhems; for, having made the reduction, that is to say, having removed the ten things positive by the ten things negative, there remains a square minus a hundred dirhems.

The following table illustrates how the computation in this excerpt might be modeled using algebra tiles to represent the negative quantities:

An interesting observation to discuss with students is how the so-called “zero principle” manifests itself as the appearance of the zero pairs “the ten things positive” and “the ten things negative.” Another interesting point to observe is the relevance to the well-known “difference of squares” identity \((A+B)(A-B) = A^2 - B^2\) to this computation and how algebra tiles involving negative quantities can be used to represent and reinforce it.

Notes

2. These figures were generated using the online algebra tiles applet mathies.ca/learningTools.php.

This section shows how the rules used by al-Khwārizmī to solve his three basic types of equations (squares = roots; squares = numbers; roots = numbers) can be simulated using algebra tiles. al-Khwārizmī presented these examples verbally in prose format. The examples in this section constitute the basis for modeling all six equations using algebra tiles in that they illustrate the essence of how to physically represent al-Khwārizmī’s prose expressions. In what follows, all descriptions are taken from [Rosen 1831, 6–7).

Example 1:  Squares are Equal to Roots (Type 1)

Example 2: Squares are Equal to Numbers (Type 2)

Example 3: Roots are Equal to Numbers (Type 3)  

Students may find it interesting that al-Khwārizmī always stated the value of the square, as well as the root, in his examples. This is the case even in Example 3, where the value of the root (x=3) is given in the problem statement.

The three examples above are all associated with monic polynomials. The next two examples illustrate al-Khwārizmī’s procedures for obtaining one entire square “whenever you meet with a multiple or sub-multiple of a square” [Rosen 1831, 13]; that is, the procedure for obtaining a leading coefficient of 1 in a non-monic quadratic equation.  As shown in these examples, the squares, roots, and numbers are simultaneously scaled by the same ratio where that ratio is chosen in order to obtain a single square. A similar procedure was used with linear equations (roots = numbers), as is shown in Example 6. While it is relatively straightforward to model this procedure with algebra tiles in the case of multiple squares or roots (Examples 4 and 6), the lack of fractional tiles makes the case of submultiples more complicated to set up (Example 5). In fact, some of al-Khwārizmī’s equations involve fractions in ways that are impossible to represent with algebra tiles, as we describe in our discussion of Example 10 in a later section.

Example 4: Squares are Equal to Roots (Type 1), beginning with a multiple of a square

Example 5: Squares are Equal to Numbers (Type 2), beginning with a submultiple of a square

Example 6: Roots are Equal to Numbers (Type 3) , beginning with a multiple of the root

This section focuses on modeling the three compound species in al-Khwārizmī’s algebra text using a step-by-step problem-solving approach, with each case treated in a separate subsection:

• Type 4: roots and squares = numbers
• Type 5: squares and numbers = roots
• Type 6: squares = roots and numbers

As al-Khwārizmī noted, the three basic equations (Types 1-2-3) discussed in the previous section “do not require that the roots be halved” [Rosen 1831, 13]. In the latter three (Types 4-5-6), on the other hand, he asserted that “halving the roots is necessary” [Rosen 1831, 13]. The utilization of moities is thus also a key feature in modeling the solutions of equations of Types 4-5-6 using algebra tiles representations. He also provided geometric constructions (based on specific equation examples) for each of these three equation types “to point out the reasons for halving” [Rosen 1831, 13]. For equations of Types 5 and 6, the diagrams that accompany these synthetic constructions are not transferrable to an algebra tile representation. As indicated by Figure 2, however, the second diagram that al-Khwārizmī gave in his demonstration for Type 4 equations (based on the example \(x^2+10x = 39\)) is reminiscent of the algebra tile models we provide in this section. Notice in particular the small square of area 25 in the lower left-hand corner, which represents the final step in the "Completing the Square" technique for solving this equation. In fact, similar diagrams are widely used with an equation of Type 4 to illustrate that solution technique.

Figure 2. Second diagram in al-Khwarizmi's geometric demonstration of Type 4
using the equation “a Square and twenty-one Dirhems are equal to ten Roots” [Rosen 1831, 16].

Following his treatment of the solution procedures for each of the six cases of equation, al-Khwārizmī went on in his treatise to present six problems “which will serve to bring the subject nearer to the understanding, to render its comprehension easier, and to make the arguments more perspicuous” [Rosen 1831, 35]. These problems were followed by a large collection of “various questions” that further illustrated how al-Khwārizmī’s procedures for “calculating by completion and reduction must always necessarily lead you to one of these cases” [Rosen 1831, 35]. In the examples below, we illustrate how to model such problems using algebra tiles. We have selected these particular examples in part because they avoid the use of fractions and mixed numbers, yet show the complexity of the equations that al-Khwārizmī treated. Because some problems involve the difference of two quantities, note that it is very natural to use subtraction within the algebra tile models. Following al-Khwārizmī’s advice to his readers that they should “reduce this [equation] according to the rules, which I have above explained to you” [Rosen, 1831, 51], we have omitted the algebra tile representations for some of the later details in our last two examples.

• Example 11: A problem leading to the case “squares = numbers”
• Example 12: A problem leading to the case “squares = numbers"
• Example 13: A problem leading to the case “squares = roots”
• Example 14: A problem leading to the case “squares and numbers = roots”
• Example 15: A problem leading to the case “squares and numbers = roots”

It is worth noting that al-Khwārizmī gave geometrical demonstrations for his solution methods for the compound equations (Types 4–6) in order to “point out the reasons for halving” [Rosen 1831, 13]. The figures he gave for equations of Type 5 and Type 6 do not, however, seem to be translatable to an algebra tile representation. Despite this and other limitations of algebra tiles (e.g., the impossibility of yielding an accurate representation of situations involving fractions), the translation of al-Khwārizmī’s verbal descriptions of his procedures to algebra tiles representations offers a multi-representational approach to equation solving that encourages diversity in algebraic reasoning and can promote reflection about its teaching.

The modeling activity proposed in this article can be implemented in an exploratory approach that prompts students to make meaningful and accurate connections between al-Khwārizmī's excerpts and the corresponding algebra tiles representation. After a basic introduction to the use of algebra tiles, the class can work individually or in small groups on Examples 1–10. To guide this work, students can be provided with a two-columned worksheet that gives the excerpts from al-Khwārizmī in the left column and asked to generate algebra tiles representations that model those excerpts in the corresponding “algebra tiles workspace” in the right column; the tables provided in this article for those examples thus serve as an “answer key” for this activity. The initial use of a multiplication mat that physically separates the dimension tiles from the binomial rectangles could be helpful in preventing any potential confusion about the dimension tiles (linear type quantities) and the binomial rectangles (area type quantities). Other obstacles or challenges that the students might encounter for particular equation types (e.g., the necessity of using either the subtraction symbol or “\(-x\)” tiles for Type 5 equations) have also been mentioned in earlier sections of this article. Whole-class discussion and instructor demonstrations may be valuable at certain junctures to overcome those challenges. Instructors who wished to extend the exploration further could then have students engage with the more complicated problems presented in Examples 11–14.

Depending on the student audience and the goals of the instructor, students could additionally be prompted to reflect on and discuss various aspects of the activity. For example, pre-service and practicing secondary teachers could be asked to focus on various components of the algebra tiles representations, such as the role of the dimension tiles, the emerging binomial rectangles, the area conservation principle, and the dynamics of viewing the binomial rectangle area alternately as a sum and as a product. Students in a history of mathematics course might be prompted to reflect on the differences between the mathematical context and assumptions that informed al-Khwārizmī’s work and those that inform today’s algebraic practices that are perhaps better captured by the algebra tile representations. Students in both these groups, as well as high school students and others, might also profitably discuss specific mathematical questions that emerge from the activity, such as whether and why Type 4 and Type 6 equations always have one positive and one negative solution.

As desired, the symbolic representation of al-Khwārizmī’s equation types and procedures could furthermore be brought into the exploration and juxtaposed against both al-Khwārizmī’s verbal descriptions and their algebra tile representations. A discussion of the roles of these different types of representations in supporting student learning of algebra—and especially the pedagogical benefits to be derived from the concrete visualization afforded by algebra tile manipulatives—could be especially fruitful with prospective and practicing secondary mathematics teachers. In these and other ways, the use of algebra tile manipulatives to explore al-Khwārizmī’s procedures opens up new avenues of influence for his Al-Kitab almukhtasar fi hisab al-jabr wa’l-muqabala.

References

Rosen, Frederic, trans. and ed. 1831. The Algebra of Mohammed ben Musa [al-Khwārizmī]. London: Oriental Translation Fund.

Otero, Daniel E. 2019. Completing the Square: From the Roots of Algebra, A Mini-Primary Source Project for Students of Algebra and Their Teachers. Convergence 16 (September). DOI:10.4169/convergence20190901.

Acknowledgements

The author wishes to thank the Convergence editorial board and anonymous reviewers for their helpful comments on earlier drafts.

About the Author

Günhan Caglayan (MR Author ID: 1116420) teaches mathematics at New Jersey City University. His main interests are mathematical visualization and student learning through modeling and visualization.