# Algebra Tiles Explorations of al-Khwārizmī's Equation Types: Modeling the Compound Equations and the Moiety Principle

Author(s):
Günhan Caglayan (New Jersey City University)

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].

Günhan Caglayan (New Jersey City University) , "Algebra Tiles Explorations of al-Khwārizmī's Equation Types: Modeling the Compound Equations and the Moiety Principle ," Convergence (October 2021)