Simplifying expressions and solving equations

This lesson asks you to simplify algebraic expressions using the properties of operations, and to create and solve equations and verify the solutions. The museum offers al-Khwarizmi's six types of equation and al-Karaji's way of multiplying composite expressions; your own equations and their contexts are for your class to bring. ¹

Put the answer back into the original problem and compare both sides. Then ask: is testing one value enough to prove a general rule? Read how the record separates doing an operation, checking an example and justifying a rule.

Read al-Khwarizmi's proof closely. He says that roots are not halved in the first three types, and that for each of the remaining three he made “a figure from which the cause is shown” ². For a square and ten roots he draws the square as a figure of unknown side, divides the ten roots into four rectangles placed on its sides, then completes the four corners with small squares totalling twenty-five, so that the large square is sixty-four with side eight ³. As the record of his algebra reports, the TDV biography describes the squares and rectangles as a geometrical representation and verification of the analytical solution, distinguishing the calculation that finds a number from the drawing that makes the relation behind it visible ⁴. He ends the theory of the six types with the rule that every problem must bring you to one of them ⁵.

For the figures, al-Khwarizmi's own stated purpose is that they show the cause ². For expressions in his book that have no figure, because they combine three different kinds, he shows their necessity in words ⁶. For al-Karaji the records preserve no stated purpose; what they show is that his al-Fakhri kept geometrical demonstrations ⁷.

Did al-Karaji's arithmetic of expressions free algebra from geometry? As MacTutor reports, Woepcke and Rashed emphasize algebraic operations becoming less dependent on geometrical representation, an important direction toward autonomous algebraic arithmetic ⁸. The account it reports from Crossley stresses numerical examples, limits of demonstration and retained Euclidean theorems in equation solving ⁸. TDV likewise notes geometrical proof in al-Fakhri ⁷. The record presents these interpretations without resolving them.

What we read of al-Khwarizmi is the Cairo edition, whose editors write the six types in modern symbols and explain that he worked only with positive numbers ^{9 10}. For al-Karaji the witnesses are textual: TDV discusses al-Fakhri, al-Badi and other works and their editions and scholarship ⁷. Woepcke's introduction describes damaged manuscript material and restoration, reminding readers that an edition has editorial mediation ¹¹. The record of al-Karaji's arithmetic says its explanatory numbers and symbols are the museum's own, not an example claimed to have been copied from the historical book ⁸.

The record says that substitution can uncover an arithmetic mistake, but that agreement at one value does not prove a rule for every value ⁸. Take a position and answer the strongest objection: does a drawing like al-Khwarizmi's squares justify a rule better than a check with one number?

• The six types of equation: the figure for each compound type, and the six model problems.
• Al-Karaji's polynomial arithmetic: operations on expressions, their limits, and the historians' views.
• al-Karaji: his works, his name and what is not known.
• The Book of al-Jabr and al-Muqabala: the chapters, and the Cairo editors' modern symbols.

Read The six types of equation (al-Khwarizmi) and then Al-Karaji's polynomial arithmetic. Solve one of al-Khwarizmi's problems in your notebook and verify each answer, then expand a product of your own the way the record shows al-Karaji's method.