Unknowns and equations

This lesson asks you to evaluate algebraic expressions with whole numbers and decimal tenths, and to solve equations with several terms, checking each solution. The museum offers a book whose title gave algebra its name, and some of its worked examples. ¹

Put your answer back into the equation and see whether both sides are equal. Then explain, step by step, what you did to each side.

Al-Khwarizmi says that reckoning by al-jabr and al-muqabala needs three kinds of number: roots, squares and simple number ². A square, he says, is what results from a root multiplied by itself ³. They combine into six types of problem. One type is “five squares equal eighty”, so one square is a fifth of eighty, sixteen ⁴. In each type, more or less than one square is reduced to a single square, and whatever equals it is treated likewise ³.

In his preface al-Khwarizmi says he wrote the book with the encouragement of the caliph al-Ma'mun, whose love of learning and closeness to scholars he praises ⁵. He says he made it for what people need in their dealings, such as inheritances and trade ². These are his words about his own purpose, as the record reports them from the preface of the book.

Al-Khwarizmi insisted that everything worked by al-jabr and al-muqabala must bring you to one of six types ⁶. What do you gain, and what might you lose, when you fit many different problems into a few types, and why?

• Algebra (al-jabr): the three kinds of number, and the two operations that name the science.
• The six types of equation: a worked example for each type.
• The Book of al-Jabr and al-Muqabala: the preface, and the chapters in order.
• al-Khwarizmi: what is known of his life.

Read the records on algebra and on the six types of equation. Take the example of four roots equal to twenty, write it with a letter for the unknown, solve it, and check your answer.