Terms, expressions, equations and checking the answer

Put the answer back into the original problem and see whether the equality holds. If an expression has like terms, where do you collect them before you evaluate it?

One of al-Khwarizmi's six types has two answers. The problem is “a square and twenty-one equal ten of its roots”, which has two answers: root three and square nine, or root seven and square forty-nine ². He says this type alone “works by both addition and subtraction” ². If the square of half the roots is less than the number, he calls the problem “impossible” ³. All of this is in the record of the six types.

The records preserve no sentence in which al-Karaji says why he extended arithmetic to expressions with an unknown. They show a historian's account of what moved the work. MacTutor explains the new direction through the encounter between Diophantine arithmetic and Arabic algebraic concepts, with an emphasis on generalization and on more widely applicable methods ⁴. TDV connects al-Fakhri with the patronage of Fakhr al-Mulk in the Buyid setting ^4 5.

“Polynomial arithmetic” is the museum's modern name for part of al-Karaji's work, not a claim that he used today's definitions or notation ⁴. The record says the work was neither algebra created from nothing nor the invention of the letters and exponents used in its examples ⁴. His division was of a composite quantity by a single term, not a method for dividing any polynomial by any other ⁴.

The record says that substitution can uncover an arithmetic mistake, but that agreement at one value does not prove a rule for every value ⁴. Why is one value not enough, and what else would you want to see before you trusted a rule?

• The six types of equation: a worked example for each type, and the figure that proves three of them.
• Al-Karaji's polynomial arithmetic: what his operations on expressions could do, and what they could not.
• al-Karaji: his works, his patron and what is not known about him.
• al-Khwarizmi: the author of the six types.