Pretty good read, but the OCD in me could avoid pointing out that the example they give has a wrong solution:
x2+10x=39. Find x2.
This actually has two solutions for x: 3 and -13, so x^2 is 9 or 169.
It is probably a good example of how referring to preconditions at every step of the proof would help catch errors. From my experience writing code, I'd also argue that this would also make proofs more beautiful, because discovering that some steps require a lot of previous references might prompt the mathematician to restructure/refactor the proof to make it simpler to write, thus making it simpler overall.
There are more solutions than just that; for example, we could take x to be the matrix [[387, 320], [-480, -397]] (so x^2 = [[-3831, -3200], [4800, 4009]]), or the split-complex number 8j - 5 (so x^2 = 89 - 80j), or 5 in the ring of integers modulo 36 (so x^2 = -11).
But al-Khwarizmi was presumably writing in the context of familiar quantities >= 0, which is a perfectly fine thing to do.
In the context of electrical engineering, j is often used to denote the imaginary unit, that is, something such that j^2 = -1; in other words it's the same as what is usually called 'i'.
But, in the context of the split-complex numbers, j is something such that j^2 = 1, and i suppose it is writen as 'j' to distinguish it from 'i'.
So, the 'j' used here is different from the 'j' used in electrical engineering.
For example, if j were the square root of -1, as in electrical engineering, then (8j - 5)^2 would equal -39 - 80j: 64j^2 - 80j + 25 = -64 -80j + 25 = -39 - 80j; but here, in the split-complex numbers, (8j - 5)^2 = 64j^2 -80j + 25 = 64 - 80j + 25 = 89 - 80j.
“A square and 10 roots are equal to 39 units. The question therefore in this type of equation is about as follows: what is the square which combined with ten of its roots will give a sum total of 39? The manner of solving this type of equation is to take one-half of the roots just mentioned. Now the roots in the problem before us are 10. Therefore take 5, which multiplied by itself gives 25, an amount which you add to 39 giving 64. Having taken then the square root of this which is 8, subtract from it half the roots, 5 leaving 3. The number three therefore represents one root of this square, which itself, of course is 9. Nine therefore gives the square.”
If you interpret square to mean a literal square, then x can't be -13 as that would give a square whose sides are negative in length.
"In the 9th and 10th century AD, Islamic mathematicians were familiar with negative numbers from the works of Indian mathematicians, but the recognition and use of negative numbers during this period remained timid. Al-Khwarizmi in his Al-jabr wa'l-muqabala (from which we get the word "algebra") did not use negative numbers or negative coefficients, although al-Karaji wrote in his al-Fakhrī that "negative quantities must be counted as terms"."
So it would indeed seem that your interpretation is what al-Khwarizmi meant.
x2+10x=39. Find x2.
This actually has two solutions for x: 3 and -13, so x^2 is 9 or 169.
It is probably a good example of how referring to preconditions at every step of the proof would help catch errors. From my experience writing code, I'd also argue that this would also make proofs more beautiful, because discovering that some steps require a lot of previous references might prompt the mathematician to restructure/refactor the proof to make it simpler to write, thus making it simpler overall.