Completion and Balancing
- Thread starter shahar
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You do realize that the language in which algebra is explained and the notation with which algebra is done have evolved over the centuries. No one would do algebra his way today. And no, you need all his techniques to solve the types of equations that he addressed. Moreover, he did not address even all linear equations. He was a very great mathematician, but the average ninth grade student today knows more algebra than he did: mathematics is a progressive discipline.
Dr.Peterson
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I have seen different explanations of his terms, so I can't be sure who is right. (Even if I could read Arabic, there would probably be questions.) But according to Wikipedia, "completion" (al-jabr) means undoing a subtraction by adding the same (positive) quantity to both sides, while "balancing" (al-muqabala) means undoing an addition by subtracting the same (positive) quantity from both sides. I'm not sure whether he expressed it in anything like the modern way, though.
As you can see, the only reason he had to separate these ideas is that he didn't use negative numbers! For the same reason, he couldn't solve all quadratic equations in the same way, but had to split them into 6 cases so that no coefficients (or variables) had to be negative.
Note also that his main focus was on quadratic, not linear, equations (though I presume he also solved the latter).
Today, we need a different pair of rules: one to handle additions and subtractions, and one to handle multiplications and divisions. Lots of other "rules", such as the commutative property, also underlie what we do in solving equations.