 An important change in mathematics occurred in the Islamic world during the 8th century. Prior to this time, problems were solved in a number of different ways. We might describe them as guess and modify the Egyptian ah-ha method, clever guessing as in diathontist, and follow these steps as in the Mesopotamian quadratics. But there is no elements-like organization of problem-solving methods. And this leads us to the man from Qarizm. Muhammad ibn Musa al-Qarizmi wrote a book on problem-solving. We know very little about his life beyond his name, Muhammad, son of Musa, from the Qarizm. That's a region in Central Asia, just south of the Aral Sea, in what is now modern-day Uzbekistan. Like Euclid, he was not the first to organize problem-solving methods. But like Euclid, his was the influential work and replaced all previous works. Qarizmi's book was titled The Book of Completion and Restoration. Now in Arabic, completion is al-jabta, while restoration is al-mukhabala. And this is why we refer to general problem-solving as al-mukha, um, algebra. Oh, and one other word from al-Qarizmi, we get another word, the process by which a problem is solved is called an algorithm. Al-Qarizmi's algebra is organized along Euclidean lines. A problem is presented, a solution is given, the solution is then proven. The basis of the proof is geometry, so we have to interpret expressions as geometric objects. This leads to the principle of homogeneity. New geometrically, an expression like x squared plus 4x plus 5 is nonsense. X squared is, as its name suggests, a square, a two-dimensional object. If x squared is a square, then x is one side, and 4x is four of those sides. And 5, well since 5 is a number, it's a bunch of units. And so what we have is we have a square, four line segments, and a bunch of units. These can't equal anything. To make sense of all of this, all terms have to represent the same type of object. They must be homogeneous. Now since x squared is a square, we can try to interpret all terms as two-dimensional objects. So x squared is a square with side length unknown. For x, well let's make that a rectangle with one side x and the other side 4. And 5, well we'll make that a rectangle with area 5. This leads to a new problem. While x squared plus 4x plus 5 makes sense as an expression, x squared plus 4x plus 5 equals 0 is nonsense as an equation. Because this says we have a bunch of figures, and if we put all these figures together, we get nothing. So instead, we have to consider several cases. Two aren't very interesting. x squared equals a, and x squared equals px. Now in the geometric language, x squared is a square, and a is a number, so this first case is called square equal to number. And that's like saying we know the area of the square, what's the side length. The second case, p times x, well x is a side, so this is referred to as square equal to sides. The three important cases, x squared plus px equals q, square and sides equal to number. That's like saying the square and this rectangle equal some area. x squared equals px plus q, that's square equal to sides and number. So our square is a rectangle with the same side and some other figure. x squared plus q equals px, square and number equal to sides. Or our square and some other figure are together equal to the rectangle. So Alcorizmi considers the problem, a square and ten sides is equal to 39. This is a square and sides equal to number, and as in the elements, Alcorizmi describes a procedure, then proves it geometrically. So Alcorizmi's procedure, take half the number of sides, five, square the result, 25, add to the number, 64, take the root, eight, then subtract half the number of sides, three. And our solution is this final result. Alcorizmi then proceeds to prove that this solution works. In particular, he has to show that this is going to work regardless of the actual numbers. So the task at hand is to draw a figure that represents a square and ten sides. Well, we know what a square looks like. And the side of the square is our unknown. We also want to represent ten sides. Well, that's going to be a rectangle where one side is our unknown and ten is our other side. And so this is the figure that is equal to 39. And we can view that as saying that the area of this figure is 39. And our goal at this point is to figure out what this unknown side is. And we'll do that by manipulating our figure. And that manipulation proceeds as follows. So let A be our square. Now we've added ten of its sides by adding this rectangle, which has one side the unknown and the other side of length ten. But let's split this in half so we can add ten of its sides by adding two rectangles, C and D, both with one side five. Notice that we have an incomplete square. So we'll complete our square by adding in this missing piece. And note that the missing piece is a square with a side of five. Now, since the area of the figure doesn't change if we move its pieces around, this object still has area 39. And if we add this square with a side of five, well, the area of the square is 25. And so the area of the whole figure is 64. But since it's a square, we know the side length is eight. And we know this part of the figure has length five, so the original square has side three.