By John, Ph.D. Tabak
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Extra info for Algebra: Sets, Symbols, and the Language of Thought (The History of Mathematics), Revised Edition
Although it is not immediately apparent, certain classes of problems cannot be solved by using straightedge and compass techniques. In fact, some of the most famous mathematical problems from antiquity are famous precisely because they cannot be solved with a straightedge and compass. Greek Algebra 29 There are three classical geometry problems that are very important in the history of algebra. Their importance in geometry is that they remained unsolved for more than 2,000 years. They were not unsolved because they were neglected.
To multiply two numbers together they used a method that consisted of repeatedly doubling one of the numbers and then adding together some of the intermediate steps. For example, to compute 5 × 80, first find 2 × 80 and then double the result to get 4 × 80. Finally, 1 × 80 would be added to 4 × 80 to get the answer, 5 × 80. This method, though it works, is awkward. Egyptian algebra employed the symbol heap for the unknown. Problems were phrased in terms of “heaps” and then solved. To paraphrase a problem taken from the most famous of Egyptian mathematical texts, the Ahmes papyrus: If 1 heap and 1/7 of a heap together equal 19, what is the value of the heap?
Heath. Great Books of the Western World, vol. 11. ) See the pictorial version of Euclid’s statement. Notice that the illustration shows three rectangles, two smaller ones and a large one. (The large rectangle is made of the four outside line segments. ) All three rectangles have the same height. We use x to represent the height of each of the rectangles. The rectangle on the left has length y and the rectangle on the right has length z. The length of the largest rectangle is y + z. Now we recall the formula for the area of a rectangle: Area = length × width.
Algebra: Sets, Symbols, and the Language of Thought (The History of Mathematics), Revised Edition by John, Ph.D. Tabak