Alex's Adventures in Numberland - Alex Bellos [74]
While using numbers for the purposes of entertainment has been a consistent theme of mathematical discovery, maths only properly got started as a tool for solving practical problems. The Rhind Papyrus, which dates back to around 1600 bc, is the most comprehensive surviving mathematical document from ancient Egypt. It contains 84 problems covering areas such as surveying, accounting, and how to divide a certain number of loaves among a certain number of men.
The Egyptians stated their problems rhetorically. Problem 30 of the Rhind Papyrus asks, ‘If the scribe says, What is the heap of which will make ten, let him hear.’ The ‘heap’ is the Egyptian term for the unknown quantity, which we now refer to as x, the fundamental and essential symbol of modern algebra. Nowadays we would say that Problem 30 is asking: What is the value of x such that multiplied by x is 10. To put it more concisely: What is x such that () x = 10?
Because the Egyptians didn’t have the symbolic tools that we have now, such as brackets, equal signs, or xs, they solved the above question using trial and error. Estimating for the heap, they then worked out the answer. The method is called the ‘rule of false position’ and is rather like playing golf. Once you are on the green, it’s easier to see how to get the ball into the hole. Similarly, once you have an answer, even a wrong answer, you know how to get closer to the right one. By comparison, the modern method of solution is to combine the fractions in the equation with the variable x, so that:
x = 10
Which is the same as:
Or:
Which further reduces to:
And finally:
Symbolic notation makes life so much easier.
The Egyptian hieroglyph for addition was , a pair of legs walking from right to left. Subtraction was , a pair of legs walking from left to right. As number symbols evolved from tally notches to numerals, so the symbols for arithmetical operations also evolved.
Still, the Egyptians had no symbol for the unknown quantity, and neither did Pythagoras or Euclid. For them, maths was geometric, tied to what was constructible. The unknown quantity required a further level of abstraction. The first Greek mathematician to introduce a symbol for the unknown was Diophantus, who used the Greek letter sigma, . For the square of the unknown number he used Y, and for the cube he used KY. While his notation was a breakthrough in its time, since it meant problems could be expressed more concisely, it was also confusing since, unlike the case with x, x2 and x3, there was no obvious visual connection between and its powers Y and KY. Despite his symbols’ shortcomings, however, he is nevertheless remembered as the father of algebra.
Diophantus lived in Alexandria sometime between the first and the third centuries ce. Nothing else is known of his personal life except for the following riddle, which appeared in a Greek collection of puzzles and is said to have been inscribed on his tomb:
God vouchsafed that he should be a boy for the sixth part of his life; when a twelfth was added, his cheeks acquired a beard; He kindled for him the light of marriage after a seventh, and in the fifth year after his marriage He granted him a son. Alas! Late-begotten and miserable child, when he had reached the measure of half his father’s life, the chill grave took him. After consoling his grief by the science of numbers for four years he reached the end of his life.
The words are perhaps less an accurate description of Diophantus’s family circumstances than they are a tribute to the man whose innovative notation presented new methods for solving problems like the one above. The ability to express mathematical sentences