## The Development of Algebra

# The Greeks

Whereas many Greeks made decisive advances in geometry, as far as we know they only produced one algebraist, Diophantus of Alexandria (c 250 A.D.). Diophantus used an abridged notation for frequently occuring operations, and a special symbol for the unknown. Thus for the unknown he wrote , if it occured once. For our 3x, he wrote , where is the plural of the unknown and represents the coefficient 3.

Addition was denoted by simply placing the summands next to each other, and subtraction was indicated by the symbol .
Instead of our sign for equality, he wrote . Also terms which were not tied to the unknown were preceded by the symbol . As an example,

for our: | x | | 1 | + | 7 | = | x | | 4 | - | 5 |

he would write: | | | | | | | | | | | |

Besides being the first to use symbols systematically in algebra, Diophantus was also the first to give general rules for the solution of an equation. An example, in our notation,

is as follows: | 8x - 11 - 2x + 5 = x - 4 + 3x + 10 |

rearranged in the form | 8 x + 5 + 4 = x + 3x + 10 + 11 + 2x |

or | 8x + 9 = 6x + 21 |

Then Diophantus gives the following rule:

"... it will be necessary to subtract like from like on both sides, until one term is found equal to one term."

What he meant was that one must subtract 6x from 8x and 9 from 21, so that there is only one term on each side. Thus:

8x - 6x = 21 - 9

2x = 12

x = 6.

Diophantus also had methods for solving simultaneous and quadratic equations. He did not recognise negative numbers, so he had to distinguish three cases for quadratics:

1. ax + bx = c

2. ax = bx + c

3. ax + c = bx (a, b, c > 0).

Each of these cases had its own method of solution, which correspond to the following expressions:

1. x =

2. x =

3. x =

The fourth possible case ax + bx + c = 0 (a, ,b, c > 0) does not occur in Diaphantus' work, since it never admits a positive solution.

Although the Greeks before Diophantus hardly touched algebraic problems, the work of Archimedes (287 - 212 B.C.) for example shows that they did have methods for solving various types of equation, but the greatest advances were made by Diophantus. His name is today commemorated in the term "Diophantine equation" which is an equation for which only positive integer solutions are required.

e.g. x = 2 + y, as a Diophantine equation, only has the solution x = 3, y = 5.

The Hindus and Arabs

Egypt and Babylonia

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