POLYNOMIALS AND ALGEBRAIC FRACTIONS 2

Today’s algebraic language is simple and easy to use. However, it took a long time to reach this point. We can divide its history into three main phases.

• rhetorical algebra It was written in everyday language. It was used in Babylon, Egypt and Greece. It was then used by Arabic mathematics in the 9th century.

• syncopated algebra Diophantus (3rd c.) pioneered this type of algebra. He made the processes easier by using abbreviations. During the Renaissance (15th and 16th centuries), it was improved through the incorporation of new symbols: operations, coefficients, powers, etc.

• symbolic algebra This is the predecessor of the algebraic language we know today and consists of a full set of symbols. It was created by Viète in the 16th century. It was then perfected by Descartes in the 17th century.

The lack of a functional form of algebra meant that mathematicians had to come up with ingenious ways to find or demonstrate algebraic relationships. In many cultures they used geometric shapes, giving rise to geometric algebra.

François Viète (1540-1603). French mathematician who published In Artem Analyticam Isagoge (1591). In the book, he introduced the use of letters in mathematical formulas as common practice.

Use what you have learned to solve problems

Diophantus’ intriguing system of notation

Let’s look at an example of how Diophantus described polynomials notation → ss7

2

5 M

4 u6 s means ‘squared’ (therefore, ss means ‘fourth power’); c, ‘cubed’; x, ‘the unknown’; u, ‘number’. M means ‘minus’ and precedes all monomials with negative coefficients. Looks odd, doesn’t it?

Using geometric algebra

With our modern form of algebraic notation, it is very easy to prove that:

But rhetorical algebra had no way of performing this operation, so mathematicians had to find another way to prove the expression.

Look how Pythagoras did it:

If a and b are segments, ab is the area of a rectangle.

1 Use modern algebraic notation to express the following polynomial, given using Diophantus’ notation:

2 Express the following polynomial using Diophantus’ notation:

3 Use Pythagoras’ reasoning to demonstrate the expression above, taking