Did you know? Notes from the history of maths

Did you know? Notes from the history of maths

‘A’ is for algebra and angst

Maths teachers know that when a class are told the new topic is to do with algebra, there is likely to be a collective groan. Algebra, however, has a rich and fascinating history. Who invented algebra? Why do we use the letter x?

Certainly, as far back as 1900 BCE (our Bronze Age), the Babylonians were solving problems such as:

The length of a rectangle exceeds its width by 7.

Its area is 60. Find its length and width.

In the classroom, we would solve this algebraically by giving a symbol for the unknown width (w, say). So that the length becomes w+7 and the area of the rectangle: width x length becomes w (w+7) = 60. Students would then solve this quadratic by a taught method to get the solution w = 5 (or -12, which is discounted as a possible length).

The Babylonian method was an algorithm, not involving symbols, and would look like this:

1. Halve the 7 => 3½

2. Square this => 12¼

3. Add the area => 72¼

4. Square root this => 8½

5. Length is last answer + answer to 1. => 12

Width is last answer – answer to 1. => 5

It isn’t clear, from the clay tablets that show these methods, to what extent the Babylonian mathematicians saw these as general methods and, as a result, whether they can be thought of as algebra.

The Greek Mathematician, Diophantus, in the 3rd century CE did use a symbol for the unknown. He is particularly known for his work on indeterminate equations (which have many solutions). We would write these as x2 + y2 = a2. In 1637, Fermat reading a Latin translation of “Arithmetica” by Diophantus , wrote that he had found a proof that xn + yn = zn (x,y,z,n all positive whole numbers), has no solutions for n > 2 but it was ‘too long to fit on the margin’. Mathematicians tried to prove this ever since and only succeeded when Andrew Wiles did so, 357 years later.

The word ‘algebra’ is Arabic and was first used in the book “The Compendious Book on Calculation by Completion and Balancing” written by al –Khwārizmi who lived from about 780 to 850 CE. The al-ğabr means “completion" or "restoring” by moving a negative quantity from one side of the equation to the other side. This technique is well known to students learning to manipulate algebra. In our notation,

x2 = 40x − 4x2 is transformed by al-ğabr into

5x2 = 40x.

al –Khwārizmi was showed how quadratics can be transformed into a set of basic types of equations which were then solved by algorithms similar to those use by the Babylonians. The main difference with this work is that the methods were generalised and proved. The use of letters in algebra was not used systematically until much later in the work of Francois Viète who lived from 1540 to 1603. He used uppercase vowels for “things sought” and uppercase consonants for “things given”. This was further developed by René Descartes

(1596-1650) who used letters from the end of the alphabet for numbers sought. The use of ‘x’ in algebra is possibly a matter of chance. The story goes that when Descartes’ book “La Géométrie” was being typeset, the printer began to run short of the last letters of the alphabet. He asked Descartes if it mattered whether x, y or z was used in each of the books many equations.

Descartes replied that it didn’t. The printer selected x since the letters y and z are used in French more frequently than x.

Don Hoyle
Mathematics Matters