The Development Of Algebra

Knowledge about the possible beginnings of human mental development comes from research on the co-evolution of language and the human brain. The central idea that comes out of this research is that mankind, uniquely, is what is called a `symbolic species`. That is, we have the ability to make ideas `visible in the mind`. This means we can create, manage, manipulate and blend images in ways that help us to form new relationships and produce new concepts.

We can only speculate on the earliest stages in the development of mathematics but we do have evidence which shows that we have the power to visualise and represent physical objects in our mind. This power is seen in the representations of counting, bartering and building activities of early civilisations. In particular, our perceptions of symmetry seem to have produced many kinds of geometrical representations in different cultures.

2. The Pythagoreans (c. 550 - c. 450 BCE) Imagine, for example, what might be done with arrangements of stones in the sand:

6x6 dots 6x6 dpts Rectangles 6x6 dots triangles In these images, a square pattern has been divided into different areas, making rectangles or triangles.

6x6 dots squares and rectangles 6x6 dots gnomon

In the above two images, other shapes have been produced, leading to speculations about relationships between numbers and areas, and it is thought that the elementary number theories of the Pythagoreans might have been generated by images like these [see Note 1 below].

Further manipulation, dissection of squares and rearrangement, leads to images of right-angled triangles and the familiar relationship between numbers and areas. For example:

Pythagoras on 6x6 dots Pythagoras 2 on 6x6 dots 3. Egypt: The Rhind Papyrus. Early methods for solving problems come from the Rhind Mathematical Papyrus, written in Egypt about 1,550 years BCE. It is a `problem text` of exercises for training scribes, who were the administrators. It has examples of methods for weighing, measuring and surveying, for finding areas and volumes, and for working out rates of pay for workers of various kinds. Part of the Rhind Papyrus In particular, the Papyrus has a series of problems showing how to solve what we could call `linear equations` by a method that became commonly used by merchants throughout the Mediterranean countries for some three thousand years, called the `Method of False Position`. It is used for all kinds of calculations involving comparison of values and quantities. This method still appeared in school text books in the early 20th century. Here is Problem 26: This belongs to the group of `h` problems, sometimes written as `heap` meaning `quantity` or `number`. 4¯ is an example of the usual way we represent Egyptian `unit` fractions. In a quantity and its 4¯ (is added) to it so that 15 results A quantity is added to a quarter of itself and the answer is 15 Calculate with 4 You shall calculate its 4¯ as 1. Total 5. You begin by assuming h=4, so, as the left hand side is h+h4 this gives 4+1=5: h=4 does not work as it gives an answer of 5 not 15.

Work with 5 to find 15. \ . 5 \ 2 10 3 shall result. Now take the 5 and divide it into 15 to get 3. Multiply 3 times 4 . 3 2 6 \ 4 12 12 shall result Multiply the 4 by 3 to find the result you want, 12. . 12 4¯ 3 total 15 The quantity 12 its 4¯ 3. Total 15 So, 12 add a quarter of 12 is 15. In modern notation, the problem looks very easy: h+h4=15

We begin by using h=4 which immediately makes the fraction h4=1 So the left hand side is 5, but we need 15. If by using 4 we obtain 5, what do we need to use to get 15?

Falso position triangle The sundial was one of the first instruments used for measuring time, and the properties of the right-angled triangle were well known to ancient people (Problem 56 deals with similar triangles). The slopes of pyramids and other sloping surfaces were measured by the `seked` - the horizontal distance measured for every cubit of height. Visualising a triangle like this would make the problem much simpler and the ratios could then easily be compared. `h`15=45

So, for the denominator of the right hand ratio to be 15, we need to multiply the numerator and denominator of the ratio 45 by 3.

4. Early Indian Mathematics (c. 1500 - 500 BCE) Similar traces of the above kind of visualisation appeared in the Indus valley in roughly the same period, although there are claims that the procedures were much older [see Note 2 below].

The Vedic people entered India about 1500 BCE. The name comes from their sacred rituals called the Vedas. These date from about the 15th to the 5th century BCE and were used for sacrificial rites which took place at an altar.

The name Sulbasutras (meaning `cord-rules`) applies to the part of the Vedas which gives the rules for building the brick altars used in ritual sacrifice. Instructions were given:

"He who desires heaven is to construct a fire-altar in the form of a falcon" and "a fire-altar in the form of a tortoise is to be constructed by one desiring to win the world of Brahmin" [see Note 3 below]. Many of these instructions contain transformations for preserving areas, such as changing a rectangle into a square of the same size. See examples below:

Two squares combined area For two squares of the same size, it is easy to see how to combine the areas to make one square, but not obvious for squares of different sizes. Two unequal squares 2

"If it is desired to combine two squares of different measures, a [rectangular] part is cut off from the larger [square] with the side of the smaller; the diagonal of the cut-off [rectangular] part is the side of the combined square."

Two unequal squares 1

"Alternatively if it is desired to combine two squares of different measures, a rectangle is formed with the side of the smaller [square] [as breadth] and that of the larger [as length]; the diagonal of the rectangle [thus formed] is the side of the combined square."

These instructions clearly show a knowledge of the `Pythagorean relation`.

The base line for the altar was always East - West and careful and precise interpretation of the instructions enabled the priests to build the falcon `fire-altar`.

Kite Altar The Sulbasutras were written in Sanscrit and the texts often used `code words` or abbreviations for lengths, areas and other quantities. For example, the word `eyes` represented number two and `teeth` represented `thirty-two`, so a sentence might read something like: `we have the eyes, so do this to find the teeth`, and the geometry was hidden under a cloak of instructions coded in number language.

5. Old Babylonian Mathematics (c. 1850 - 1600 BCE) The earliest mathematics appeared in Mesopotamia some 3,500 or more years BCE with a variety of specific number systems for trading with different things like grain and cereals, milk and dairy produce, or things made out of clay or wood. The `sexagesimal system` as it is popularly known, appeared before 2000 BCE and was well-established by the time of Hammurabi, (1795-1750 BCE) the first king of Babylon. Most of the well-known evidence we have on clay tablets comes from the Babylonian Kingdom period that ended about 1600 BCE.

ration tablet 2ration tablet 1

Accounts for cereals, beer or oil - about 3,000 BCE

The `Old Babylonian` Empire had a well-organised society with an efficient administration which enabled scribes to administer social justice through a system of fair management and distribution of goods. Central to this system were the schools where scribes were trained. The school culture was basically oral, depending on the memorisation of texts and routines for calculation; tables for multiplication, division, reciprocals and other useful units were available. Emphasis on administration, labour management and quantity surveying led to the use of `word problems` that could be solved using standard algorithms such as: `Cut and Paste` Geometry: Removing a Square from a larger Square. Removing a square

The claim that the Babylonians were doing `arithmetical algebra` (or even quadratic equations) is now seen as an error. Recent research into the language has tried to recover the original thought processes, and shows that the underlying concepts were essentially geometrical. We don`t really need the algebraic labels here, because the geometrical procedure is quite general. The size of the smaller red square does not matter, and the numerical instructions hid the underlying geometrical visualisation [see Note 5 below].

Word Problems This example of a `sum and product` problem is from a Babylonian `school tablet`. The problem has been rewritten to convey the geometrical nature of the original procedure [see Note 6 below].

Length, Width. I have raised, length and width. Surface: 252 I have put together length and width: 32 32 put together, 252 surface: 18 length, 14 width The problem and answers are stated, and the method is described. This is the way of doing it: Here is the arithmetic: break off half of 32: this gives 16 322=16 raise 16 by itself: 256 162=256 leave out the surface: 256?252=4 256?252=4 find the side of this square 4: it is 2 4?=2 put together 16 and 2: 18 length 16+2=18 tear out 2 from 16: 14 width 16?2=14 I have raised length and width. 252 (16+2)(16?2)=252 The visual thinking behind the calculation appears to have gone like this: We have the area of a rectangle, 252; We have the sum of its length and width, 32 (the semi-perimeter); We know from practical experience (see diagram above); that by removing a small (red) square from a larger (blue) square, we can make a rectangle. So, if we make the side of the large square half of 32; then the area of the large square will be 256. Now take away the rectangle, (area 252), and we are left with the small (red) square.

Geometrically, this will always work whatever the lengths of the sides concerned. Arithmetically , the problems were arranged so that the numbers used in the calculation always resulted in a perfect (red) square.

Many more examples like this have been discovered, and as well as the underlying geometrical visualisation, there is also the clear emergence of an algorithmic process . The Babylonians had established one of the important essential aspects of problem-solving. They had established a method for solving particular types of area problems , and it was this method that was taken over by later generations of scholars and eventually developed into the algorithm we use in classrooms today.

6. Greek Geometry By about 350 BCE the numerical and intuitive geometry of these cultures was changed into the deductive science we know as Euclidean Geometry. Book II of the Elements, (still often labelled `geometric algebra`) contains much of the knowledge discussed above in terms of geometric theorems. The transformation of rectangles and parallelograms into other shapes of equal area is demonstrated in general terms from a deductive point of view. The myth that the Greeks could not deal with irrational magnitudes has no basis in fact. The first diagram below shows how, from rational measurements of the sides of the rectangle, it is possible to create a square with non-rational sides [see Note 7 below].

Constructing a square equivalent in area to a rectangle Constructing a square The red side of the rectangle is brought into line with the upper blue edge by drawing an arc (radius r ). The (red + blue ) line is bisected to find the centre of a circle, and a semi-circle is drawn. The side of the square required is the green vertical line. xy=z^2 In this diagram there are three similar right-angled triangles, APB in the semicircle, AXP and PXB. The coloured sides of the triangles correspond to the diagram above. The relationship shown is the equivalence of corresponding ratios. Thomas Heath, in his translation of Euclid [see Note 8 below], states:

"For want of the necessary notation the Greeks had no algebra in our sense. They were obliged to use geometry as a substitute for algebraical operations; and the result is that a large part of their geometry may appropriately be called `geometrical algebra`."

So for Heath, referring to the diagram above, and in modern notation, if AX=a, XB=b and PX=x, then ab=x2 This attitude influenced the way we interpreted much of Greek mathematics as algebra until quite recently.