Algebra History (An Overview)

Source: Mathematics History Topics - John K. Baumgart

Strange and intriguing is the origin of the word "algebra". It is not subject to a clear etymology such as the word "arithmetic", which derives from the Greek arithmos ("number"). Algebra is a Latin variant of the Arabic word al-jabr (sometimes transliterated al-jebr), used in the title of a book, Hisab al-jabr w'al-muqabalah, written in Baghdad around the year 825 by the Arabic mathematician Mohammed ibn-Musa al Khowarizmi (Mohammed son of Moses of Khowarizm). This work of algebra is often referred to in abbreviation as Al-jabr.

A literal translation of the complete title of the book is "science of restoration (or reunion) and reduction", but mathematically it would be better to "science of transposition and cancellation" - or, according to Boher, "the transposition of subtracted terms to the other member of the book. equation "and" the cancellation of similar (equal) terms in opposite members of the equation ". Thus, given the equation:

x2 + 5x + 4 = 4 - 2x + 5x3
al-jabr provides
x2 + 7x + 4 = 4 + 5x3
and al-muqabalah provides
x2 + 7x = 5x3
Perhaps the best translation was simply "the science of equations".
Although originally "algebra" refers to equations, the word today has a much broader meaning, and a satisfactory definition requires a two-phase approach:
(1) Ancient (elementary) algebra is the study of equations and methods of solving them.
(2) Modern (abstract) algebra is the study of mathematical structures such as groups, rings, and bodies - to name but a few.
Indeed, it is convenient to trace the development of algebra in terms of these two phases, since the division is both chronological and conceptual.

Algebraic equations and notation

The ancient (elementary) phase, which covers the period from 1700 BC to about 1700 AD, was characterized by the gradual invention of symbolism and the resolution of equations (generally numerical coefficients) by various methods, showing little progress until the resolution. "general" of cubic and quartic equations and the inspired treatment of polynomial equations in general by François Viète, also known as Vieta (1540-1603).

The development of algebraic notation has evolved over three stages: the rhetoric (or verbal), the syncopated (in which word abbreviations were used) and the symbolic. In the last stage, the notation underwent several modifications and changes until it became reasonably stable at the time of Isaac Newton. It is interesting to note that even today, there is no complete uniformity in the use of symbols. For example, Americans write "3.1416" as an approximation of Pi, and many Europeans write "3.1416". In some European countries, the symbol "÷" means "minus". Since algebra probably originated in Babylon, it seems appropriate to illustrate the rhetorical style with an example from that region. The following problem shows the relative degree of sophistication of Babylonian algebra. It is a typical example of problems encountered in cuneiform writing, on clay tablets dating back to the time of King Hammurabi. The explanation, of course, is made in Portuguese; and Indo-Arabic decimal notation is used instead of cuneiform sexagesimal notation. The right column provides the corresponding passages in modern notation. Here's the example:

1 Length, width. I multiplied length by width, thus obtaining the area: 252. I added length and width: 32. One asks: length and width.

2 Given 32 sum; 252 area.x + y = k

xy = P}… (A)

3 Answer 18 length; 14 width.
4 Following is this method: Take half of 32 which is 16.k / 2
16 x 16 = 256(k / 2)2
256 - 252 = 4(k / 2)2 - P = t2 }… (B)
The square root of 4 is 2.
16 + 2 = 18 length.(k / 2) + t = x.
16 - 2 = 14 width(k / 2) - t = y.
5 Proof I multiplied 18 length by 14 width.

18 x 14 = 252 area

((k / 2) + t) ((k / 2) -t)

= (k2/ 4) - t2 = P = xy.

Note that in step 1 the problem is formulated, in 2 the data are presented, in 3 the answer is given, in 4 the solution method is explained. with numbers and finally at 5 the answer is tested.

The above "recipe" is used repeatedly for similar problems. It has historical significance and current interest for several reasons.

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First of all, this is not the way we would solve the system today (A). The standard procedure in current algebra school texts is to solve, say, the first equation for y (in terms of x), substitute in the second equation and then solve the resulting quadratic equation in x; that is, we would use the substitution method. The Babylonians also knew how to solve systems by substitution, but often preferred to use their parametric method. That is, using modern notation, they conceived x and y in terms of a new unknown (or parameter) t doing x = (k / 2) + t and y = (k / 2) -t.

Then the product:

xy = ((k / 2) + t) ((k / 2) - t) = (k / 2)2 - t2 = P

led them to relationship (B):

(k / 2)2 - P = t2

Second, the problem above has historical significance because the Greek (geometric) algebra of the Pythagoreans and Euclid followed the same method of solution - translated, however, in terms of line segments and areas and illustrated by geometric figures. A few centuries later, another Greek, Diophantus, also used the parametric approach in his work with "diophantine" equations. He began modern symbolism by introducing word abbreviations and avoiding the somewhat intricate style of geometric algebra.

Third, Arab mathematicians (including al-Khowarizmi) did not use the method employed in the above problem; They preferred to eliminate one of the unknowns by substitution and to express everything in terms of words and numbers.

Before leaving Babylonian algebra, let us note that they were able to solve a surprising variety of equations, including certain special types of cubics and quartics - all with numerical coefficients, of course.

Algebra in Egypt

Algebra arose in Egypt at about the same time as in Babylon; but Egyptian algebra lacked the sophisticated methods of Babylonian algebra, as well as the variety of equations solved by Papyrus Moscow and Papyrus Rhind - Egyptian documents dating from about 1850 BC and 1650 BC respectively, but reflecting mathematical methods of an earlier period. For linear equations, the Egyptians used a resolution method consisting of an initial estimate followed by a final correction - a method to which Europeans later named it both an abstruse "false position rule". The algebra of Egypt, like that of Babylon, was rhetorical.

The relatively primitive Egyptian numbering system compared to that of the Babylonians helps explain the lack of sophistication of Egyptian algebra. Sixteenth-century European mathematicians had to extend the Indo-Arabic notion of number before they could significantly advance beyond the Babylonian results of solving equations.

Greek geometric algebra

Greek algebra as formulated by the Pythagoreans and Euclid was geometric. For example, what we write as:

(a + b)2 = a2 + 2ab + b2

was conceived by the Greeks in terms of the diagram presented in Figure 1 and was curiously stated by Euclid in Elements, book II, proposition 4:

If a straight line is divided into any two parts, the square over the entire line is equal to the squares over the two parts, along with twice the rectangle that the parts contain. This is, (a + b)2 = a2 + 2ab + b2.

We are tempted to say that for the Greeks of Euclid's time the2 It was really a square.

There is no doubt that the Pythagoreans knew Babylonian algebra well and indeed followed standard Babylonian equation-solving methods. Euclid has recorded these Pythagorean results. To illustrate it, we chose the theorem corresponding to the Babylonian problem considered above.

From book VI of Elements, we have proposition 28 (a simplified version):

Given a straight line AB that is, x + y = k, build along this line a rectangle with a given area xy = P, assuming the rectangle "falls short" in AB by an amount "filled" by another rectangle the square BF in Figure 2, similar to a given rectangle which here we admit to be any square.

In the solution of this requested construction (Fig. 2) Euclid's work is almost exactly parallel to the Babylonian solution of the equivalent problem. As indicated by T.L.Heath / EUCLID: II, 263 /, the steps are as follows:

Bisecte AB in M:k / 2
Build the square MBCD:(k / 2)2
Using VI, 25, construct the DEFG square with an area equal to the excess of MBCD over the given area P:t2 = (k / 2)2 - P
So of coursey = (k / 2) - t

As he often did, Euclid left the other case to the student - in this case x = (k / 2) + t, which Euclid certainly realized but did not formulate.

It is indeed remarkable that most of the Babylonian standard problems were "redone" in this way by Euclid. But why? What led the Greeks to give their algebra this awkward formulation? The answer is basic: they had conceptual difficulties with irrational fractions and numbers.

Even if Greek mathematicians were able to circumvent fractions by treating them as integer ratios, they had insurmountable difficulties with numbers such as the square root of 2, for example. We recall the Pythagoreans' "logical scandal" when they discovered that the diagonal of a unit square is incommensurable with the side (ie, dia / side is different from the ratio of two integers).

Thus, it was their strict mathematical rigor that forced them to use a set of line segments as a convenient domain of elements. For even though square root of 2 cannot be expressed in terms of integers or their ratios, it can be represented as a line segment that is precisely the diagonal of the unit square. Perhaps it is not just a joke to say that the linear continuum was literally linear.

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In passing we should mention Apollonius (c. 225 BC), who applied geometric methods to the study of conic sections. In fact, his great treatise Conical sections it contains more analytic conical geometry - all phrased in geometric terminology - than today's university courses.

Greek mathematics came to a sudden halt. The Roman occupation had begun, and did not encourage mathematical scholarship, even if it stimulated some other branches of Greek culture. Due to the heavy style of geometric algebra, it could not survive only in written tradition; I needed a lively, oral medium. It was possible to follow the flow of ideas as long as an instructor pointed to diagrams and explained; but direct schools did not survive.

Algebra in Europe

The algebra that entered Europe (via Liber abaci de Fibonacci and translations) had regressed in both style and content. Diophantus and Brahmagupta's semi-symbolism (syncopation) and their relatively advanced achievements were not intended to contribute to an eventual eruption of algebra.

The renaissance and rapid flowering of algebra in Europe was due to the following factors:

  1. ease of manipulating numerical works through the Indo-Arabic numbering system, far superior to systems (such as Roman) that required the use of abacus;

  2. invention of the movable type press, which accelerated the standardization of symbolism by improving communications, based on widespread distribution;

  3. resurgence of the economy, sustaining intellectual activity; and the resumption of commerce and travel, facilitating the exchange of ideas as well as goods.

Commercially strong cities first emerged in Italy, and it was there that the algebraic renaissance in Europe effectively began.

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