DOCUMEN The History of Algebra

HISTORY OF ALGEBRA

PENDIDIKAN MATEMATIKA

UNIVERSITAS NEGERI JAKARTA

“ALGEBRA”

The word “algebra” is derived from the Arabic “ al-jabr ”* that can be

translated as “restoration” or “completion” and refers to the operation of

“transposing” a subtracted quantity on one side of an equation to the other side where it becomes an added quantity.

*The word comes from Al-Kitab al-mukhtasar fi hisab al-jabr wa'l-muqabala, written in 830 by Al-Khwārizmī. The word 'muqabalah' is said to refer to 'reduction' or 'balancing'—that is, the cancellation of like terms on opposite sides of the equation.

https://www.youtube.com/watch?v=Ok5BGimzUkg

3 STAGES OF THE PHYLOGENETIC DEVELOPMENT OF ALGEBRA:

 The Rhetorical algebra

(early traces of algebraic thinking in ancient Babylonia, ca 2000BCE )

 The Syncopated algebra

^{(250CE – End1600)}

 The Symbolic algebra

(first introduced in France in the 16^th century)

RHETORICAL ALGEBRA

Algebraic expressions or problems were written (expressed) in full sentences. Rhetorical algebra was first developed by the ancient Babylonians and remained dominant up to the 16th century.

For example:

 The rhetorical form of x + 1 = 2 is “the thing plus one equals two“, or

possibly “the thing plus 1 equals 2”

BABYLONIAN ALGEBRA

Babylonians used a geometric form of algebra that involved a geometric

unknown and reasoning to find its value. The unknown was a line, and

they performed geometrical operations on this unknown line to get the

answer.

QUADRATIC EQUATIONS IN BABYLONIAN ALGEBRA

𝒙

^𝟐

+ 𝟔𝒙 − 𝟏𝟔 = 𝟎

𝒙 𝟐 + 𝟔𝒙 − 𝟏𝟔 = 𝟎

𝒙 𝟐 + 𝟔𝒙 = 𝟏𝟔

_{𝒙𝟐 𝟔𝒙}

𝒙 𝒙

6

𝒙^𝟐 3𝒙

3𝒙 9

A SYSTEM OF TWO EQUATIONS IN BABYLONIAN ALGEBRA

“ I added the area of my two squares: 1,525. The side of the second square equals 2/3 of the side of the first and another 5 more . “

(Note: the Babylonians expressed this using base-60 numbers )

Write the problem in modern notation!

(The solution given was in the form of a step-by-step geometric procedure.)

Similar problems were also found in the Egyptians (Rhind Papyrus); the Chinese (The Nine Chapters of the Mathematical Art); early Greeks (Elements).

LINEAR EQUATIONS IN PAPYRUS

“Find the number such that if it is taken 1 Τ

¹ ₂

times and then 4 is added, the sum is 10.”



In modern notation, the equation is simply (1 Τ

¹ ₂

) x + 4 = 10.



The scribe proceeded as follows: “Calculate the excess of this 10 over 4. The result is 6. You operate on 1 1/2 to find 1. The result is 2/3. You take 2/3 of this 6. The result is 4. “



The Egyptians also had the method of “false position”, i.e., the method of

assuming a convenient but probably incorrect answer and then adjusting it by

using proportionality

EXAMPLES OF RHETORICAL ALGEBRA IN THE CHINESE ( THE NINE CHAPTERS OF THE MATHEMATICAL ART ), AND IN THE EARLY GREEKS ( ELEMENTS ).

The ‘Nine Chapters on the Mathematical Art’ :

By selling 2 cows and 5 goats to buy 13 pigs, there is a surplus of 1000 cash. The money obtained from selling 3 cows and 3 pigs is just enough to buy 9 goats. By selling 6 goats and 8 pigs to buy 5 cows, there is a deficit of 600 cash. What is the price of a cow, a goat, and a pig?”

PROPOSITION II–4 in Book II of Elements:

If a straight line is cut at random, the square on the whole is equal to the squares on the segments and twice the rectangle contained by the segments.

SYNCOPATED ALGEBRA



This phase involved the use of abbreviations for those quantities and

operations that were frequently recurring, for example letters as unknowns.

The main concern is to discover the unknowns and no attempt to express the

‘general rule’.



Syncopated algebraic expression first appeared in

Diophantus'

Arithmetica (3rd century CE), followed by

Brahmagupta's

Brahma Sphuta Siddhanta (7th

century).

 Syncopated algebra did not immediately become the standard method until much later in the sixteenth century

 https://www.youtube.com/watch?v=YC1RzcMhtBw&t=183s (Diophantus and Syncopated Notation)

DIOPHANTUS OF ALEXANDRIA (250CE)

• He wrote Arithmetica, a treatise; six among thirteen books have survived. He lived in Alexandria (Egypt), between 150 and 250 CE.

• Diophantus was the first to introduce symbols for unknown numbers abbreviations for powers of numbers, relationships, and operations as used in Syncopated algebra.

• He showed how to solve equations by using restoration and

confrontation. In modern terms, these correspond to (1) moving a

quantity from one side to the other with a change in sign, and (2)

eliminating like terms from both sides.

BRAHMAGUPTA (ca 600 CE)

The algebra of Brahmagupta, like that of Diophantus, was syncopated.

Addition was indicated by juxtaposition, subtraction by placing a dot over the subtrahend, and division by placing the divisor below the dividend, as in our fractional notation but without the bar. The

operations of multiplication and evolution (the taking of roots), as

well as unknown quantities, were represented by abbreviations of

appropriate words.

BRAHMAGUPTA (ca 600 CE)

He was the first one to give a general solution of the linear Diophantine equation ax + by = c. where a, b, and c are

integers.

For this equation to have integral solutions, the gcd (a, b)

must divide c: and Brahmagupta knew that if a and b are

relatively prime, all solutions of the equation are given by

x = p + mb, y = q – ma, where m is an arbitrary integer

AL-KHAWARIZMI

• Al'Khwarizmi was an Islamic

mathematician who wrote on Hindu- Arabic numerals.

• The word algorithm derives from his name.

• His algebra treatise Hisab al-jabr w'al-muqabala gives us the

word algebra and can be considered as the first book to be written on

algebra.

𝑥² + 10 𝑥 = 39

SOLVE A QUADRATIC EQUATION: 𝑥 2 + 𝑏𝑥 = 𝑐

Muhammad ibn Musa al-Khwarizmi (9

^th

century)

He presented six types of equations that can be written using three kinds of quantities: the square (of the unknown), the root of the square (the

unknown itself ), and the absolute number (the constant in the equation).

Katz, 2009, p.272

OMAR KHAYYAM

• He discovered a geometrical method of solving cubic equations by intersecting a parabola with a circle.

• He stated that the solution of this cubic

requires the use of conic sections and that it cannot be solved by ruler and

compass methods.

• It was the mathematician and poet ‘Umar ibn Ibrahim al-Khayyami (1048–1131), who first systematically classified and then proceeded to solve all types of cubic equations by this general method.

CHINESE PASCAL TRIANGLE

The Chinese Mathematician, Jia Xian devised a triangular representation of the coefficients of the binomial theorem in the 11th century. Later, another Chinese Mathematician Yang Hui

further studied Jui Xian's triangle, finding more

properties in it. His study and discoveries found

in the triangle further popularized the triangle. In

China they call the triangle Yanghui's Triangle.

SYMBOLIC ALGEBRA

 This stage was marked by the development of the current algebra symbols that we use nowadays.

 Early steps toward this can be seen in the work of several Islamic mathematicians such as Ibn al-Banna (13th-14th centuries) and al-Qalasadi (15th century), although fully

symbolic algebra was developed by François Viète (16th century). Later, René

Descartes (17th century) introduced the modern notation (for example, the use of x) and showed that the problems occurring in geometry can be expressed and solved in terms of algebra (Cartesian geometry).

 The use of symbols in algebra enhanced the learning of algebra mathematics greatly. It was greatly adopted in the mid-seventeenth century.

FRANÇOIS VIÈTE(1540–1603).

• He introduced the first systematic

algebraic notation in his book In artem analyticam isagoge .

• He demonstrated the value of symbols introducing letters to represent

unknowns. He suggested using letters as symbols for quantities, both known and unknown. He used vowels for the

unknowns and consonants for known

quantities.

“AN EQUATION IS NOT CHANGED BY ANTITHESIS”

Rene Descartes (17

^th

century)

 In 1637, Descartes in his Geometrie, produced a new field of mathematics, namely analytic geometry*, that incorporated algebraic methods into geometry problems.

 Descartes showed how equations are built up from their solutions. If a polynomial equation has, e.g., (x + 5) as one of the factors, then the equation will have a

negative root or stated as ‘false’ root, since x + 5 = 0 will give a negative root.

 Descartes was the one who introduced the modern algebraic notation where letters chosen from near the end of the alphabet, x, y, and z to designate unknown

quantities.

 The Cartesian coordinate system is named after him.

*Analytic geometry was born in 1637 of two fathers, Ren´e Descartes and Pierre de Fermat.

ALGEBRA IN THE 18 TH CENTURY

 In the eighteenth century, algebra meant the solving of equations. Thus, Newton, Maclaurin, and Euler each presented their own ideas on this subject

 Newton’s text began at a very elementary level, but by the end he had given a rather comprehensive course with many interesting details on the solution of algebraic equations. By the end of his text, however, Newton had also solved much more difficult problems, including problems in physics and astronomy, and had developed Descartes’

rule of signs, the relationships between the coefficients of a polynomial and its roots, and formulas to determine the sums of various integral powers of the roots of a polynomial equation.

 Maclaurin began his A Treatise of Algebra in Three Parts not only with algorithms for calculation but also with attempts to explain the reasoning behind the algorithms. For example, in dealing with negative numbers, he noted that any quantity can enter algebraic computation as either an increment or a decrement. As examples of these two forms, he included such concepts as excess and deficit, value of money due to a man and due by him, a line drawn to the right and one to the left, and elevation above horizon and depression below

Newton’s word problems

 If two couriers A and B, 59 miles apart, set out one morning to meet each other, and of these A completes 7 miles in 2 hours and B 8 miles in 3 hours, while B starts his journey 1 hour later than A: how far a distance has A still to travel before he meets B?

 If a scribe can copy out 15 sheets in 8 days, how many scribes of the same output are needed to copy 405 sheets in 9 days?

An even better introduction to algebra, perhaps, was provided by Euler’s Vollst¨andige Anleitung zur Algebra (Complete Introduction to Algebra).

Euler, like Maclaurin, began his text by providing a definition of the subject: “The foundation of all the mathematical sciences must be laid in a complete treatise on the science of numbers, and in an accurate examination of the different possible methods of calculation. This fundamental part of mathematics is called Analysis, or Algebra. In Algebra, then, we consider only numbers, which

represent quantities, without regarding the different kinds of quantity.” Later on in the text, he made the definition somewhat more specific: Algebra is “the science which teaches how to determine

unknown quantities by means of those that are known.”

FOUR CONCEPTUAL PHASES OF ALGEBRA DEVELOPMENT

Algebra development was attributed to the simultaneous development of four other conceptual phases:

(1) the geometric development that comprised most of the algebra concepts being geometrical.

(2) the static equation–solving stage. This represented the objective of numbers that satisfied certain relations.

(3) the dynamic phase concentrated on motion,

(4) the abstract phase focused on mathematical structures (Boyer, 1991).