Algebra
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Algebra,
branch of mathematics in which
arithmetical operations and formal manipulations are applied to abstract
symbols rather than specific numbers. The notion that there exists such a
distinct subdiscipline of mathematics, as well as the term algebra to
denote it, resulted from a slow historical development. This article presents
that history, tracing the evolution over time of the concept of the equation,
number systems, symbols for conveying and manipulating mathematical statements,
and the modern abstract structural view of algebra.
Algebra can
essentially be considered as doing computations similar to those of arithmetic but
with non-numerical mathematical objects. However, until the 19th century,
algebra consisted essentially of the theory of equations. For example, the fundamental
theorem of algebra belongs to the theory of equations and is
not, nowadays, considered as belonging to algebra (in fact, every proof must
use the completeness of the real numbers, which
is not an algebraic property).
The word
"algebra" is derived from the Arabic word الجبر al-jabr,
and this comes from the treatise written in the year 830 by the medieval
Persian mathematician, Muhammad ibn Mūsā
al-Khwārizmī, whose Arabic
title, Kitāb al-muḫtaṣar fī ḥisāb al-ğabr wa-l-muqābala,
can be translated as The Compendious Book on Calculation by Completion
and Balancing. The treatise provided for the systematic solution of linear and quadratic equations. According to one history, "It is not certain
just what the terms al-jabr and muqabalah mean,
but the usual interpretation is similar to that implied in the previous
translation. The word 'al-jabr' presumably meant something like 'restoration'
or 'completion' and seems to refer to the transposition of subtracted terms to
the other side of an equation; the word 'muqabalah' is said to refer to 'reduction'
or 'balancing'—that is, the cancellation of like terms on opposite sides of the
equation. Arabic influence in Spain long after the time of al-Khwarizmi is
found in Don Quixote, where the word 'algebrista' is
used for a bone-setter, that is, a 'restorer'." The term
is used by al-Khwarizmi to describe the operations that he introduced, "reduction" and "balancing", referring to the
transposition of subtracted terms to the other side of an equation, that is,
the cancellation of like terms on opposite sides of the equation.
1. Algebraic Expression
Algebra
did not always make use of the symbolism that is now ubiquitous in mathematics;
instead, it went through three distinct stages. The stages in the development
of symbolic algebra are approximately as follows:
a. Rhetorical algebra, in which equations
are written in full sentences. For example, the rhetorical form of x +
1 = 2 is "The thing plus one equals two" or possibly "The thing
plus 1 equals 2". Rhetorical algebra was first developed by the ancient Babylonians and
remained dominant up to the 16th century.
b. Syncopated algebra, in which some
symbolism is used, but which does not contain all of the characteristics of
symbolic algebra. For instance, there may be a restriction that subtraction may
be used only once within one side of an equation, which is not the case with
symbolic algebra. Syncopated algebraic expression first appeared in Diophantus' Arithmetica (3rd
century AD), followed by Brahmagupta's Brahma Sphuta Siddhanta (7th
century).
c. Symbolic algebra, in which full
symbolism is used. 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—see below)
and showed that the problems occurring in geometry can be expressed and solved
in terms of algebra (Cartesian geometry).
Equally
important as the use or lack of symbolism in algebra was the degree of the
equations that were addressed. Quadratic equations played an important
role in early algebra; and throughout most of history, until the early modern
period, all quadratic equations were classified as belonging to one of three
categories.
a. x² + px = q
b. x² = px + q
c. x² + q = px
where
p and q are positive. This trichotomy comes about because quadratic equations
of the form x² + px + q = 0, with p and q positive, have no positive
roots. In between the rhetorical and syncopated stages of symbolic algebra, a
geometrc constructive algebra was developed by classical Greek and Vedic Indian
mathematicians in which algebraic equations were solved through geometry. For
instance, an equation of the form x² = A was solved by finding the side of area
A.
2. Conseptual Stages
In addition to the three stages of
expressing algebraic ideas, some authors recognized four conceptual stages in
the development of algebra that occurred alongside the changes in expression.
These four stages were as follows:
·
Geometric stage, where the concepts of algebra are
largely geometric. This dates back to the Babylonians and
continued with the Greeks, and was later revived by Omar Khayyám.
·
Static equation-solving stage, where the objective
is to find numbers satisfying certain relationships. The move away from
geometric algebra dates back to Diophantus and Brahmagupta, but algebra didn't
decisively move to the static equation-solving stage until Al-Khwarizmi introduced
generalized algorithmic processes for solving algebraic problems.
·
Dynamic function stage, where motion is an underlying idea.
The idea of a function began
emerging with Sharaf al-Dīn
al-Tūsī, but algebra did not decisively move to the dynamic function
stage until Gottfried Leibniz.
·
Abstract stage, where mathematical structure plays a
central role. Abstract algebra is
largely a product of the 19th and 20th centuries.
3. The Influence of
Arabic Mathematicians
In the middle east, there were many
developments from mathematicians who have had a huge impact on how we see and
use algebra today. Muhammad ibn Musa al-Khwarizmi is the most prominent and
most important of the arabic mathematicians and is is known as the father of
algebra to this day. The world ‘algorithm' is taken from the Latin version of
his name, which shows just how important his influence on mathematics
was. The Compendious Book on Calculation by Completion and
Balancing is the book that marked his major contribution to algebra.
This book outlined how polynomial
equations should be solved up to the second degree. The transposition of terms
to the other side of an equation was also discussed at length in the book. As
anyone who has studied algebra will know that this is a vital concept, and it
all goes back to al-Khwarizmi. His method of solving quadratic and linear
equations worked by reducing the equation to a simpler form. It's something
that is still done by all students of algebra.
Muhammad ibn Musa al-Khwarizmi wasn't
the only Arab mathematician to do important work relating to algebra at this
time. The other key figure was Omar Khayyám. He wrote books and included
information on how to solve equations to the third degree. This took his work
further than any of the other Arabic mathematicians that had come before him.
His method of solving cubic equations is very important and is used today.
Written by:
Fitria Rahmawati (4218007)
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