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PART 3
THINKING
MATHEMATICALLY
3.1 MATHEMATICS AS AN
AXIOMATIC-DEDUCTIVE
SYSTEM
ALGEBRA: The Language of Mathematics
• Algebra may be described as a
generalization and extension of arithmetic.
• Arithmetic is concerned primarily with the
effect of certain operations, such as addition
or multiplication, on specified numbers.
• Arithmetic becomes algebra when general
rules are stated regarding these operations
(like the commutative law for addition).
• Algebra began in ancient Egypt and
Babylon, where people learned to solve
linear equations (ax = b) and quadratic
equations (ax2 + bx + c = 0), as well as
indeterminate equations such as x2 + y2 = z2,
whereby several unknowns are involved.
• The Alexandrian mathematicians Hero of
Alexandria and Diophantus (c. AD 250)
continued the traditions of Egypt and
Babylon, but it was Diophantus’ work
Arithmetica that was regarded as the earliest
treatise on algebra.
Diophantus’ work was devoted mainly to
problems in the solutions of equations,
including difficult indeterminate
equations. Diophantus invented a
suitable notation and gave rules for
generating powers of a number and for
the multiplication and division of simple
quantities. Of great significance is his
statement of the laws governing the use
of the minus sign, which did not,
however, imply any idea of negative
quantities.
• During the 6th century, the ideas of
Diophantus were improved on by Hindu
mathematicians. The knowledge of
solutions of equations was regarded by the
Arabs as “the science of restoration and
balancing” (the Arabic word for restoration,
al-jabru, is the root word for algebra).
• In the 9th century, the Arab mathematician
al-Khwarizmi wrote one of the first Arabic
algebras, a systematic expose of the basic
theory of equations.
• By the end of the 9th century, the Egyptian
mathematician Abu Kamil (850-930) had
stated and proved the basic laws and
identities of algebra and solved many
complicated equations such as x + y + z =
10, x2 + y2 = z2, and xz = y2.
• During ancient times, algebraic expressions
were written using only occasional
abbreviations.
• In the medieval times (A.D. 476 to 1453),
Islamic mathematicians were able to deal
with arbitrarily high powers of the unknown
x, and work out the basic algebra of
polynomials (without yet using modern
symbolism). This included the ability to
multiply, divide, and find square roots of
polynomials as well as a knowledge of the
binomial theorem.
• The Persian Omar Khayyam showed how to
express roots of cubic equations by line
segments obtained by intersecting conic
sections, but he could not find a formula for
the roots.
• In the 13th century appeared the writings of the
great Italian mathematician Leonardo
Fibonacci (1170-1230), among whose
achievements was a close approximation to the
solution of the cubic equations x3 + bx2 + cx +
d = 0.
• Early in the 16th century, the Italian
mathematicians Scipione del Ferro (14651526), Niccolo Tartaglia (1500-57), and
Gerolamo Cardano (1501-76) solved the
general cubic equations in terms of the
constants appearing in the equation.
• Cardano's pupil, Ludovico Ferrari (152265), soon found an exact solution to
equations of the fourth degree.
• As a result, mathematicians for the next
several centuries tried to find a formula for
the roots of equations of degree 5 or higher.
• The development of symbolic algebra by
the use of general symbols to denote
numbers is due to 16th century French
mathematician Francois Viete, a usage that
led to the idea of algebra as a generalized
arithmetic. Sir Isaac Newton gave it the
name Universal Arithmetic in 1707.
• The main step in the modern development
of algebra was the evolution of a correct
understanding of negative quantities,
contributed
in
1629
by
French
mathematician, Albert Girard.
• His results though were later overshadowed by
that of his contemporary, Rene Descartes, whose
work is regarded as the starting point of modern
algebra.
• Descartes’ most significant contribution to
mathematics, however, was the discovery of
analytic geometry, which reduces the solution of
geometric problems to the solution of algebraic
ones.
• His work also contained the essentials of a course
on the theory of equations which includes
counting the “true” (positive) and “false”
(negative) roots of an equation.
Efforts continued through the 18th century
on the theory of equations. Then German
mathematician Carl Friedrich Gauss in
1799 gave the first proof (in his doctoral
thesis) of the Fundamental Theorem of
Algebra which states that every
polynomial equation of degree n with
complex coefficients has n roots in the
complex numbers.
• But by the time of Gauss, algebra had entered its
modern phase. Early in the 19th century, the
Norwegian Niels Abel and the French Evariste
Galois (1811-32) proved that no formula exists for
finding the roots of equations of degree 5 or
higher.
• But some quintic and higher degree equations are
found to be solvable by radicals, and the
conditions under which a polynomial equation is
solvable by radicals were first discovered by
Galois. In order to do this he had to introduce the
concept of a group.
• Thus attention shifted from solving polynomial
equations to studying the structure of abstract
mathematical systems (such as groups) whose axioms
were based on the behavior of mathematical objects,
such as the complex numbers.
• Modern algebra is concerned with the formulation and
properties of quite general abstract systems of this
type.
• Groups became one of the chief unifying concepts of
19th century mathematics. Important contributions to
their study were made by French mathematicians
Galois and Augustin Cauchy, the British
mathematician Arthur Cayley, and the Norwegian
mathematicians Niels Abel and Sophus Lie.
• Gradually, other sets of mathematical objects with
certain operations were recognized to have similar
properties, and it became of interest to study the
algebraic structure of such systems, independently
of the type of the underlying mathematical objects.
• The widespread influence of this abstract approach
led George Boole to write The Laws of Thought
(1854), an algebraic treatment of basic logic.
• Since that time, modern algebra – also called
abstract algebra – has continued to develop. The
subject has found applications in all branches of
mathematics and in many of the sciences as well.
Thus we see two main phases in the
development of algebra
• Classical Algebra - concerned mainly with the
solutions of equations using symbols instead of
specific numbers, and arithmetic operations to
establish procedures for manipulating these symbols
• Modern Algebra - arose from classical algebra by
increasing its attention to abstract mathematical
structures. Mathematicians consider modern algebra
a set of objects with rules for connecting or relating
them. As such, in its most general form, algebra may
fairly be described as the language of mathematics.
Properties of groups
• The distinguishing properties which makes a set G
with a given binary operation * a group are:
that for any two elements a and b of G, the element
a*b is also in G;
* is associative, that is, a*(b*c) = (a*b)*c for all a,b,c
in G;
there is an element e in G such that e* x = x*e = x for
all x in G. The element e is called the identity element
for *;
to each element a in G, there exists an element b in G
such that a*b = b*a = e. The element b is called the
inverse element of a in *.
• A group is called abelian if * is commutative, that is,
a*b = b*a for all elements a and b in G.
• Among the elementary properties of a group are the
following:
• Left and right cancellation laws hold G, that is,
a*b = a*c implies b=c, and b*a = c*a implies
b=c.
• The identity element and the inverse of an
element are unique.
• The linear equations a * x = b and y * a = b have
unique solutions in a group.
• These properties apply to any set possessing
a group structure such as:
the real numbers under addition;
the nonzero complex numbers under
multiplication
the integers modulo m
the permutations on the set {1,2,3}; and
the complex roots (called fourth-roots of
unity) of the equation z4 = 1 under
multiplication.
• Further properties of such a system could
then be derived algebraically from those
assumed (called the axioms) or those
already proved, without referring to the
types of object the members of G actually
were. This was effectively proving a fact
about any set G which had the
distinguishing properties, thus producing
many theorems for one proof.
Groups with the same structure
• It is possible that two groups (with the same
cardinality as sets) may be structurally alike, that
is, although they may be different sets with
different binary operations defined on them, these
operations combine or manipulate the elements in
exactly the same manner. If this happens, we say
that the two groups are isomorphic.
• Two groups G and H with binary operations * and
o, respectively, are said to be isomorphic if there
exists a one-to-one correspondence f from G onto
H, such that f(a * b) = f(a) o f(b). We call f an
isomorphism, a structure-preserving map.
• Isomorphic
groups
possess
common
properties which are preserved by the
isomorphism f. Thus, they are seen to be
essentially the same groups.
• Abstract algebra has been concerned with the
study of the distinguishing properties of
isomorphic groups, which eventually leads to
the classification of groups.
• Examples of isomorphic groups are the set of
integers and the set of even integers both
under integer addition; and the group of
integers modulo 4 and the fourth-roots of
unity under multiplication.
• The group of integers modulo 4 is also
isomorphic to the group defined by the four
military commands A (Attention), LF (Left
Face), RF (Right Face) and AF (About
Face). The binary operation is defined in the
following table.
A LF RF AF
A
A LF RF AF
LF LF AF A
RF
RF RF A
AF LF
AF AF RF LF AF
Other algebraic structures
• Other algebraic structures which proved
rewarding were given names such as ring,
field, semi-group, and module. Rings are
systems with two binary operations (instead
of one) defined on them. However, the
algebraic structure becomes more restrictive
because we deal with more operations and
more axioms.
• A ring is a set R together with two binary
operations + and * defined on R such that:
• R is an abelian group under +
• * is associative
• for all a, b and c in R, a * (b + c) = a*b + a* c, and
(a + b)*c = a*c + b*c.
• A ring is said to be commutative if * is
commutative. A ring with a multiplicative identity
is called a ring with unity. A field is a
commutative ring with unity in which every
nonzero element has a multiplicative inverse in R.
Examples of fields are the field of real numbers,
the field of complex numbers and the integers
modulo p, p a prime.