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 (1465-1526), 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 (1522-65), 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.

• 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 *.

Properties of groups

• 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.

• 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.

Groups with the same structure

• 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.