3. 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).
4. • 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.
5. In the set of
natural numbers,
ax = b
does not always
have a solution.
( )
a
b
x
a
b
x
a
b
xa
a
b
a
ax
a
bax
=
=⋅
=
⋅
⋅=
=
1
1
11
6. ax2
+ bx + c = 0, a ≠ 0
The graph of
is a parabola.
The parabola opens
a. upward if a > 0.
b. downward if a < 0.
8. 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.
9. 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.
10. • 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.
11. • 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
.
12. • 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.
13. • 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 cubic equations of the form
x3
+ bx2
+ cx + d = 0.
14. • 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.
15. • 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.
16. • 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.
17. 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 set of
complex numbers.
18. • 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.
19. • 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.
20. • 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.
21. 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.
22. Properties of groups
1. for any two elements a and b of G, the element
a•b
is also in G;
2. • is associative, that is,
a •(b • c) = (a • b) • c for all a,b,c in G;
3. 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,
4. 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 •.
23. • A group G is called abelian if the operation • in G is
commutative, that is, a • b = b • a for all elements a
and b in G.
• Among the properties of a group are the following:
Left and right cancellation laws hold, 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.
24. • These properties apply to any set possessing a
group structure such as:
- the set of real numbers under addition;
- the set of non-zero complex numbers under
multiplication
- the set of + integers under addition 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.
25. Addition modulo m
a + b = r mod m,
if r is the remainder when
we divide the ordinary
sum of a and b by m.
Illustrations:
1. What is 6 + 5 modulo 4?
Ans. 3
6 + 5 = 3 mod 4
26. 2. What is 11 + 7 modulo 12?
Ans. 6
11 + 7 = 6 mod 12
3. What is 24 + 9 modulo 16?
Ans. 1
24 + 9 = 1 mod 16
30. The complex roots (called fourth-roots of
unity) of the equation z4
= 1 under
multiplication
1 i -1 -i
1
i
-1
-i
1 -1 -i
i -i
i
1
-i
i
1 -1
e = 1
1-1
= 1
(-1)-1
= -1
(i)-1
= - i
(-i)-1
= i
-1 1
-1 -i i
31. • 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.
32. 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.
33. • 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.
34. 0 1 2 3
0
1
2
3
S= {0, 1, 2, 3} under
addition mod 4
0 1 2 3
1 2 3 0
2 3 0 1
3 0 1 2
S= {1,i,-1,-i} under
multiplication
1 i -1 -i
1
i
-1
-i
1 1 -1 -i
i -1 -i 1
-1 -i 1 i
-i 1 i -1
35. • 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 RF AF LF
A A RF AF LF
RF RF AF LF A
AF AF LF A RF
LF LF A RF AF