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ALGEBRA
Adreina Shamelia bt Shamsul Anuar
Nur Mawaddah bt. Mohd Azhar
Nurul Izza Syahira bt. Abd Majid
Rowenna Renee Anak Rigen
DEFINITION
OF
ALGEBRA
• Algebra is one of the broad parts of mathematics, together withnumber theory, geometry and analysis.
• For historical reasons, the word "algebra" has several related
meanings in mathematics, as a single word or with qualifiers. As a
single word without article, "algebra" names a broad part of
mathematics. As a single word with article or in plural, "algebra" denotea specific mathematical structure.
• Algebra can essentially be considered as doing computationssimilar to that of arithmetic with non-numerical mathematicalobjects. Initially, these objects represented either numbers thatwere not yet known (unknowns) or unspecified numbers(indeterminate or parameter), allowing one to state and proveproperties that are true no matter which numbers are involved. Forexample, in the quadratic equation
• a, b,c are indeterminates and x is the unknown. Solving thisequation amounts to computing with the variables to express theunknowns in terms of the indeterminates. Then, substituting anynumbers for the indeterminates, gives the solution of a particularequation after a simple arithmetic computation.
HISTORY
OF
ALGEBRA
Algebra
Arabic word, “al-jabr” Study of structure, relation
and quantity
Restoration, Completion
“muqabalah” means
Reduction/Balancing
Classical
Algebra
Abstract
Algebra
Egyptian
Algebra
Abstract
/Modern Algebra
European
Algebra
after 1500
Arabic/Islamic
Algebra
Egyptian Algebra.
-Based on Rhind papyrus.
-Rhetorical.
-Method of false position.
-Problems were stated and solved verbally.
Babylonian Algebra.
-More advanced.
-Excellent sexagesimal.
-General procedure equivalent to solving quadraticequations.
-Quadratic formula.
-Some use symbols.
Greek Geometrical Algebra.
-Solving problems in geometric form.
-Applying deductive reasoning and describing general
procedures.
Diophantine Algebra.
-The father of Algebra.
-Introduced the syncopated style of writing equations.
-Introduced Arithmetica.
-Gives a treatment of indeterminate equations.
-Each of the 189 problems in the Arithmetica is solved by a
different method.
-Accepted only positive rational roots and ignored all others.
Hindu/Indian Algebra.
-Successors of the Greeks in the history of mathematics.
-Motivated by astronomy and astrology.
-Treated zero as a number and discussed operations involving this number.
-Introduced negative numbers to represent debts.
-Brahmagupta recognized that a positive number has two square roots.
-Developed correct procedure for operating with irrational numbers.
-Did not recognize square roots of negative numbers.
Arabic/Islamic Algebra.
-Improved the Hindu number symbols and idea of positional notation.
-Contributed first of all the name, Hisab al-jabr w-al muqabala.
-They could solve quadratic equations, recognizing two solutions, possiblyirrational, but usually rejected negative solutions.
-Rhetorical.
European Algebra after 1500.
-Zero had been accepted as a number and irrationalswere used freely.
-Renaissance mathematics was to be characterized bythe rise of algebra.
-Subsequent efforts to solve polynomial equations ofdegrees higher than four by methods.
Chinese Algebra.
-Oldest Chinese mathematical documents.
-Nine Chapters on the Mathematical Art, Sea-Mirror ofthe Circle Measurements, Magic Squares, and PreciousMirror of the Four Elements.
Abstract/Modern Algebra.
-Various sorts of mathematical objects (vectors,
matrices, transformations, etc.)
-Scope of algebra was expanded to the study of
algebraic form and structure.
-Peacock was the founder of axiomatic thinking
in arithmetic and algebra. (Euclid of Algebra)
-Gibbs developed algebra of matrices.
-Peano created an axiomatic treatment of the
natural numbers in 1889.
TOPICS
OF
ALGEBRA
The Language of Algebra
Real Numbers
Solving Linear Equations
Graphing Relations and Functions
Analyzing Linear Equations
Solving Linear Inequalities
Solving Systems of Linear Equations and Inequalities
Polynomials
Factoring
Quadratic and Exponential Functions
Radical Expressions and Triangles
Rational Expressions and Equations
Statistics
Probability
Fibonacci Numbers
When we look a Fibonacci Numbers, we can quickly see the
pattern.
1, (1+0) 1, (1+1) 2, (1+2) 3, (2+3) 5, (3+5) 8, ....
Many flower species have been found that produce petals that
follow this sequence.
For example:
Enchanter's Nightshade flowers = 2 petals
Lilies = 3 petals
Wild Geranium = 5 petals
Delphinium = 8 petals
Corn Merigold = 13 petals
Also, pineapple scales and pine cones spiral in two different
directions. The number of spirals are Fibonacci numbers.
Pineapple = 5 & 8, 8 & 13
Pine cones = 5 & 8, 8 & 13
Algebra in Nature
Finite Space
The packing industry has surely spent much time and effort
trying to find the best was to pack products into boxes for
shipment. The goal usually is to allow for the least amount of
wasted space and hold maximum capacity. All along they had
to do was turning to bee keeping.
Bees have chosen what appears to be the most efficient and
economically shaped packing container, a regular hexagonal
prism. When calculating the densities of this tessellation and
comparing it with those of a square prism or an equilateral
triangular prism, you will find the bee made the correct choice
by sticking with the regular hexagonal prism
Finite Space
APPLICATIONS
OF
ALGEBRA
Astronomy
Astronomer use math all the time. One way it is used is when we
look at objects in the sky with a telescope. The camera that is
attached to the telescope basically records a series of numbers -
those numbers might correspond to how much light different
objects in the sky are emitting, what type of light, etc. In order to
be able to understand the information that these numbers contain,
we need to use math and statistics to interpret them. Another waythat astronomers use math is when they are forming and testing
theories for the physical laws that govern the objects in the sky.
Also, in addition to flying and maneuvering a spacecraft,
astronauts are often involved in conducting scientific experiments
aboard the spacecraft, which would involve math in other waystoo.
Biology
Algebraic biology applies the algebraic methods of symbolic computation
to the study of biological problems, especially in genomics, proteomics,
analysis of molecular structures and study of genes. Computations in the
field of biology are done in order to simulate experiments and/or calculate
features of a biologic process or structure. Such as for example
calculating mathematical predictions of intercellular features, cellular
interaction, body reaction to chemicals and analysis of heritage. In recent
years, methods from algebra, algebraic geometry, and discrete
mathematics have found new and unexpected applications in systems
biology as well as in statistics, leading to the emerging new fields of
"algebraic biology" and "algebraic statistics." Furthermore, there are
emerging applications of algebraic statistics to problems in biology. This
year-long program will provide a focus for the further development and
maturation of these two areas of research as well as their
interconnections. The unifying theme is provided by the common
mathematical tool set as well as the increasingly close interaction between
biology and statistics.
Geometry
Algebraic geometry is a branch of mathematics which, as the name
suggests, combines techniques of abstract algebra, especially
commutative algebra, with the language and the problems of geometry.
Initially a study of polynomial equations in many variables, the subject of
algebraic geometry starts where equation solving leaves off, and it
becomes at least as important to understand the totality of solutions of a
system of equations, as to find some solution; this leads into some of the
deepest waters in the whole of mathematics, both conceptually and in
terms of technique.
The fundamental objects of study in algebraic geometry are algebraic
varieties, geometric manifestations of solutions of systems of polynomial
equations. Plane algebraic curves, which include lines, circles, parabolas,
lemniscates, and Cassini ovals, form one of the best studied classes of
algebraic varieties. A point of the plane belongs to an algebraic curve if its
coordinates satisfy a given polynomial equation. Basic questions involve
relative position of different curves and relations between the curves given
by different equations.
Cryptology
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