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

1.

2.

3.

 

4.

5.