Algebra Essay, Research Paper

ALGEBRA REPORT

Algebra is a branch of mathematics in which letters are used to represent basic arithmetic relations. As in arithmetic, the basic operations of algebra are addition, subtraction, multiplication, division, and the extraction of roots. Arithmetic cannot generalize mathematical relations such as the Pythagorean theorem: which states that the sum of the squares of the sides of any right triangle is also a square. Arithmetic can only produce specific instances of these relations (for example 2 + 3 + 4 ). But algebra can make a purely general statement that fulfills the conditions of the theorem: a + b = c . Any number multiplied by itself is termed squared and is indicated by a superscript number 2.

Classical algebra, which is concerned with solving equations, uses symbols instead of numbers and uses arithmetic operations to establish ways of handling symbols. Modern algebra has evolved from classical algebra by increasing its attention to the structures within mathematics. Mathematicians consider modern algebra to be a set of objects with rules for connecting them. Algebra may be described as the language of mathematics.

HISTORY

The history of algebra started in Egypt and Babylon, where people learned how to solve linear and quadratic equations. They also learned to solve indeterminate equations, where several unknowns are involved. Babylonians used basically the same procedures that are used today. In Alexandrian times, Diophantus book Arithmetica led the way in finding difficult solutions to indeterminate equations. This early knowledge was embraced by the Islamic world, where it was known as the “science of restoration and balancing.” The Arabic word for restoration is al-jabru, hence the word 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, with both examples and proofs. By the end of the 9th century the Egyptian mathematician Abu Kamil had stated and proved the basic laws and identities of algebra and solved problems in finding x,y,and z so that x + y + z = 10, x + y = z , and xz=y .

The symbols of algebra include numbers, letters and signs that indicate various arithmetic operations. Numbers are constants, but letters can represent either constants or variables. Letters that are used to represent constants are taken from the beginning of the alphabet; those used to represent variables are taken from the end of the alphabet.

SYMBOLS

The grouping of algebraic symbols and the sequence of arithmetic operations rely on grouping symbols to ensure that the language of algebra is clearly read. Grouping symbols include parentheses (), brackets [], braces {}, and horizontal bars (also called vinculums) that are used most often in division and roots.

DEFINITIONS

Equation

Any statement involving the equality relation (=) is called an equation. An equation is called an identity if the equality is true for

all values of its variables; if the equation is true for some values of its variables and false for others the equation is conditional.

Variable

A symbol that represents a number is called a variable.

Term

A term is any algebraic expression consisting only of products of constants and variables. The numerical part of a term is called its coefficient.

Prime Numbers

Any integer (whole #) that can be evenly divided only by itself and by the #1 are prime numbers. Example, 2,3,5,7,11, are all prime numbers.

Powers of a number

Powers of a number are formed by successively multiplying the number by itself. The term a raised to the third power would be shown as a or a*a*a*.

Prime Factors

Prime factors of any number are those factors in which it can be reduced so that the number is expressed only as the product of primes and their powers. Example, the prime factor of 15 are 3 and 5.

Ch.9 Definitions

Sets-a well defined member of objects which is also called members or elements.

Roster notation-lists the members of a set.

Set-builder notation-gives a description of how a set is built.

Intersection-is the set of all members that are comman to both sets. Using an upside down U.

Empty set-no members common to both sets.

Union-The set of all members that are equal to both A and B. Use a U to show the union.

Conjunction- x>5 and x6. Using OR

Half-planes- regions above and below the line.

Boundry line- The line that it forms

System of inequalities- 2 or more linear inequalities for which is a common solution is sought.