Friday, October 3, 2014

Algebra

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Algebra
1.
Algebra (from Arabic al-jebr meaning "reunion of broken parts") is one of the broad parts of mathematics, together with number theory, geometry and analysis. In its most general form algebra is the study of symbols and the rules for manipulating symbols and is a unifying thread of all of mathematics. As such, it includes everything from elementary equation solving to the study of abstractions such as groups, rings, and fields. The more basic parts of algebra are called elementary algebra, the more abstract parts are called abstract algebra or modern algebra. Elementary algebra is essential for any study of mathematics, science, or engineering, as well as such applications as medicine and economics. Abstract algebra is a major area in advanced mathematics, studied primarily by professional mathematicians. Much early work in algebra, as the Arabic origin of its name suggests, was done in the Near East, by such mathematicians as Omar Khayyam (1048-1131).

Elementary algebra differs from arithmetic in the use of abstractions, such as using letters to stand for numbers that are either unknown or allowed to take on many values. For example, in x + 2 = 5 the letter x is unknown, but the law of inverses can be used to discover its value: x=3. In E=mc^2, the letters E and m are variables, and the letter c is a constant. Algebra gives methods for solving equations and expressing formulas that are much easier (for those who know how to use them) than the older method of writing everything out in words.
The word algebra is also used in certain specialized ways. A special kind of mathematical object in abstract algebra is called an "algebra", and the word is used, for example, in the phrases linear algebra and algebraic topology(see below).

A mathematician who does research in algebra is called an algebraist.

2.
Simply put, Algebra is about finding the unknown or it is about putting real life problems into equations and then solving them. Unfortunately many textbooks go straight to the rules, procedures and formulas, forgetting that these are real life problems being solved.
A branch of mathematics that substitutes letters for numbers. An algebraic equation represents a scale, what is done on one side of the scale with a number is also done to the other side of the scale. The numbers are the constants. Algebra can include real numbers, complex numbers, matrices, vectors etc.

In the simplest of form, I could say:
A clown was carrying a handful of balloons. Along came the wind and blew 8 away, leaving him only with 9. How many did he start with?

In algebra, this problem would then be converted to:
x - 8 = 9

The x replaces the unknown that we are trying to find out, we know the wind blew 8 away and we know that the man was left with 9 balloons.

Remember, in Algebra x seems to be the favorite letter to substitute for the unknown.
The goal in algebra is to find out the unknown. Therefore, we often end up with x = something. In this problem, when you apply algebra, you'll learn that it's like using a scale, you want to isolate x and in so doing you'll do the same thing on each side of the = sign. To isolate x in x-8=9, I will need to add 8 to the left side of the = sign and add 8 to the right side of the = sign. I am then left with x = 8 + 9 therefore solving my real life problem that x = 17, meaning that the balloon man started with 17 balloons.

Why Do I Need Algebra?
Only you can answer this question. I've always said math is an opportunity gateway and you can't get to higher maths without taking algebra. Algebra develops your thinking, specifically logic, patterns, problem solving, deductive and inductive reasoning. The more math you have, the greater the opportunity for jobs in engineering, actuary, physics, programming etc. Higher math is often an important requirement for entrance to college or universities.
Ultimately, you need to do your own homework to determine if your goals mean sticking it out in math, however, you can never go wrong if you do.

3.
What is Algebra?
Algebra is a branch of mathematics that uses mathematical statements to describe relationships between things that vary over time. These variables include things like the relationship between supply of an object and its price. When we use a mathematical statement to describe a relationship, we often use letters to represent the quantity that varies, since it is not a fixed amount. These letters and symbols are referred to as variables. (See the Appendix One for a brief review of constants and variables.)

The mathematical statements that describe relationships are expressed using algebraic terms, expressions, or equations (mathematical statements containing letters or symbols to represent numbers). Before we use algebra to find information about these kinds of relationships, it is important to first cover some basic terminology. In this unit we will first define terms, expressions, and equations. In the remaining units in this book we will review how to work with algebraic expressions, solve equations, and how to construct algebraic equations that describe a relationship. We will also introduce the notation used in algebra as we move through this unit.
Algebraic Terms
The basic unit of an algebraic expression is a term. In general, a term is either a number or a product of a number and one or more variables. Below is the term –3ax.
The numerical part of the term, or the number factor of the term, is what we refer to as the numerical coefficient. This numerical coefficient will take on the sign of the operation in front of it. The term above contains a numerical coefficient, which includes the arithmetic sign, and a variable or variables. In this case the numerical coefficient is –3 and the variables in the term are a and x. Terms such as xz may not appear to have a numerical coefficient, but they do. The numerical coefficient is 1, which is assumed.

Algebraic Expressions
An expression is a meaningful collection of numbers, variables, and signs, positive or negative, of operations that must make mathematical and logical sense. Expressions:
  • contain any number of algebraic terms
  • use signs of operation—addition, subtraction, multiplication, and division.
  • do not contain an equality sign (=)

An example of an expression is:
–3ax + 11wx2y
In an expression, the signs of operation separate it into terms. The sign also becomes part of the term that it follows. The expression above contains two terms, the first term is –3ax and the second term is +11wx2y. The addition sign separates the two terms. For example, in the expression given above the plus sign (+) separates the –3ax from 11wx2y and is also part of the second term. Terms that do not have a sign listed in front of them are understood to be positive.

Below are several examples that are not expressions.
x + • y
This statement tells us "x plus multiplied by y". This does not make mathematical or logical sense. This collection of symbols is nonsense.
y = 2x – 1
This statement is not an expression because expressions are not allowed to contain the equal sign.
NOTE: The operation of multiplication can be represented by using a x, •, or by placing items to be multiplied in parentheses, brackets or braces, or in the case of variables, just written next to one another. The statements a xb, ab, (a)( b), and ab are equivalent. In this booklet we will use the latter three representations.

Algebraic Equations
An equation is a mathematical statement that two expressions are equal. The following three statements are equations:
4 + 5 = 9

x – 35 = 56k2 + 3

x + 3 = 15
The first equation, 4 + 5 = 9, contains only numbers; the other two, however, also contain variables. All three contain two expressions separated by an equal sign:
When an equation contains variables you will often have to solve for one of those variables. Using equations to solve for a variable will be discussed later in this booklet.

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