User:Fr33kman/Underconstruction/Algebra

Algebra is a branch of mathematics (math) that deals with ways of solving equations and inequalities. Although many different problems can be solved with algebra, the search for at least one unknown value (or answer) is a part of all of them. Some of these ways of solving problems mean that a person needs to require understanding the properties of numbers and math operations used on numbers (adding, subtracting, multiplying or dividing, square roots, raising a number to a power, and taking logarithms). Algebra does this by using letters (a,b,c,...) or other symbols to represent numbers, either because the numbers are unknown or because the numbers change during the course of the problem, in which case the letters are called variables.

The purpose of algebra was originally to solve equations, of which linear equations and quadratic equations are examples. The techniques developed to solve such equations often had to do with the properties of the participants in an equation, like how to factor polynomials, or how to solve systems of many equations at once using things like a matrix. The study of abstract algebra is the study of things like polynomials and matrices that arise in solving equations—these entities are therefore known as algebraic structures. Understanding the properties of more complicated algebraic structures can help us solve more complicated equations.

Of particular importance to the study of algebra is the study of functions, since functions often appear in equations that we are trying to solve. A function is like a box you can put a number or numbers into and get a certain number out. When a function takes one variable in, like f(x), it is called a univariate function. When a function takes many variables in, like Q(x,y,z), it is called a multivariate function.

The study of algebra is heavily concerned with the subject of graphing functions, as graphs can be powerful tools in helping us to study the solutions to equations.

In some math problems one variable is changed. The variable that is changed is called the "independent" variable. After the variable is changed the math operations in the equation are performed to make a number. The number that is made is called the "dependent" variable. Most of the time the independent variable is written as x and the dependent variable is written as y, for example, in y = 3x + 1. This is because we do not know what number they are until we are told.

Many math problems are about physics. In many of these physics problems time is a variable. Time is written t. Using the basic ideas in algebra can help reduce a math problem to its simplest form making it easier to solve difficult problems. Algebra is taught in school to help in harder mathematics, science, and engineering classes.

Here is a simple example of an algebra problem:

Sue has 12 jellybeans, Ann has 24 jellybeans. They decide to share so that they have the same number of jellybeans. Let x represent the number of jellybeans Ann gives to Sue. Then we want 12 + x = 24 - x.

Here are steps you can use to solve the problem:

1. Subtract 12 from both sides of the equation. This gives x = 12 - x.
2. Add x to both sides of the equation. This gives 2x = 12.
3. Divide both sides of the equation by 2. This gives x = 6. If Ann gives Sue 6 jellybeans, they will have the same number of jellybeans.

Of course, this problem could be solved without algebra. The purpose of simple story problems such as this one is to teach algebra. Once a student is comfortable with simple problems like this, it is easier to try harder ones. With practice, the students can use algebra when faced with a problem that is too hard to solve any other way. Problems such as building a freeway, designing a cell phone, or finding the cure for a disease all require algebra.

In addition to "elementary algebra", or basic algebra, there are advanced forms of algebra, taught in colleges and universities, such as abstract algebra, linear algebra, and universal algebra. With abstract algebra and universal algebra, it is hard to see how studying them helps solve problems, but with linear algebra it is much clearer.

Algebra can be used to solve real problems because the rules of algebra work in real life and numbers can be used to represent the values of real things.

Writing algebra

In algebra, adding z to y (or y plus z) is written as y + z. Subtracting z from y (or y minus z) is written as y − z.

In algebra, multiplying y by z (or y times z) can be written in 4 ways: y × z, y*z, y·z, or yz. yz is the most usual form of writing the product of y and z in algebra.

When we multiply a number and a letter in algebra, we write the number in front of the letter: 5 × y = 5y. When the number is 1, then the 1 is not written because 1 times any number is that number (1 × y = y) and so is not needed.

When we multiply 2 numbers in algebra, the only way is usually 3·4. × is not used, because it looks too much like the letter x.

In algebra, dividing y by z (or y over z) is written as y ÷ z or y/z. y/z is more commonly used.

Graphing algebra

Algebra also introduces graphing, or drawing a picture that shows all the values of the variables that make the equation true. Usually this is easy to do when there are only one or two variables. The graph is often a line, and if the line does not bend or go straight up-and-down it can be described by the basic formula $y=mx+b$ where b is the y-intercept of the graph and m is the slope. This formula applies to the coordinates of the graph or $(x,y)$.

History

The word "algebra" is a Latin form of the Arabic word Al-Jabr ("casting") and comes from a mathematics book Al-Maqala fi Hisab-al Jabr wa-al-Muqabilah, ("Essay on the Computation of Casting and Equation") written in the 9th century by a famous Persian mathematician, Muhammad ibn Mūsā al-Khwārizmī, who was a Muslim born in Khwarizm in Uzbekistan. He flourished under Al-Ma'moun in Baghdad, Iraq through 813-833 AD, and died around 840 AD. The book was brought into Europe and translated into Latin in the 12th century. The book was then given the name 'Algebra'.

Other pages

• List of mathematics topics
• Order of Operations