Algebra

Algebra is a part of mathematics (frequently just called math or maths). It is used to find values and variables that make equations and inequalities true. It often needs knowledge of the rules of numbers and mathematics operations used on numbers (adding, subtracting, multiplying, dividing, and other operations). Algebra uses letters (a,b,c,...) and other symbols to represent numbers. The letters and symbols in algebra are called variables. We use variables because the numbers are unknown or can change.

Algebra was first used to solve equations. Two examples are linear equations and quadratic equations.

The ways developed to solve equations frequently made use of the facts of the parts of an equation. For example, how to factorize polynomials, or how to use a matrix to solve many linear equations at once. Abstract algebra is the study of things that are found in equations. For example, polynomials and matrices. These things are known as algebraic structures. Understanding the properties of more complex algebraic structures can help us solve more complex 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.

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.

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 two numbers in algebra, the only way is usually 3·4. 3(4) can also be used. The symbol "×" 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 $(x, y)$. There may be linear or simultaneous equations of algebra. E.g. A linear graph is drawn, equations stating x=ky is to be drawn on the graph line.This is called inverse or direct proportion. Inverse proportion=Downwards sloping line,Direct proportion=Upwards sloping line with a circle at the origin.

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 CE, and died around 840 CE. 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
• Parabola
• Computer Algebra System