Algebra

Algebra is a branch of mathematics of Arabian origin or transmission which may be roughly characterized as a generalization and extension of arithmetic, in which symbols are employed to denote operations, and letters to represent number and quantity; it also refers to a particular kind of abstract algebra structure, the algebra over a field.

History

The origins of Algebra trace to the cultures of the ancient Egyptians and Babylonians who used algebra to solve linear, quadratic, and indeterminate equations more than 3,000 years ago.

Greek mathematician Euclid (300 BC) in his book 2 of his Elements which deals with quadratic equations.

Algebraic equations are also treated in Chinese mathematician Jiuzhang suanshu's, The Nine Chapters of Mathematical Art around 100 BC.

Hero of Alexandria, in early Greece (circa mid-1st century), treated algebraic equations in his 3 volumes books.

Greek Mathematician Diophantus around 200 AD, often referred to as the "father of algebra", is best known for his Arithmetica, a work on the solution of algebraic equations and on the theory of numbers.

The word "Algebra" itself comes from the name of the treatise first written by a persian mathematician Al-Khwarizmi 700 AD, who wrote a treatise titled: Kitab al-mukhtasar fi Hisab Al-Jabr wa-al-Moghabalah meaning The book of summary concerning calculating by transposition and reduction. The word Al-jabr (which Algebra is derived from) means "reunion", "connection" or "completion".

Classification

Algebra may be roughly divided into the following categories:

elementary algebra, where the properties of operations on the real number system are recorded, symbols are used as "place holders" to denote constants and variables, and the rules governing mathematical expressions and equations involving these symbols are studied,

abstract algebra, where algebraic structures such as groups, rings and as fieldss are axiomatically defined and investigated.

The specific properties of vector spaces are studied in linear algebra.

universal algebra, where those properties common to all algebraic structures are studied.

computer algebra, where algorithms for the symbolic manipulation of mathematical objects are collected

In advanced studies axiomatic algebraic systems like groups, rings, fields, and algebras over a field are investigated in the presence of a natural topology compatible with algebraic structure. The list includes

Normed linear spaces
Banach spaces
Hilbert spaces
Banach algebras
Normed algebras
Topological algebras
Topological groups
Forms of algebra

There are many forms of algebraic equations. Some are listed below:

Linear equations
Quadratic equations
Cubic equations
Exponential equations
Linear equations
There are three basic ways in which you could write linear equations: slope-intercept, standard form, and point slope form.
The first way that can be written is in the form . This is called slope-intercept.
To graph slope-intercept, substitute a number for and solve for slope. Graph the results on the graph for for the x-axis and for the y-axis.
is value for the y-axis in relation to , , and
is the coefficient of the variable, and it represents the slope. The slope is the steepness of the line produced when the equation is graphed. can be found by using this formula:
is the variable in the equation. The variable is the part that can be changed. When changes, so does . When the equation is graphed, the line shows what is for each value of .
is the number added to the equation. In the expression , . also represents the intercept on a graph.
The intercept is where the line crosses the axis.
The second way that can be written is the standard form. It is written in the form of
To graph standard form, find the x and y intercepts and connect.
and and are constants, as in, they don't change in the same line (opposite of variables)
and are the x-intercepts and the y-intercepts respectively.
Quadratic equations
Quadratic equations are written in the form .
When a quadratic equation is graphed, it produces a curved line called a parabola.
is the coefficient of the variable squared
is the coefficient of the variable
is the extra added number. It is the same as the in Linear equations
Cubic equations
Cubic equations are written in the form y = ax³ + bx² + cx + d. In this form, there are three x-intercepts. When graphed, the line will start going up, then curve to go down, then switch again to go up. (the opposite can occur with negative variables)
a is the coefficient of the variable cubed
b is the coefficient of the variable squared
c is the coefficient of the variable
d is the non-variable
Exponential equations

Exponential equations are written in the form y = m^x + b.

Factoring trinomials

Simple factoring

Trinomials are algebraic expressions consisting of three unlike terms, such as x^{2^{+ 3x + 2. They can be factored using the "FOIL" technique. You factor the expression by using two sets of parentheses, each consisting of two terms, where the first, outside, inside, and last numbers of both sets multiplied together and added equal the trinomial. E.g.,
x² + 5x + 6
is equivalent to
(x + 3)(x + 2).
Firsts (x times x) + Outsides (x times 2) + Insides (3 times x) + Lasts (3 times 2) = The trinomial (x^{2^{+ 5x + 6). The last numbers in each set of parenthesis have another relationship. When multiplied together, they always equal the last number (3 times 2 equals 6), and when added, they equal the coefficient of the variable (3 plus 2 equals 5). The coefficient is the number in front of the variable that you multiply it by. This is because they're both multiplied by the variable, and then added.
Two variables

Sometimes, you get expressions such as:
3x^{2^{+ 8xy + 4y^{2^{.
In this situation, the factored form will look like:
(3x + 2y)(x + 2y).
3x times x is 3x^{2^{, 3x times 2y is 6xy, 2y times x is 2xy, and 2y times 2y is 4y^{2^{. This time, the coefficients of x have to be multiplied with the coefficient of x^{2^{, and same with x.
Symbols

Depending on whether the numbers are added or subtracted, you may need to use different symbols in the parenthesis.}}}}}}}}}}}}}}

If you add the mx and add the b, the symbols are both plus.

If you add the mx and subtract the b, the symbols are one plus and one minus.

If you subtract the mx and add the b, the symbols are both minus

If you subtract the mx and subtract the b, the symbols are one plus and one minus.

Symbolic method

The symbolic method is a way to figure out a variable when it's on both sides of the equation. E.g.,

3x + 25 = 5x + 5

The first step is to isolate the variable. By subtracting 3x from both sides, you get 25 = 2x + 5.
The second step is to get only the variable on one side. To do this, you subtract 5 from both sides to get 20 = 2x.
The last step is to get just 1 x. Divide both sides by the coefficient, in this case 2, and you have 10 = x.

The word algebra is also used for various algebraic structures: