Algebra
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Algebra is a branch of mathematics 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. The word algebra is of Arabic origin.
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History
The origins of algebra trace to the cultures of the ancient Egyptians and Babylonians who used an early type of algebra to solve linear, quadratic, and indeterminate equations more than 3,000 years ago.
Around 300 BC Greek mathematician Euclid in book 2 of his Elements addresses quadratic equations, although in a strictly geometrical fashion.
Around 100 BC Algebraic equations are treated in the Chinese mathematics book Jiuzhang suanshu, The Nine Chapters of Mathematical Art.
Around 150 AD Greek mathematician Hero of Alexandria treats algebraic equations in his 3 volumes mathematics tomes.
Around 200 AD Greek mathematician Diophantus , often referred to as the "father of algebra", writes his famous Arithmetica, a work featuring solutions of algebraic equations and on the theory of numbers.
The word algebra itself is derived from the name of the treatise first written by Persian mathematician Al-Khwarizmi in 820 AD 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 (from which algebra is derived) 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 fields 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 geometric structure (a topology) which is compatible with the algebraic structure. The list includes
- Normed linear spaces
- Banach spaces
- Hilbert spaces
- Banach algebras
- Normed algebras
- Topological algebras
- Topological groups
Algebraic equations
There are many forms of algebraic equations. Some are listed below:
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 <math>y=Mx+B<math>. This is called the slope-intercept form.
- To graph the slope-intercept form of a line, substitute a number for <math>x<math> and solve for <math>y<math> slope. Graph the results on the graph for <math>x<math> for the x-axis and <math>y<math> for the y-axis.
- <math>y<math> is value for the y-axis in relation to <math>x<math>, <math>m<math>, and <math>b<math>
- <math>M<math> 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. <math>M<math> can be found by using this formula:
- <math>(y2-y1)/(x2-x1)<math>
- <math>x<math> is the variable in the equation. The variable is the part that can be changed. When <math>x<math> changes, so does <math>y<math>. When the equation is graphed, the line shows what <math>y<math> is for each value of <math>x<math>.
- <math>B<math> is the number added to the equation. In the expression <math>2x+3<math>, <math>B=3<math>. <math>B<math> also represents the <math>y<math> intercept on a graph.
The <math>y<math> intercept is where the line crosses the <math>y<math> axis.
- The second way that can be written is the standard form. It is written in the form of <math>ax+by=c.<math>
- To graph standard form, find the x and y intercepts and connect.
- <math>a<math> and <math>b<math> and <math>c<math> are constants, as in, they don't change in the same line (opposite of variables)
- <math>x<math> and <math>y<math> are the x-intercepts and the y-intercepts respectively.
Quadratic equations
Polynomialdeg2.png
y = x2 - x - 2 = (x+1)(x-2)
Quadratic equations are written in the form <math>y=ax^2+bx+c<math>. When a quadratic equation is graphed, it produces a curved line called a parabola.
- <math>a<math> is the coefficient of the variable squared
- <math>b<math> is the coefficient of the variable
- <math>c<math> is the extra added number. It is the same as the <math>B<math> in Linear equations
Cubic equations
Polynomialdeg3.png
y = x3/5 + 4x2/5 - 7x/5 - 2
= 1/5 (x+5)(x+1)(x-2)
Cubic equations are written in the form <math>y=ax^3+bx^2+cx+d<math>. In this form, there are up to three x-intercepts. When graphed, the curve will change direction twice.
- 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 <math>y=m^x+b<math>.
Factoring trinomials
Simple factoring
Trinomials are algebraic expressions consisting of three unlike terms, such as <math>x^2+3x+2<math>. 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.,
- <math>x^2+5x+6<math>
is equivalent to
- <math>(x+3)(x+2)<math>.
Firsts (x times x) + Outsides (x times 2) + Insides (3 times x) + Lasts (3 times 2) = The trinomial (<math>x^2+5x+6<math>).
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: <math>3x^2+8xy+4y^2<math>. In this situation, the factored form will look like: (3x + 2y)(x + 2y). 3x times x is 3x2, 3x times 2y is 6xy, 2y times x is 2xy, and 2y times 2y is <math>4y^2<math>. This time, the coefficients of x have to be multiplied with the coefficient of <math>x^2<math>, 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:
- algebra over a field
- algebra over a set
- Boolean algebra
- sigma-algebra
- F-algebra and F-coalgebra in category theory
References
- Ziauddin Sardar, Jerry Ravetz, and Borin Van Loon, Introducing Mathematics (Totem Books, 1999).
- Donald R. Hill, Islamic Science and Engineering (Edinburgh University Press, 1994).
- George Gheverghese Joseph, The Crest of the Peacock : The Non-European Roots of Mathematics (Princeton University Press, 2000).
See also
- Fundamental theorem of algebra (which is really a theorem of mathematical analysis, not of algebra)
- Diophantus, "father of algebra"
- Mohammed al-Khwarizmi, also known as "father of Algebra". [1] (http://www.math.umd.edu/~czorn/hist_algebra.pdf)
External links
- Algebra Lessons of Life (http://kagawasan.netfirms.com/english/ilearnedmorethanjustalgebra.shtml)
- Curriculum and Assessment in an Age of Computer Algebra Systems (http://www.ericdigests.org/2003-1/age.htm)
- Algebra solvers for students with graphs and work shown. Learning instead of cheating (http://www.algebra.com/)af:Algebra
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