is the most basic form of algebra
taught to students who are presumed to have no knowledge of mathematics
beyond the basic principles of arithmetic
. While in arithmetic only numbers
and their arithmetical operations occur, in algebra one also uses symbols (such as a
) to denote numbers. This is useful because
- it allows the general formulation of arithmetical laws (such as a + b = b + a for all a and b), and thus is the first step to a systematic exploration of the properties of the real number system
- it allows to talk about "unknown" numbers, the formulation of equations and the study of how to solve these (for instance "find a number x such that 3x + 2 = 10)
- it allows to formulate functional relationships (such as "if you sell x tickets, then your profit will be 3x - 10 dollars")
These three are the main strands of elementary algebra, which should be distinguished from abstract algebra
, a much more advanced topic only taught to college seniors.
In algebra, an "expression" may contain numbers, variables and arithmetical operations; examples are a + 3 and x2 - 3. An "equation" is the claim that two expressions are equal. Some equations are true for all values of the involved variables (such as a + (b + c) = (a + b) + c); these are also known as "identities". Other equations contain symbols for unknown values and we are then interested in finding those values for which the equation becomes true: x2 - 1 = 4. These are the "solutions" of the equation.
As in arithmetic, in algebra it is important to know precisely how mathematical expressions are to be interpreted. This is governed by the order of operations rules.
It is then necessary to be able to simplify algebraic expressions. For example, the expression
- -4(2a + 3) - a
can be written in the equivalent form
- -9a - 12.
The simplest equations to solve are the linear ones, such as
- 2x + 3 = 10
The central technique is add/subtract/multiply or divide both sides of the equation by the same number, and by repeating this process eventually arrive at the value of the unknown x
. For the above example, if we subtract 3 from both sides, we obtain
- 2x = 7
and if we then divide both sides by 2, we get our solution
- x = 7/2
- x2 + 3x = 5
are known as quadratic equations
and can be solved using the quadratic formula
Expressions or statement may contain many variables, from which you may or may not be able to deduce the values for some of the variables. For example:
- (x - 1) × (x - 1) = y × 0
After some algebraic steps (not covered here), we can deduce that x = 1, however we cannot deduce what the value of y is. Try some values of x and y (which may lead to either true or false statements) to get a feel for this.
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