Elementary algebraElementary algebra
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 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 )
- it allows to formulate functionalal relationships (such as "if you sell x tickets, then your profit will be dollars")
These three are the main strands of elementary algebra, which should be distinguished from abstract algebra
, a much more advanced topic generally taught to college seniors.
In algebra, an "expression" may contain numbers, variables and arithmetical operations; examples are and . An "equation" is the claim that two expressions are equal. Some equations are true for all values of the involved variables (such as ); 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: . 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
can be written in the equivalent form
The simplest equations to solve are the linear
ones, such as
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
and if we then divide both sides by 2, we get our solution
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:
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.
However, if we had another equation where the values for x and y were the same, we could deduce the answer in a process known as systems of equations. For example (assume x and y are the same values in both equations):
Now, multiply the second by 2, and you have the following equations:
Because we multiplied the entire equation by 2, it actually represents the same statement. Now we can combine the two equations:
You can see that since we multiplied the second equation by 2, we can cancel out y when combining the equations, and then we can solve for x, which is 2. Note that you can multiply by negative numbers, or multiply both equations to get to a point where a variable cancels out (you can also cancel out x).
Now choose one of the equations from the beginning.
Substitute in 2 for x.
And solve for y, which equal 3. The answer to this problem is x=2 and y=3, or (2,3).
Laws of elementary algebra
Example: if then .
- Multiplication is a commutative operation.
- Division is the reverse of multiplication.
- To divide is the same as to multiply by a reciprocal:
See also: binomial, distributivity, vulgar fraction.
- Exponentiation is not a commutative operation.
- Therefore exponentiation has a pair of reverse operations: logarithm and exponentiation with reciprocal exponents (e.g. square roots).
- Examples: if then . If then .
- The square root of minus one is i.
- Distributive property of multiplication with respect to addition: .
- Distributive property of exponentiation with respect to multiplication: .
- How to combine exponents: .
- If a = b and b = c, then (Transitivity of Equality).
- (Reflexivity of Equality).
- If then (Symmetry of Equality).
- If and then .
- If then for any c, due to Reflexivity of Equality.
- If and then = .
- If then for any c due to Reflexivity of Equality.
- If two symbols are equal, then one can be substituted for the other at will.
- If and then (Transitivity of Inequality).
- If then for any c.
- If and then .
- If and then .