Elementary algebra


Elementary algebra, also known as high school algebra or college algebra, encompasses the basic concepts of algebra. It is often contrasted with arithmetic: arithmetic deals with specified numbers, whilst algebra introduces numerical variables. In arithmetic, operations can only be performed on numbers. In algebra, operations can be performed on numbers, variables, and terms. Unlike abstract algebra, elementary algebra is not concerned with algebraic structures outside the realm of real and complex numbers.
It is typically taught to secondary school students and at introductory college level in the United States, and builds on their understanding of arithmetic. The use of variables to denote quantities allows general relationships between quantities to be formally and concisely expressed, and thus enables solving a broader scope of problems. Many quantitative relationships in science and mathematics are expressed as algebraic equations.

Algebraic operations

Algebraic notation

Algebraic notation describes the rules and conventions for writing mathematical expressions, as well as the terminology used for talking about parts of expressions. For example, the expression has the following components:
A coefficient is a numerical value, or letter representing a numerical constant, that multiplies a variable. A term is an addend or a summand, a group of coefficients, variables, constants and exponents that may be separated from the other terms by the plus and minus operators. Letters represent variables and constants. By convention, letters at the beginning of the alphabet are typically used to represent constants, and those toward the end of the alphabet are used to represent variables. They are usually printed in italics.
Algebraic operations work in the same way as arithmetic operations, such as addition, subtraction, multiplication, division and exponentiation, and are applied to algebraic variables and terms. Multiplication symbols are usually omitted, and implied when there is no space between two variables or terms, or when a coefficient is used. For example, is written as, and may be written.
Usually terms with the highest power, are written on the left, for example, is written to the left of. When a coefficient is one, it is usually omitted. Likewise when the exponent is one,. When the exponent is zero, the result is always 1. However, being undefined, should not appear in an expression, and care should be taken in simplifying expressions in which variables may appear in exponents.

Alternative notation

Other types of notation are used in algebraic expressions when the required formatting is not available, or can not be implied, such as where only letters and symbols are available. As an illustration of this, while exponents are usually formatted using superscripts, e.g.,, in plain text, and in the TeX mark-up language, the caret symbol represents exponentiation, so is written as "x^2". This also applies to some programming languages such as Lua. In programming languages such as Ada, Fortran, Perl, Python and Ruby, a double asterisk is used, so is written as "x**2". Many programming languages and calculators use a single asterisk to represent the multiplication symbol, and it must be explicitly used, for example, is written "3*x".

Concepts

Variables

Elementary algebra builds on and extends arithmetic by introducing letters called variables to represent general numbers. This is useful for several reasons.
  1. Variables may represent numbers whose values are not yet known. For example, if the temperature of the current day, C, is 20 degrees higher than the temperature of the previous day, P, then the problem can be described algebraically as.
  2. Variables allow one to describe general problems, without specifying the values of the quantities that are involved. For example, it can be stated specifically that 5 minutes is equivalent to seconds. A more general description may state that the number of seconds,, where m is the number of minutes.
  3. Variables allow one to describe mathematical relationships between quantities that may vary. For example, the relationship between the circumference, c, and diameter, d, of a circle is described by.
  4. Variables allow one to describe some mathematical properties. For example, a basic property of addition is commutativity which states that the order of numbers being added together does not matter. Commutativity is stated algebraically as.

    Simplifying expressions

Algebraic expressions may be evaluated and simplified, based on the basic properties of arithmetic operations. For example,
  • Added terms are simplified using coefficients. For example, can be simplified as .
  • Multiplied terms are simplified using exponents. For example, is represented as
  • Like terms are added together, for example, is written as, because the terms containing are added together, and the terms containing are added together.
  • Brackets can be "multiplied out", using the distributive property. For example, can be written as which can be written as
  • Expressions can be factored. For example,, by dividing both terms by the common factor, can be written as

    Equations

An equation states that two expressions are equal using the symbol for equality, . One of the best-known equations describes Pythagoras' law relating the length of the sides of a right angle triangle:
This equation states that, representing the square of the length of the side that is the hypotenuse, the side opposite the right angle, is equal to the sum of the squares of the other two sides whose lengths are represented by and.
An equation is the claim that two expressions have the same value and are equal. Some equations are true for all values of the involved variables ; such equations are called identities. Conditional equations are true for only some values of the involved variables, e.g. is true only for and. The values of the variables which make the equation true are the solutions of the equation and can be found through equation solving.
Another type of equation is inequality. Inequalities are used to show that one side of the equation is greater, or less, than the other. The symbols used for this are: where represents 'greater than', and where represents 'less than'. Just like standard equality equations, numbers can be added, subtracted, multiplied or divided. The only exception is that when multiplying or dividing by a negative number, the inequality symbol must be flipped.

Properties of equality

By definition, equality is an equivalence relation, meaning it is reflexive, symmetric, and transitive. It also satisfies the important property that if two symbols are used for equal things, then one symbol can be substituted for the other in any true statement about the first and the statement will remain true. This implies the following properties:
  • if and then and ;
  • if then and ;
  • more generally, for any function, if then.

    Properties of inequality

The relations less than and greater than have the property of transitivity:
  • If and then ;
  • If and then ;
  • If and then ;
  • If and then .
By reversing the inequation, and can be swapped, for example:
  • is equivalent to

    Substitution

Substitution is replacing the terms in an expression to create a new expression. Substituting 3 for in the expression makes a new expression with meaning. Substituting the terms of a statement makes a new statement. When the original statement is true independently of the values of the terms, the statement created by substitutions is also true. Hence, definitions can be made in symbolic terms and interpreted through substitution: if is meant as the definition of as the product of with itself, substituting for informs the reader of this statement that means. Often it's not known whether the statement is true independently of the values of the terms. And, substitution allows one to derive restrictions on the possible values, or show what conditions the statement holds under. For example, taking the statement, if is substituted with, this implies, which is false, which implies that if then cannot be.
If and are integers, rationals, or real numbers, then implies or. Consider. Then, substituting for and for, we learn or. Then we can substitute again, letting and, to show that if then or. Therefore, if, then or, so implies or or.
If the original fact were stated as " implies or ", then when saying "consider," we would have a conflict of terms when substituting. Yet the above logic is still valid to show that if then or or if, instead of letting and, one substitutes for and for . This shows that substituting for the terms in a statement isn't always the same as letting the terms from the statement equal the substituted terms. In this situation it's clear that if we substitute an expression into the term of the original equation, the substituted does not refer to the in the statement " implies or."

Solving algebraic equations

The following sections lay out examples of some of the types of algebraic equations that may be encountered.

Linear equations with one variable

Linear equations are so-called, because when they are plotted, they describe a straight line. The simplest equations to solve are linear equations that have only one variable. They contain only constant numbers and a single variable without an exponent. As an example, consider:
To solve this kind of equation, the technique is add, subtract, multiply, or divide both sides of the equation by the same number in order to isolate the variable on one side of the equation. Once the variable is isolated, the other side of the equation is the value of the variable. This problem and its solution are as follows:
In words: the child is 4 years old.
The general form of a linear equation with one variable, can be written as:
Following the same procedure, the general solution is given by