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Elementary algebra encompasses the basic concepts of
algebra. It is often contrasted with
arithmetic
Arithmetic () is an elementary part of mathematics that consists of the study of the properties of the traditional operations on numbers— addition, subtraction, multiplication, division, exponentiation, and extraction of roots. In the 19th ...
: arithmetic deals with specified
numbers, whilst algebra introduces
variables (quantities without fixed values).
This use of variables entails use of algebraic notation and an understanding of the general rules of the
operations
Operation or Operations may refer to:
Arts, entertainment and media
* ''Operation'' (game), a battery-operated board game that challenges dexterity
* Operation (music), a term used in musical set theory
* ''Operations'' (magazine), Multi-Man ...
introduced in arithmetic. Unlike
abstract algebra, elementary algebra is not concerned with
algebraic structure
In mathematics, an algebraic structure consists of a nonempty set ''A'' (called the underlying set, carrier set or domain), a collection of operations on ''A'' (typically binary operations such as addition and multiplication), and a finite set of ...
s outside the realm of
real and
complex numbers.
It is typically taught to
secondary school
A secondary school describes an institution that provides secondary education and also usually includes the building where this takes place. Some secondary schools provide both '' secondary education, lower secondary education'' (ages 11 to 14) ...
students and builds on their understanding of
arithmetic
Arithmetic () is an elementary part of mathematics that consists of the study of the properties of the traditional operations on numbers— addition, subtraction, multiplication, division, exponentiation, and extraction of roots. In the 19th ...
. 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
equation
In mathematics, an equation is a formula that expresses the equality of two expressions, by connecting them with the equals sign . The word ''equation'' and its cognates in other languages may have subtly different meanings; for example, in ...
s.
Algebraic notation
Algebraic notation describes the rules and conventions for writing
mathematical expressions
In mathematics, an expression or mathematical expression is a finite combination of symbols that is well-formed according to rules that depend on the context. Mathematical symbols can designate numbers ( constants), variables, operations, fun ...
, as well as the terminology used for talking about parts of expressions. For example, the expression
has the following components:
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A ''coefficient'' is a numerical value, or letter representing a numerical constant, that multiplies a variable (the operator is omitted). 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 (e.g.
) are typically used to represent
constants, and those toward the end of the alphabet (e.g.
and ) are used to represent
variables. They are usually printed in italics.
Algebraic operations work in the same way as
arithmetic operations, such as
addition
Addition (usually signified by the Plus and minus signs#Plus sign, plus symbol ) is one of the four basic Operation (mathematics), operations of arithmetic, the other three being subtraction, multiplication and Division (mathematics), division. ...
,
subtraction
Subtraction is an arithmetic operation that represents the operation of removing objects from a collection. Subtraction is signified by the minus sign, . For example, in the adjacent picture, there are peaches—meaning 5 peaches with 2 taken ...
,
multiplication
Multiplication (often denoted by the cross symbol , by the mid-line dot operator , by juxtaposition, or, on computers, by an asterisk ) is one of the four elementary mathematical operations of arithmetic, with the other ones being additi ...
,
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
In mathematics, a coefficient is a multiplicative factor in some term of a polynomial, a series, or an expression; it is usually a number, but may be any expression (including variables such as , and ). When the coefficients are themselves var ...
is used. For example,
is written as
, and
may be written
.
Usually terms with the highest power (
exponent
Exponentiation is a mathematical operation, written as , involving two numbers, the '' base'' and the ''exponent'' or ''power'' , and pronounced as " (raised) to the (power of) ". When is a positive integer, exponentiation corresponds to re ...
), are written on the left, for example,
is written to the left of . When a coefficient is one, it is usually omitted (e.g.
is written
). Likewise when the exponent (power) is one, (e.g.
is written
). When the exponent is zero, the result is always 1 (e.g.
is always rewritten to ). 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"., as well as 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
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Elementary algebra builds on and extends arithmetic by introducing letters called variables to represent general (non-specified) numbers. This is useful for several reasons.
#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
.
#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 (algebraic) description may state that the number of seconds,
, where m is the number of minutes.
#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
.
#Variables allow one to describe some mathematical properties. For example, a basic property of addition is
commutativity
In mathematics, a binary operation is commutative if changing the order of the operands does not change the result. It is a fundamental property of many binary operations, and many mathematical proofs depend on it. Most familiar as the name of ...
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 (
addition
Addition (usually signified by the Plus and minus signs#Plus sign, plus symbol ) is one of the four basic Operation (mathematics), operations of arithmetic, the other three being subtraction, multiplication and Division (mathematics), division. ...
,
subtraction
Subtraction is an arithmetic operation that represents the operation of removing objects from a collection. Subtraction is signified by the minus sign, . For example, in the adjacent picture, there are peaches—meaning 5 peaches with 2 taken ...
,
multiplication
Multiplication (often denoted by the cross symbol , by the mid-line dot operator , by juxtaposition, or, on computers, by an asterisk ) is one of the four elementary mathematical operations of arithmetic, with the other ones being additi ...
,
division and
exponentiation). For example,
*Added terms are simplified using coefficients. For example,
can be simplified as
(where 3 is a numerical coefficient).
*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
can be written as
Equations
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An equation states that two expressions are equal using the symbol for equality, (the
equals sign). One of the best-known equations describes Pythagoras' law relating the length of the sides of a
right angle
In geometry and trigonometry, a right angle is an angle of exactly 90 Degree (angle), degrees or radians corresponding to a quarter turn (geometry), turn. If a Line (mathematics)#Ray, ray is placed so that its endpoint is on a line and the ad ...
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 (addition) 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 as
); 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
In mathematics, an equivalence relation is a binary relation that is reflexive, symmetric and transitive. The equipollence relation between line segments in geometry is a common example of an equivalence relation.
Each equivalence relation ...
, meaning it has the properties (a)
reflexive (i.e.
), (b)
symmetric (i.e. if
then
) (c)
transitive (i.e. if
and
then
). 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
In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every real ...
, 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 ( 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 (and with , substituting 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
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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 equation
In mathematics, a linear equation is an equation that may be put in the form
a_1x_1+\ldots+a_nx_n+b=0, where x_1,\ldots,x_n are the variables (or unknowns), and b,a_1,\ldots,a_n are the coefficients, which are often real numbers. The coefficien ...
s that have only one variable. They contain only constant numbers and a single variable without an exponent. As an example, consider:
: Problem in words: If you double the age of a child and add 4, the resulting answer is 12. How old is the child?
:Equivalent equation:
where represent the child's age
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:
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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 (i.e. subtract from both sides, and then divide by ), the general solution is given by
Linear equations with two variables
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A linear equation with two variables has many (i.e. an infinite number of) solutions. For example:
:Problem in words: A father is 22 years older than his son. How old are they?
:Equivalent equation:
where is the father's age, is the son's age.
That cannot be worked out by itself. If the son's age was made known, then there would no longer be two unknowns (variables). The problem then becomes a linear equation with just one variable, that can be solved as described above.
To solve a linear equation with two variables (unknowns), requires two related equations. For example, if it was also revealed that:
; Problem in words
: In 10 years, the father will be twice as old as his son.
;Equivalent equation
:
Now there are two related linear equations, each with two unknowns, which enables the production of a linear equation with just one variable, by subtracting one from the other (called the elimination method):
:
:
In other words, the son is aged 12, and since the father 22 years older, he must be 34. In 10 years, the son will be 22, and the father will be twice his age, 44. This problem is illustrated on the associated plot of the equations.
For other ways to solve this kind of equations, see below,
System of linear equations
In mathematics, a system of linear equations (or linear system) is a collection of one or more linear equations involving the same variable (math), variables.
For example,
:\begin
3x+2y-z=1\\
2x-2y+4z=-2\\
-x+\fracy-z=0
\end
is a system of three ...
.
Quadratic equations
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A quadratic equation is one which includes a term with an exponent of 2, for example,
, and no term with higher exponent. The name derives from the Latin ''quadrus'', meaning square. In general, a quadratic equation can be expressed in the form
, where is not zero (if it were zero, then the equation would not be quadratic but linear). Because of this a quadratic equation must contain the term
, which is known as the quadratic term. Hence
, and so we may divide by and rearrange the equation into the standard form
:
where
and
. Solving this, by a process known as
completing the square, leads to the
quadratic formula
:
where
the symbol "±" indicates that both
:
are solutions of the quadratic equation.
Quadratic equations can also be solved using
factorization (the reverse process of which is
expansion, but for two
linear terms is sometimes denoted
foiling
A sailing hydrofoil, hydrofoil sailboat, or hydrosail is a sailboat with wing-like foils mounted under the hull. As the craft increases its speed the hydrofoils lift the hull up and out of the water, greatly reducing wetted area, resulting in dec ...
). As an example of factoring:
:
which is the same thing as
:
It follows from the
zero-product property
In algebra, the zero-product property states that the product of two nonzero elements is nonzero. In other words, \textab=0,\texta=0\textb=0.
This property is also known as the rule of zero product, the null factor law, the multiplication prope ...
that either
or
are the solutions, since precisely one of the factors must be equal to
zero. All quadratic equations will have two solutions in the
complex number system, but need not have any in the
real number system. For example,
:
has no real number solution since no real number squared equals −1.
Sometimes a quadratic equation has a root of
multiplicity 2, such as:
:
For this equation, −1 is a root of multiplicity 2. This means −1 appears twice, since the equation can be rewritten in factored form as
:
Complex numbers
All quadratic equations have exactly two solutions in
complex numbers (but they may be equal to each other), a category that includes
real numbers,
imaginary number
An imaginary number is a real number multiplied by the imaginary unit , is usually used in engineering contexts where has other meanings (such as electrical current) which is defined by its property . The square of an imaginary number is . Fo ...
s, and sums of real and imaginary numbers. Complex numbers first arise in the teaching of quadratic equations and the quadratic formula. For example, the quadratic equation
:
has solutions
:
Since
is not any real number, both of these solutions for ''x'' are complex numbers.
Exponential and logarithmic equations
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An exponential equation is one which has the form
for
, which has solution
:
when
. Elementary algebraic techniques are used to rewrite a given equation in the above way before arriving at the solution. For example, if
:
then, by subtracting 1 from both sides of the equation, and then dividing both sides by 3 we obtain
:
whence
:
or
:
A logarithmic equation is an equation of the form
for
, which has solution
:
For example, if
:
then, by adding 2 to both sides of the equation, followed by dividing both sides by 4, we get
:
whence
:
from which we obtain
:
Radical equations
A radical equation is one that includes a radical sign, which includes
square roots,
cube root
In mathematics, a cube root of a number is a number such that . All nonzero real numbers, have exactly one real cube root and a pair of complex conjugate cube roots, and all nonzero complex numbers have three distinct complex cube roots. Fo ...
s,