Elementary algebra

Elementary algebra encompasses some of the basic concepts of algebra, one of the main branches of mathematics. It is typically taught to secondary school students and builds on their understanding of arithmetic. Whereas arithmetic deals with specified numbers, algebra introduces quantities without fixed values, known as variables. This use of variables entails a use of algebraic notation and an understanding of the general rules of the operators introduced in arithmetic. Unlike abstract algebra, elementary algebra is not concerned with algebraic structures outside the realm of real and complex numbers.equivalentlyequivalently Elementary algebra encompasses some of the basic concepts of algebra, one of the main branches of mathematics. It is typically taught to secondary school students and builds on their understanding of arithmetic. Whereas arithmetic deals with specified numbers, algebra introduces quantities without fixed values, known as variables. This use of variables entails a use of algebraic notation and an understanding of the general rules of the operators introduced in arithmetic. Unlike abstract algebra, elementary algebra is not concerned with algebraic structures outside the realm of real and complex numbers. 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 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 3 x 2 − 2 x y + c {displaystyle 3x^{2}-2xy+c} has the following components: 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. a , b , c {displaystyle a,b,c} ) are typically used to represent constants, and those toward the end of the alphabet (e.g. x , y {displaystyle x,y} and z) are used to represent variables. They are usually written 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, 3 × x 2 {displaystyle 3 imes x^{2}} is written as 3 x 2 {displaystyle 3x^{2}} , and 2 × x × y {displaystyle 2 imes x imes y} may be written 2 x y {displaystyle 2xy} . Usually terms with the highest power (exponent), are written on the left, for example, x 2 {displaystyle x^{2}} is written to the left of x. When a coefficient is one, it is usually omitted (e.g. 1 x 2 {displaystyle 1x^{2}} is written x 2 {displaystyle x^{2}} ). Likewise when the exponent (power) is one, (e.g. 3 x 1 {displaystyle 3x^{1}} is written 3 x {displaystyle 3x} ). When the exponent is zero, the result is always 1 (e.g. x 0 {displaystyle x^{0}} is always rewritten to 1). However 0 0 {displaystyle 0^{0}} , being undefined, should not appear in an expression, and care should be taken in simplifying expressions in which variables may appear in exponents. 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. For example, exponents are usually formatted using superscripts, e.g. x 2 {displaystyle x^{2}} . In plain text, and in the TeX mark-up language, the caret symbol '^' represents exponents, so x 2 {displaystyle x^{2}} is written as 'x^2'. In programming languages such as Ada, Fortran, Perl, Python and Ruby, a double asterisk is used, so x 2 {displaystyle x^{2}} 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, 3 x {displaystyle 3x} is written '3*x'. Elementary algebra builds on and extends arithmetic by introducing letters called variables to represent general (non-specified) numbers. This is useful for several reasons. Algebraic expressions may be evaluated and simplified, based on the basic properties of arithmetic operations (addition, subtraction, multiplication, division and exponentiation). For example,