SYMBOLS AND SPECIAL TERMS In Algebra

• The symbols of algebra include numbers, letters, and signs that indicate various arithmetic operations.

• Numbers are constants (values that do not change), but letters can represent either unknown constants or variables (values that vary).
• Letters that are used to represent constants are taken from the beginning of the alphabet; those used to represent variables are taken from the end of the alphabet

Operation Symbols

• The basic operational signs of algebra are familiar from arithmetic: addition (+), subtraction (-), multiplication (×), and division (÷).

• The multiplication symbol × is often omitted or replaced by a dot, as in a · b. A group of consecutive symbols, such as abc, indicates the product (the result of multiplication) of a, b, and c. Division is commonly indicated by a horizontal bar (also called a vinculum), as in

A

/

C

• A virgule, or slash (/), may also be used to indicate division: a/c.

• A power is the product of a number multiplied by itself.
  
• The notation 4² (read “four squared”), for example, is used as an abbreviation for 4 · 4 (4 times 4); thus 4² = 16.
  
• The 4 in 4² is called the base, and the small raised number 2 is called the exponent.
  
• An exponent indicates how many times the number is multiplied by itself: x³ (read “x cubed”) means x · x · x.
  
• More generally xⁿ (read “x to the nth power” or “x to the nth” where n is any number) means the product of x multiplied by itself n times. Fractions can take exponents as well: (1/2)² = ‚1/4.

• A number whose nth power is equal to x is an nth root of x. When n is 2 the term “square root” is used and when n is 3 the term “cube root” is used. For example, 3 and -3 are both square roots of 9 since 3² = 9 and (-3)² = 9; 2 is a cube root of 8 since 2³ = 8; -2 is a cube root of -8; y is a cube root of ˆ. The square root of x is denoted like this

• The number of times the root is multiplied by itself is called the index. The index is usually omitted for square roots, but appears as a small raised number just before the root symbol for higher roots

• The two possible values of square roots, one positive and one negative, are often written using the plus or minus symbol: ±. The equation = 2 or -2, for instance, can be abbreviated = ±2.

Order of Operations and Grouping

• Algebra relies on an established sequence for performing arithmetic operations. This ensures that everyone who executes a string of operations arrives at the same answer. Multiplication is performed first, then division, followed by addition, then subtraction. For example:

1 + 2 · 3

equals 7 because 2 and 3 are multiplied first and then added to 1. Exponents and roots have even higher priority than multiplication:

3 · 2² = 3 · 4 = 12

• Grouping symbols override the order of operations. All operations within a group are carried out first. Grouping symbols include parentheses ( ), brackets [ ], braces { }, and horizontal bars that are used most often for division and roots. Adding parentheses to a previous example:

(1 + 2) · 3

indicates that 1 should be added to 2 first, and then the result multiplied by 3 for a total of 9 rather than 7. Brackets and braces are used in more complicated combinations that require multiple nested (one inside the other) groups. Operations within the innermost group are carried out first:

{2[5 + 3(1 + 4)]} =

{2[5 + 3 · 5]} =

{2[5 + 15]} =

{2 · 20} = 40

• When a slash is used to indicate division, care must be taken to group the terms appropriately. For example,

ax+b

------------------

c-dy

Special Definitions

• Any statement that contains the equality relation (=), such as 3x = 9, is called an equation. An equation is called an identity if the equality is true for all values of its variables; if the equation is true for some values of its variables and false for others, the equation is conditional. The equation x + 0 = x, for example, is an identity while 3x = 9 is conditional because it is only true when x = 3. A term is any algebraic expression consisting only of products of constants and variables; 2x, -a, and ‚s⁴x are all examples of terms. The numerical part of a term is called its coefficient. The coefficients of each term above are, respectively, 2, -1, and ‚.

• An expression containing one term, such as 2x³, is called a monomial. An expression involving the addition or subtraction of two terms, as in 2x² + 3x, is called a binomial, while an expression with three terms, such as 4x⁵ – x⁴ + 7x, is known as a trinomial. Polynomial is the general name for expressions in which any number of terms are added or subtracted. The degree of a polynomial refers to the largest exponent of the variables in the polynomial. For example, if the largest exponent of a variable is 3, as in ax³ + bx², the polynomial is said to be of degree 3. Similarly, the expression xⁿ + x^n-1 + x^n-2 is of degree n.

• A linear equation with one variable is a polynomial equation of degree one—that is, of the form ax + b = 0. These are called linear equations because graphing these equations results in straight lines. A quadratic equation in one variable is a polynomial equation of degree two—that is, of the form ax² + bx + c = 0.

• A prime number is any integer (the counting numbers: 1, 2, 3, …; their negatives; and zero) that can be evenly divided only by itself and by the number 1 or the number -1. Thus, 2, 3, 5, 7, 11, 13, 17, and 19 are all prime numbers..

• A factor of a number is any integer by which the number can be divided evenly, with no remainder. The factors of 6, for example, are 1, 2, 3, and 6, because 6 ÷ 1 = 6, 6 ÷ 2 = 3, 6 ÷ 3 = 2, and 6 ÷ 6 = 1. The prime factors of any number are those factors to which it can be reduced such that the number is expressed only as the product of primes and their powers. For example, the prime factors of 6 are 2 and 3. Similarly, because 60 = 2² × 3 × 5, the prime factors of 60 are 2, 3, and 5.

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Algebra

• Algebra, branch of mathematics in which symbols (usually letters) represent unknown numbers in mathematical equations.
• Algebra allows the basic operations of arithmetic, such as addition, subtraction, and multiplication, to be performed without using specific numbers.
• People use algebra constantly in everyday life, for everything from calculating how much flour they need to bake a certain number of cookies to figuring out how long it will take to travel by car at a certain speed to a destination that is a specific distance away.
• Arithmetic alone cannot deal with mathematical relations such as the Pythagorean theorem, which states that the sum of the squares of the lengths of the two shorter sides of any right triangle is equal to the square of the length of the longest side.
• Arithmetic can only express specific instances of these relations.
• A right triangle with sides of length 3, 4, and 5, for example, satisfies the conditions of the theorem: 3² + 4² = 5². (3² stands for 3 multiplied by itself and is termed “three squared.”) Algebra is not limited to expressing specific instances; instead it can make a general statement that covers all possible values that fulfill certain conditions—in this case, the theorem: a² + b² = c².
• This article focuses on classical algebra, which is concerned with solving equations, uses symbols instead of specific numbers, and uses arithmetic operations to establish ways of handling symbols.
• The word algebra is also used, however, to describe various modern, more abstract mathematical topics that also use symbols but not necessarily to represent numbers.
• Mathematicians consider modern algebra a set of objects with rules for connecting or relating them.
• As such, in its most general form, algebra may fairly be described as the language of mathematics.

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