1. 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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Symbols And Special Terms
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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. See also Mathematical 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 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: (y)2 = ‚.
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.
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Order of Operations and Grouping
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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,
cannot be written ax + b/c – dy. The second notation indicates that b should be divided by c before b is added to ax. Grouping symbols can be used to correctly represent the fraction when using a slash: ( ax + b)/( c - dy).
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