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Line 31: − − Just as in traditional [[algebra]], [[exponent]]s can be represented as a series of multiplications in boolean algebra, but the results are different. A<sup>n</sup> can be represented as the value A multiplied n times, or A•A•A•A... = C. Boolean multiplication returns a "true" (or a 1) if all values are "true" (or 1), but in this case, all values are always the same. So if A = 0, C = 0, and if A = 1, C = 1. This means that A<sup>n</sup> simply reduces to A! Line 31:
Line 41: − The [[order of operations]] for boolean algebra is the same as that for traditional algebra, except that there are fewer functions for boolean algebra: parenthesis are evaluated first, followed by multiplication then addition. Bars over multiple variables are treated at the same level as parenthesis.+
Rewrite intro, explaining what's going on with computers.
'''Boolean algebra''' is a division of mathematics which deals with boolean variables, which are variables which can only have two values. Typically these values are 0 and 1, especially within the fields of [[electronics]] or [[computers]], however boolean variables can have other values, such as "true" and "false". Boolean algebra was invented in the nineteenth century by an [[English]] mathematician named George Boole.
'''Boolean algebra''' or '''[[boolean logic]]''' is the formal mathematical discipline that deals with "truth values"—"true" or "false". Its fundamental operations are "and", "or" and "not". One can write "propositions" (equations) of boolean algebra, such as
P = (Q+R)•(T')
and manipulate them the way one would manipulate ordinary algebraic equations.
Boolean algebra as a formal mathematical study was pioneered by (and is named after) [[English]] mathematician [[George Boole]] in the 1830's.
The terms "boolean algebra" and "boolean logic" are used interchangeably. The words are not capitalized (except at the beginning of a sentence, of course) even though they are named after a person.
==Boolean Algebra and Computers==
The thing that elevates boolean algebra from a somewhat obscure branch of mathematics to one of the driving forces of modern society is that it is the basis for computers. Computers do everything in boolean logic. All numbers are represented internally in [[binary]] (base 2) notation, with digits ("bits") 1 and 0, corresponding to "true" and "false", respectively. Computers are then designed in terms of boolean algebra equations. For example, addition is performed by 32 copies of these equations:
S = (A•B•C) + (A•B'•C') + (A'•B•C') + (A'•B'•C)
D = (A•B) + (A•C) + (B•C)
A modern computer processing chip has tens of millions of transistors, all calculating boolean operations.
==Three Basic Operations==
==Three Basic Operations==
The three basic operations of boolean algebra are AND ([[multiplication]]), OR ([[addition]]), and NOT. See the table below for a numerical description. From these functions, the other functions of boolean algebra can be derived.
The three basic operations of boolean algebra are AND (analogous to [[multiplication]]), OR (analogous to [[addition]]), and NOT. See the table below for a numerical description. From these functions, the other functions of boolean algebra can be derived. In the following descriptions, we will use the 1-and-0 notation rather than the true-and-false notation.
{| class="wikitable" align="left"
{| class="wikitable" align="left"
===AND===
===AND===
AND is the boolean equivalent of multiplication. The product of two numbers are non-zero if both numbers are non-zero. In words, the AND function states: "If A AND B are true then C is true". The AND function is commutative, so it results in the same answer no matter what order the values are in. For example, A • B = B • A, and A • (B • C) = (A • B) • C.
AND is the boolean equivalent of multiplication. The product of two numbers are non-zero if both numbers are non-zero. In words, the AND function states: "If A AND B are true then C is true". The AND function is commutative, so it results in the same answer no matter what order the values are in. For example, A • B = B • A, and A • (B • C) = (A • B) • C.
===OR===
===OR===
===Order of Operations===
===Order of Operations===
By convention, the [[order of operations]] (sometimes called "operator precedence") for boolean algebra is the same as that for traditional algebra, except that there are fewer functions for boolean algebra: parenthesis are evaluated first, followed by multiplication then addition. Bars over multiple variables are treated at the same level as parenthesis. In practice, considerations of operation order are never a problem.
==Derived Functions==
==Derived Functions==