Boolean algebra is a form of mathematics developed by English mathematician George Boole (1815–1864). Boole created a system by which certain logical statements can be expressed in mathematical terms. The consequences of those statements can then be discovered by performing mathematical operations on the symbols.
As a simple example, consider the following two statements: "I will be home today" and "I will be home tomorrow." Then let the first statement be represented by the symbol P and the second statement be represented by the symbol Q. The rules of Boolean algebra can be used to find out the consequences of various combinations of these two propositions, P and Q.
In general, the two statements can be combined in one of two ways:
P or Q: I will be home today OR I will be home tomorrow.
P and Q: I will be home today AND I will be home tomorrow.
Now the question that can be asked is what conclusions can one draw if P and Q are either true (T) or false (F). For example, what conclusion can be drawn if P and Q are both true? In that case, the combination P or Q is also true. That is, if the statement "I will be home today" (P) is true, and the statement "I will be home tomorrow" (Q) is also true, then the combined statement, "I will be home today OR I will be home tomorrow" (P or Q) must also be true.
By comparison, suppose that P is true and Q is false. That is, the statement "I will be home today" (P) is true, but the statement "I will be home tomorrow" (Q) is false. Then the combined statement "I will be home today OR I will be home tomorrow" (P or Q) must be false.
Most problems in Boolean algebra are far more complicated than this simple example. Over time, mathematicians have developed sophisticated mathematical techniques for analyzing very complex logical statements.
Two things about Boolean algebra make it a very important form of mathematics for practical applications. First, statements expressed in everyday language (such as "I will be home today") can be converted into mathematical expressions, such as letters and numbers. Second, those symbols generally have only one of two values. The statements above (P and Q), for example, are either true or false. That means they can be expressed in a binary system: true or false; yes or no; 0 or 1.
Binary mathematics is the number system most often used with computers. Computer systems consist of magnetic cores that can be switched on or switched off. The numbers 0 and 1 are used to represent the two possible states of a magnetic core. Boolean statements can be represented, then, by the numbers 0 and 1 and also by electrical systems that are either on or off. As a result, when engineers design circuitry for personal computers, pocket calculators, compact disc players, cellular telephones, and a host of other electronic products, they apply the principles of Boolean algebra.
[ See also Algebra ; Computer, digital ]