Computer Architecture
The function of a computer is to carry out instructions, this occurs within the Central Processing Unit in loop known as a the Fetch-Execute Cycle.
Binary Arithmetic
We can apply the normal rules of arithmetic to binary, just as we can to any other number system – and the only time this can get a little complicated is when considering binary subtraction – or negative binary since strictly the negative sign should not be used.
Number Bases
We are used to working with the concept that there are 10 unique numerical digits, 0 to 9, however due to the boolean nature of logic circuits, computers tend to operate using just the digits 0 and 1, the binary number system.
Circuit Design
Ultimately the point of boolean algebra is to facilitate good circuit design and ultimately to produce electronic equipment. Simplification and conversion to functionally complete sets are just a means to this end. All of the operators we have encountered can exist as logic gates in a circuit, and by combining these gates we can produce useful circuits – or “components”.
Simplification of Boolean Algebra (If Only…)
Some of these algebraic expressions are starting to get quite complicated, but fear not there are tools we can use to simplify them by the boolean equivalent of factorisation.
More Boolean Operators
Using the three primary boolean operators (AND, NOT and OR) it is possible to derive a number of other operators, these are not unique – in the sense that they can be replicated by combining one or more of the primary operators in order to form them. Examples of these derived operators are XOR, NAND and NOR. Any set of operators which can be used to perform any desired operation is known as a functionally complete set, AND/OR/NOT is a functionally complete set since any derived operator can be formed from the base set. NAND and NOR are also individually functionally complete sets as either of them can be used to perform any operation.