By James L. Peterson and Werner Rheinboldt (Auth.)
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Extra resources for Computer Organization and Assembly Language Programming
Since we are using odd parity, this is an error, and is called a 45 46 1 BASIC COMPUTER ORGANIZATION parity error. Although we know that the character is illegal, we do not know what the original character was. It might have been an " E " , but it could also have been an " I " (01001001) whose bit 3 changed, or an "A" whose parity bit changed. Also, notice that if two bits change, the parity of the result is still correct. Hence, parity bits allow us to detect some, but not all, errors. Hopefully, the more common errors will be detected.
Longer words mean longer waiting for carries to propagate. 3 to 2 microseconds. Addition of two π-bit words may produce a number requiring n + 1 bits for its representation. This is too large to fit into one computer word and is known as overflow. For example, if we were working with a 6-bit computer word, we could represent the numbers from 000000 (base 2) (00 (octal) = 0 (base 10)) up to 111111 (base 2) (77 (octal) = 63 (base 10)). If we add 010110 (base 2) (26 (octal)) to 100101 (base 2) (45 (octal)), we have a sum of 0 1 0 1 1 0 10 0 1 0 1 1 1 1 0 11 (26 (octal)) (45 (octal)) (73 (octal)) which is fine, but if we add 101010 (base 2) (52 (octal)) to 110001 (base 2) (61 (octal)) 10 1 0 1 0 1 1 0 0 0 1 10 1 1 0 11 (52 (octal)) (61 (octal)) (133 (octal)) and we have a carry into a seventh bit position, which does not exist in a six-bit word.
Separating this from the remaining fraction, we can derive the second bit by multiplying by two again, and so on. 44 first bit = next bit = = = = 0 1 1 0 1 = 1 = 1 = 1 = 0 = 1 and so on. 435 (base 10) = 0 . 33333333333333333. This illustrates both the conversion process and that not all decimal numbers can be exactly represented in a finite number of bits. Some nonrepeating decimal fractions are repeating fractions in binary, just as the fraction 1/3 is a repeating fraction in decimal. No matter how many bits we allow in a word to represent a binary fraction, we may not be able to represent some decimal fractions exactly.