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 SUMMARY Now that you've completed this chapter, you should have a basic understanding of number systems. The number systems that were dealt with are used extensively in the microprocessor and computer fields. The following is a summary of the emphasized terms and points found in the "Number Systems" chapter. The UNIT represents a single object. A NUMBER is a symbol used to represent one or more units. The RADIX is the base of a positional number system. It is equal to the number of symbols used in that number system. A POSITIONAL NOTATION is a system in which the value or magnitude of a number is defined not only by its digits or symbol value, but also by its position. Each position represents a power of the radix, or base, and is ranked in ascending or descending order. The MOST SIGNIFICANT DIGIT (MSD) is a digit within a number (whole or fractional) that has the largest effect (weighing power) on that number. The LEAST SIGNIFICANT DIGIT (LSD) is a digit within a number (whole or fractional) that has the least effect (weighting power) on that number. The BINARY NUMBER SYSTEM is a base 2 system. The symbols 1 and 0 can be used to represent the state of electrical/electronic devices. A binary 1 may indicate the device is active; a 0 may indicate the device is inactive. The OCTAL NUMBER SYSTEM is a base 8 system and is quite useful as a tool in the conversion of binary numbers. This system works because 8 is an integral power of 2; that is, 23 = 8. The use of octal numbers reduces the number of digits required to represent the binary equivalent of a decimal number. The HEX NUMBER SYSTEM is a base 16 system and is sometimes used in computer systems. A binary number can be converted directly to a base 16 number if the binary number is first broken into groups of four digits. The basic rules of ADDITION apply to each of the number systems. Each system becomes unique when carries are produced. SUBTRACTION in each system is based on certain rules of that number system. The borrow varies in magnitude according to the number system in use. In most computers, subtraction is accomplished by using the complement (R's or R's-1) of the subtrahend and adding it to the minuend. To CONVERT A WHOLE BASE 10 NUMBER to another system, divide the decimal number by the base of the number system to which you are converting. Continue dividing the quotient of the previous division until it can no longer be done. Extract the remainders - the remainder from the first computation will yield the LSD; the last will provide the MSD. To CONVERT DECIMAL FRACTIONS, multiply the fraction by the base of the desired number system. Extract those digits that move to the left of the radix point. Continue to multiply the fractional product for as many places as needed. The first digit left of the radix point will be the MSD, and the last will be the LSD. The example to the right shows the process of converting 248.3210 to the octal equivalent (370.2438). BINARY numbers are converted to OCTAL and HEX by the grouping method. Three binary digits equal one octal digit; four binary digits equal one hex digit. To CONVERT binary, octal, and hex numbers to DECIMAL use the POWERS of the base being converted. BINARY-CODED DECIMAL (BCD) is a coding system used with some microprocessors. A correction factor is needed to correct invalid numbers