Digital Systems Flashcards
positional numbering system
- position of a digit in the number determines that digit’s contribution to the total value of the number
base, r
- also known as the radix
- determined by the position of the digit
radix point
separates the integer part of a number from the fractional part
expansion method
method of calculating an equivalent decimal value from a base-r number
binary number system
- base- 2 number system
- only 2 binary digits (bits)L 0 and 1
- number consist of string of bits
counting in base-r
each time:
- count up one
- increment the least significant digit
carry
if the least significant digit reaches the base, r, it is reset to zero and a carry is generated into the next LSD
Binary bit addition
bits can be added, subtracted, multiplied and divided
only digits 0 and 1 are allowed in the results
octal number system
base-8 number system
alternative to working with long binary numbers
uses digits 0 through 7
(2)^3 = 8 so three binary digits can be represented by a singe octal
hexadecimal number system
- shorter method of representing the value of four binary digits at a time, (2)^4 =16
- requires 16 unique characters
- generally uses capital letter A through F for digits 10 through 15
converting base 10 numbers to base-r
- remainder method used
- repeated division by base, r, until quotient is zero
- base-r number found by taking remainders in reverse order in which they were found
radix (base) complement, R(M)
- given M, an N-bit, base-r argument, R(M) is the number which when added to M results in a sum of r^N
- R(M) depends on the machine being used. N is the maximum number of digits used by the machine to store an integer
diminished radix complement
- also called r - 1 complement
- equal to R(M) - 1
for example, if working in base-10 R(M) is the ten’s complement and one less than this is the nine’s complement
computer representation of negative numbers
- the typical representation of a negative number ( a minus sign) is not possible in a machine
- one N digit, usually the MD is reserved for sign representation
- this reduces the machine’s capacity to represent numbers N -1 bits per number
- it is arbitrary whether the sign bit for negative numbers is 0 or 1, as long as the MSD is different for positive and negative numbers
1’s complement
- obtained by inverting each bit of the number, M
- ideal for forming a negative number,
21 = 0001 0101
-21 = 1110 1010
2’s complement
for an N-bit binary integer, M, the 2’s complement is
R(M) = 2^N - M
An alternative way to calculate this to take the 1’s complement and add 1.
21= 0001 0101 -21= 1110 1011
Boolean algebra
a system for representing and evaluating logical variables (Boolean variable) and their combinations
- logical variables are typically represented by uppercase letters (A, B, and so on)
- in a digital environment, Boolean variables are confined for two values representing true and false, typically 1 and 0, respectively
logic gate
- implements logic operators digitally
- has one or more inputs and a single output
- dot (“bubble”, or small circle) designates use of NOT operation
truth tables
represents results of Boolean function for all combinations of the input variables
order of operations
- defines order of precedence for which digital calculations are performed
- analogous to the order of operations for mathematical operations
binary digit
also known as bit, or scalar
binary scalar, B
- typically takes on value of either 0 or 1
- represents one bit of information
- Boolean states are associated with bit values
n-tuple
- grouping of n binary scalars
- represented by [ A1, A2, A3, .. An]
- value of n is called word length
- total number of values that can be represented is N =2^N
cell
- smallest information storage unit
- often referred to as a word or byte