Digital Systems and Binary Numbers Mano & Ciletti Chapter 1 By Suleyman TOSUN Ankara University
Digital Systems Binary Numbers Number-Base Conversions Octal and Hexadecimal Numbers Complements Signed Binary Numbers Binary Codes Binary Storage and Registers Binary Logic
Digital computer is the best-known example of a digital system Others are telephone switching exchanges, digital voltmeters, digital calculators, etc. A digital system manipulates discrete elements of information Discrete elements: electric impulses, decimal digits, letters of an alphabet, any other set of meaningful symbols
In a digital system, discrete elements of information are represented by signals Electrical signals (voltages & currents) are the most common Present day systems have only two discrete values (binary) Alternative, many-valued circuits are less reliable A lot of information is already discrete and continuous values can be quantized (sampled)
A digital computer is an interconnection of digital modules
To understand each module, it is necessary to have a basic knowledge of digital systems
7392 represents a quantity that is equal to
Decimal number system is of base (or radix) 10 In binary system, possible values are 0 and 1 and each digit is multiplied by E.g. 11010.11 is
Hexadecimal (base 16) numbers use digits 09 and letters A, B, C, D, E, F to represent values 10-15
Operations work similarly in all bases
Converting a number from base x to decimal is simple (as shown before) Decimal to base x is easier if number is separated into integer and fraction parts Convert 41 to binary
Divide 41 by 2, quotient is 20 and remainder is 1. Continue dividing the quotient until it becomes 0. Remainders give us the binary number as follows:
Conversion of a fraction is similar but the number is multiplied by to instead of dividing
Octal and Hexadecimal Numbers
Conversions between binary, octal and hexadecimal numbers are easier Each octal digit corresponds to 3 binary digits and each hexadecimal digit corresponds to 4 binary digits
the subtraction operation Logical operations
Two types exist
The radix complement (r’s complement)
10’s complement, 2’s complement
The diminished radix complement ((r-1)’s complement)
9’s complement, 1’s complement
Diminished Radix (r-1) complement
Given a number N in base r having n digits:
When r=10, (r-1)’s complement is called 9’s complement. 10n-1 is a number represented by n 9’s. 9’s complement of 546700 is (n=6)
(r-1)’s complement of N is (rn-1)-N
9’s complement of 012398 is (n=6)
For binary numbers, r=2 and r-1=1. 1’s complement of N is (2n-1)-N If n=4, 2n=10000. So, 2n-1=1111. To determine the 1’s complement of a number, subtract each digit from 1. Or, bit flip!!! Replace 0’s with 1’s, 1’s with 0’s!!!
If N= 1011000, 1’s comp.= 0100111 If N= 010110, 1’s comp.= 101001
Radix (r’s) complement
Given a number N in base r having n digits:
When r=10, r’s complement is called 10’s complement. 10’s complement of 546700 is (n=6)
r’s complement of N is rn-N
10’s complement of 012398 is (n=6)
For binary numbers, r=2, 2’s complement of N is 2n-N To determine the 2’s complement of a number, determine 1’s complement and add 1 to it. Example:
If N= 1011001, 1’s comp.= 0100110, 2’s comp.=0100111 If N= 1101100, 2’s comp.= 0010100
Another way of finding 2’s comp.: Leave all least significant 0’s and the first 1 unchanged, bit flip the remaning digits.
Subtraction with Complements Minuend: 101101 Subtrahend: 100111 Difference: 000110 1.
Add the minuend M to r’s complement of the subtrahend N. M + (rn-N) = M - N+rn If M>=N, the sum will produce an end carry. Discard it and what is left is the result M-N. If M
Signed Binary Numbers
Negative numbers is shown with a minus sign in math. In digital systems, the first bit decides the sign of the number.
If the first bit 0, the number is positive. If the first bit 1, the number is negative.
This is called signed magnitude convention.
Signed complement systems
To represent negative number, 1’s complement and 2’s complements are also used.
Represent +9 and -9 in eight bit system
+9 is same for all systems: 00001001 -9
To determine negative number
Signed magnitute: Take the positive number, change the most significant bit to 1 One’s complement: Take the one’s complement of the positive number. Two’s complement: Take the two’s complement of the positive number. (Or add 1 to one’s complement)
Take the 2’s complement of subtrahend. Add it to the minuend. Discard cary if there is any.
Examples: 10-5 (8 bits), -3-5, 18-(-9)
Binary Codes – BCD Codes
n bit can code upto 2n combinations.
Other Decimal Codes
Only one bit changes when going from one number to the next. How to determine the gray code equivalent of a number:
Add 0 to the left of number. XOR every two neigboring pair in order. The result is the gray code.
Example: 1 1 0 0 0 0 -> 0 1 1 0 0 0 0 101000
Error Detecting Codes
Add an extra bit (parity bit) to make the total number of one’s either even or odd.
More than two inputs