Digital Systems and Binary Numbers

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Digital Systems and Binary Numbers Mano & Ciletti Chapter 1 By Suleyman TOSUN Ankara University

Outline         

Digital Systems Binary Numbers Number-Base Conversions Octal and Hexadecimal Numbers Complements Signed Binary Numbers Binary Codes Binary Storage and Registers Binary Logic

Digital Systems 







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

Digital Systems 









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)

Digital Systems

Digital Systems



A digital computer is an interconnection of digital modules



To understand each module, it is necessary to have a basic knowledge of digital systems

Binary Numbers 

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





Binary Numbers 

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

Number-Base Conversions 





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:

Number-Base Conversions

Number-Base Conversions 

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

Complements 

Simplifies  



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

999999-546700=453299

9’s complement of 012398 is (n=6) 

999999-012398=987601

1’s complement



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!!!



Example:

   

 

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

1000000-546700=453300

10’s complement of 012398 is (n=6) 

1000000-012398=987602

2’s complement 





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.

2.

3.

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
Example

Example

Example

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.

Example 

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)

Arithmetic Addition

Aritmetic subtraction



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.

BCD Addition

Example

Other Decimal Codes

Gray 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.

Binary logic

Truth tables

Gate sysbols

Timing diagrams

More than two inputs

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Digital Systems and Binary Numbers

Digital Systems and Binary Numbers Mano & Ciletti Chapter 1 By Suleyman TOSUN Ankara University Outline          Digital Systems Binary Nu...

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