Digital Fundamentals - Helsinki

Digital Fundamentals

Number systems,

Operations, and codes

Objectives

•Review the decimal number system

•Count in the binary number system

•Convert from decimal to binary and from binary to decimal

•Apply arithmetic operations to binary numbers

•Determine the 1's and 2's complements of a binary number

•Express signed binary numbers in sign-magnitude, 1's complement, 2's complement, and floating-point format

•Carry out arithmetic operations with signed binary numbers

•Convert between the binary and hexadecimal number systems

Number systems, operations, and codes

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•Convert between the binary and hexadecimal number systems

•Add numbers in hexadecimal form

•Convert between the binary and octal number systems

•Express decimal numbers in binary coded decimal (BCD) form

•Add BCD numbers

•Convert between the binary system and the Gray code

•Interpret the American Standard Code for Information Interchange (ASCII)

•Use binary numbers and codes in a system application

Binary numbers


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Application example

How can you detect a passing tennis ball?


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Binary weights


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Repeated division-by-2 method


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Decimal fractions to binary


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Binary arithmetic – addition (ADD) & subtraction (SUB)


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Binary arithmetic – multiplication (MUL) & division (DIV)


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2’S complements of binary numbers


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Signed numbers


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Conversion from signed binary to decimal


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2’s complement to decimal


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Range of signed interger numbers that can be represented

N2

)12()2( 11 −+− −− NN tofrom

total combinations

2’s complement reduces the maximum absolute value to approximately half


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with 8 bits the range is –128 to +128with 16 bits the range is –32768 to +32767

Floating point numbers in binary format

single-precision


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The exponent is expressed with a bias of 127, exponent has thus the

range of –126 to +128


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Arithmeticoperationswith signednumbers


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Subtraction


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Multiplication


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Division


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Hexadecimal numbers


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binary to hexadecimal


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hexadecimal to binary


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hexadecimal to decimal


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hexadecimal to decimal


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decimal to hexadecimal


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hexadecimal addition


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hexadecimal subtraction

Method 1. Convert the hexadecimalnumber to binary. Take the 2’s comp-lement of the binary number. Convertthe result to hexadecimal.


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Method 2. Subtract the hexadecimalnumber from the maximum hexadeci-mal number and add 1.

Method 3. Write the sequence of single hexadecimal digits. Write the sequence inreverse below the forward sequence. The 1’s complement of each hex digit is thedigit directly below it. Add 1 to the resulting number to get the 2’s complement.


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Hexadecimal subtraction


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octal-to-decimal conversion


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decimal-to-octal conversion


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binary-to-octal


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decimal-to-BCD


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BCD-to-decimal


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BCD addition


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binary-to-gray


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gray-code application example


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ASCII


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ASCII control characters


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Extended ASCII characters


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Parity method for error detection


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Digital system application


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Date post:	24-Jan-2022
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Digital Fundamentals - Helsinki

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