Transcript Chapter 2 - MyWeb at WIT

Digital Logic Chapter 2
Number Conversions
Digital Systems by Tocci
Binary  Decimal

Convert a binary number, 1001012, to a
decimal number by summing the positional
weights that contain “1”.
1
1
0
0
0
1
1* 2  0 * 2  0 * 2  1* 2  0 * 2  1* 2
5
4
3
2
1
0
32 + 0 + 0 + 4 + 0 + 1 = 3710
How about Decimal  Binary?
Decimal  Binary

Use repeated division:




Divide the decimal number by 2. The remainder of
this division is the LSB
Continue dividing the results, adding the remainders
to the left of the LSB until a quotient of zero is
obtained.
The last division is always two into 1 giving a result
of 0 with a remainder of 1. This 1 is the MSB.
Additional zeros can be added padding the binary
number so the total digits are some multiple of 8.
Repeated Division: Example 1
Until a quotient of “0” is obtained
Repeated Division: Example 2
Repeated Division: Flow Chart
•
Similar procedure
can be used to
convert from decimal
to other number
systems.
Check your solutions by converting back
to decimal.
Hexadecimal Number System


Hexadecimal number system uses base-16
The characters used in hex are:



Digits 0~9
Letters A, B, C, D, E, F
The digit positions are weighted as powers of 16, rather
than as powers of 10 as in the decimal system
Counting in Hex


Why Hexadecimal?
It is useful to represent long strings of bits.
Each character in hex can represent 4 bits
reducing the length of a number to a quarter
of the original size. It makes binary numbers
more “readable”.
Counting in hex restarts at zero and produces
a carry after the count reaches F in order to
increment to the next value.
Hex  Decimal Conversion
Multiplying each hex digit by its positional weight.
Example:
16316  1 (162 )  6  (161 )  3  (160 )
 1 256  6 16  31
 35510
Decimal  Hex Conversion



Remember the repeated division?
Divide the decimal number by 16
The 1st remainder is the LSB and the last is the MSB.


Note, when done on a calculator, a decimal remainder
can be multiplied by 16 to get the result.
If the remainder is greater than 9, the letters A~F are
used.
Until a quotient of “0” is obtained
Decimal  Hex Conversion
Hex  Binary Conversion

Hex  Binary:
Each Hex digit is converted to its four-bit binary equivalent
9F216 = 9
F
2
1001 1111 0010 = 1001111100102

Binary  Hex:




Convert from binary to hex by grouping bits in four starting with the LSB.
Each group is then converted to the hex equivalent
Leading zeros can be added to the left of the MSB to fill out the last
group.
Note the addition of leading zeroes
Example: 11101001102 = 0011 1010 0110
=
3
A
6
= 3A616
http://www.learn-programming.za.net/articles_decbinhexoct.html
Conversion among Decimal, Binary, Hex
Decimal
Binary
Hexadecimal
How to do all the conversions
?
BCD Code



Binary Coded Decimal (BCD) is another way
to present decimal numbers in binary form.
BCD is widely used and combines features of
both decimal and binary systems.
Each BCD digit is converted to a binary
equivalent.
Decimal  BCD

To convert the number 87410 to BCD:
8
7
4
0100 0111 0100 = 010001110100BCD

Each decimal digit is represented using 4 bits.
Each 4-bit group can never be greater than 9.
Reverse the process to convert BCD to decimal


BCD




BCD is NOT a number system.
BCD is a decimal number with each digit
encoded to its binary equivalent.
The primary advantage of BCD: easy to convert
to and from binary.
A BCD number is NOT the same as a straight
binary number.
BCD Review Questions
Is “1001 1011 0101” a valid BCD?
BYTE, Nibble, WORD

Byte:



Most microcomputers handle and store binary
data in groups of 8 bits.
So, special name is given to a string of 8 bits,
called a byte.
Two common questions:


How many bytes in a 32-bit string (a string of
32 bits)?
What is the largest decimal number that can
be represented in binary using two bytes?
BYTE, Nibble, WORD

Byte = 8 bits
Nibble = 4 bits

Word:




Word size in a simple system may be one byte (8
bits)
Word size in a PC is 8 bytes (64 bits)
Word size is specific to particular machines.
Alphanumeric Codes – ASCII Code


Represents characters and functions found on a
computer keyboard.
ASCII – American Standard Code for Information
Interchange.



Seven bit code: 27 = 128 possible code groups
Table 2-4 lists the standard ASCII codes
Applications:
 To transfer information between computers,
between computers and printers, and for internal
storage.
Parity Method for Error Detection

Binary data and codes are frequently moved
between locations.
For example:





Digitized voice over a microwave link.
Storage and retrieval of data from hard disks.
Communication between computer systems over
telephone lines using a modem.
Electrical noise can cause errors during
transmission.
Many digital systems employ methods for error
detection (and sometimes correction).
Parity Method for Error Detection




The parity method of error detection requires
the addition of an extra bit to a code group.
This extra bit is called the parity bit.
The bit can be either a 0 or 1, depending on the
number of 1s in the code group.
There are two methods: even and odd.
Parity Method for Error Detection

Even Parity Method:
The total number of “1”s in a group, including the
parity bit, must add up to an even number.
 The binary group 1 0 1 1 would require the
addition of a parity bit 1 1 0 1 1


The parity bit may be added at either end of a group.
Odd Parity Method:
The total number of “1”s in a group, including the
parity bit, must add up to an odd number.
Parity Method for Error Detection



The transmitter and receiver must “agree” on
the type of parity-checking being used.
Two bit errors would not indicate a parity
error.
Both odd and even parity methods are used,
but even seems to be used more often.
Schematic for Even Parity Generator