Chapter 2
Number Systems,
Operations, and Codes
Prepared by
MIR MUNTASIR HOSSAIN
Lecturer, EEE
Northern University Bangladesh
Chapter Outline
Binary Numbers
Binary Arithmetic
Complements of Binary Numbers
Hexadecimal Numbers
Binary Coded Decimal
The Gray Code
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Binary Numbers
Binary system has only two digits.
The two binary digits (bits) are 1 and 0.
Four bits are required to count from zero
to 15. In general, with n bits you can count
up to a number equal to 2n - 1.
So, Largest decimal number = 2n – 1
For example, with five bits (n = 5) you can
count from zero to thirty-one.
25 - 1 = 32 - 1 = 31
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The Weighting Structure of Binary Numbers
The right-most bit is the LSB (least significant bit) in a binary whole number and
has a weight of 20 = 1.
The weights increase from right to left by a power of two for each bit.
The left-most bit is the MSB (most significant bit); its weight depends on the
size of the binary number.
Fractional numbers can also be represented in binary by placing bits to the right
of the binary point.
The left-most bit is the MSB in a binary fractional number and has a weight of
2-1 = 0.5.
The weight structure of a binary number is
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Binary-to-Decimal Conversion
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Decimal-to-Binary Conversion (Repeated Division-by-2 Method)
EXAMPLE 2–6
Convert the
following decimal
numbers to binary:
(a) 19 (b) 45
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Converting Decimal Fractions to Binary (Repeated Multiplication
by 2 Method)
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Binary Arithmetic
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Additional Example
11000
− 111
10001
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Complements of Binary Numbers
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Hexadecimal Numbers
The hexadecimal number system has a
base of sixteen; that is, it is composed of
16 numeric and alphabetic characters.
It is used primarily as a compact way of
displaying or writing binary numbers
because it is very easy to convert
between binary and hexadecimal.
Also, long binary numbers are difficult to
read and write because it is easy to drop
or transpose a bit.
Hexadecimal is widely used in computer
and microprocessor applications.
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Binary-to-Hexadecimal Conversion
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Hexadecimal-to-Binary Conversion
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Hexadecimal-to-Decimal Conversion
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Another way to convert a hexadecimal number to its decimal equivalent:
For a 4-digit hexadecimal number, the weights are
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Decimal-to-Hexadecimal Conversion
EXAMPLE 2–28
Convert the decimal number 650 to hexadecimal by repeated division by 16.
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Binary Coded Decimal (BCD)
Binary coded decimal (BCD) is a way to express each of the decimal digits with a binary
code.
There are only ten code groups in the BCD system, so it is very easy to convert
between decimal and BCD.
Because we like to read and write in decimal, the BCD code provides an excellent
interface to binary systems.
Examples of such interfaces are keypad inputs and digital readouts.
The 8421 code is a type of BCD (binary coded decimal) code.
The designation 8421 indicates the binary weights of the four bits (23, 22, 21, 20).
The ease of conversion between 8421 code numbers and the familiar decimal numbers
is the main advantage of this code.
Applications
Digital clocks, digital thermometers, digital meters, and other devices with seven
segment displays typically use BCD code to simplify the displaying of decimal numbers.
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The Gray Code
The important feature of the Gray code is that it exhibits only a single bit change
from one code word to the next in sequence.
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Additional Examples
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Reference Book
Digital Fundamentals, 11th Edition
Thomas L. Floyd
©2015 |Pearson |
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