ECE 103
DIGITAL LOGIC DESIGN
CHAPTER I
NUMBER SYSTEMS AND
CODES
Reference: M. Morris Mano & Michael D. Ciletti, "Digital Design", Fourth
Edition, Prentice Hall of India Pvt. Ltd., Chapter – Chapter -1
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Number systems and codes
Representation of numbers
Decimal - Octal - Hexadecimal number
systems
Representation of negative numbers
Complement of a number
Binary arithmetic
Binary codes for decimal numbers
Error detecting and correcting codes
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Representation of numbers
A number in base-r has coefficients multiplied
by powers of r and is of the form
an r n  an1r n1  ..... a1r1  a0r 0  a1r 1  a2r 2  .... amr m
Range of aj is from 0 to r-1
r is also called radix of the number system
If r = 2, binary number system
If r = 8, octal number system
If r = 16, hexadecimal number system
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Number base conversions
To convert a number in base r to decimal is
done by expanding the number in a power
series and adding all the terms
If the number includes a radix point, it is
necessary to separate the number k into an
integer part and a fraction part.
Decimal number is converted to number in base
r by dividing the number and all successive
quotients by r and accumulating the remainders.
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Number base conversions
Example 1: Convert decimal 153 to octal
Example 2: Convert (0.6875)10 to binary
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Complement of a number
Used for
Simplifying subtraction
Logical manipulation
Two types
Radix complement
Diminished radix complement
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Complement of a number
Diminished Radix complement
Given a number N in base r having n digits, its
diminished radix complement is (r n 1)  N
Radix complement
Given a number N in base r having n digits, its
n
((
r
 1)  N )  1
diminished radix complement
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Representation of negative
numbers
Two ways of representation
Sign magnitude form
Sign complement form
signed 1’s complement form
signed 2’s complement form
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Sign magnitude form
The number consists of magnitude bits
and a sign bit
Used in ordinary arithmetic
It is simple
Drawbacks:
Hardware limitations
Two representations of zero
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Signed complement form
A negative number is represented by its
complement
positive numbers always start with 0 in the
leftmost position.
The complement will always start with a 1,
indicating a negative number.
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Representation of negative
numbers
Eg: Represent -9 using 8 bits in both Sign
magnitude form and sign complement
form
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Binary arithmetic
Addition
Subtraction
Multiplication
Addition
Similar to normal decimal addition
Rules of addition:
1 + 1 = 0 CY = 1
1+0=0+1=1
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Binary subtraction
The subtraction of two n-digit unsigned numbers
M - N in base r can be done as follows:
1. Add the minuend M to the r's complement of the
subtrahend N. Mathematically, M + (rn - N) = M N + rn
2. If M >= N. the sum will produce an end carry rn
which can be discarded. what is left is the result
M - N.
3. If M < N. the sum does not produce an end carry
and is equal to rn - (N - M) which is r's
complement of (N - M).
4. To obtain the answer, take the r's complement of
the sum and place a negative sign in front.
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Binary subtraction
Eg: Perform the subtraction a) X – Y and b) Y – X if X = 1010100 and Y
= 1000011 using two’s complement form.
There is no end carry so the answer is – (two’s complement of 1101111)
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Binary multiplication
Just like normal decimal multiplication
Eg: Find (1 0 1)2 × (1 1 0)2
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Binary codes for decimal numbers
A binary number of n digits gives 2n
distinct combinations which can be used to
represent distinct group of quantities
Different binary codes available are
weighted codes, un-weighted codes, selfcomplementing codes, reflecting codes
Weighted codes: Each bit position is
assigned a weighing factor and each digit
is evaluated by adding the weights of all
the ones in the coded combination
Eg: BCD code, 2-4-2-1 code, (8, 4,-2,-1)
code, etc
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Binary codes for decimal numbers
Un-weighted codes: Weight is not
assigned to the bit positions
Eg: Excess-3 code
Self complementing code: 9’s complement
of the decimal number is obtained by
changing 1’s to 0’s and viceversa.
Eg: 2-4-2-1 code, Excess-3 code
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BCD code
Decimal numbers 0 – 9 can be
represented using 4 bits.
There are 6 unused combinations in this
coding scheme.
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Different binary codes for decimal
numbers
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Gray Code
Reflection code
Advantage of Gray
code over the straight
binary number
sequence is that only
one bit in the code
group changes in
going from one
number to the next.
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ASCII Character code
The “American Standard Code for
Information Interchange“ ASCII was
suggested in 1968
Represents alphanumeric character set.
Uses 7 bits to represent 128 characters
There are special symbols which can be
represented by this code
The coding is given in next slide
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Error detection and correction
 Error detection and correction code
 An error-correcting code generates multiple parity check
bits that are stored with the data word in memory. Each
check bit is a parity over a group of bits in the data word
 When the word is read back from memory, the associated
parity bits are also read back and compared with a new set
of check bits generated from the data that have been read
lf the check bits are correct, no error has occurred.
 If the check bits do not match the stored parity, they
generate a unique pattern, called a syndrome, that can be
used to identify the bit that is in error.
 A single error occurs when a bit changes in value from 1 to
0 or from 0 to 1 during the write or read operation.
 If the specific bit in the error is misidentified, then the error
can be corrected by complementing the erroneous bit.
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Parity
Simplest form of error detection is
achieved by using parity bits.
A parity bit is an extra bit included with a
message to make the total number of 1's
either even or odd.
Eg:
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Hamming code
k parity bits are added to an n-bit data
word forming a new word of n + k bits.
The bit positions are numbered in
sequence from 1 to n + k.
The relation between the number of
message bits and parity bits is
Those positions numbered as a power of 2
are reserved for the parity bits
The remaining bits are the data bits
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Hamming code
Construction of hamming code for
11000100
Calculating parity bits
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Hamming code
Message bit sequence
Calculation of check bits
Message sequences with no error, error in bit 1 and error in bit 5
Calculation of check bits for the above message sequences
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Assignments Problems…
 1. Convert the following binary numbers in decimal: 101110; 1110101; and
110110100.
 2. Convert the following decimal numbers to the bases indicated.
a. 7562 to octal
b. 1938 to hexadecimal
c. 175 to binary
 3. Show the value of all bits of a 12-bit register that hold the number
equivalent to
decimal 215 in (a) binary; (b) octal; (c) hexadecimal; (d) binary-coded
decimal (BCD).
 4. Show the following operations using 2s complement:
a. 10000111 – 1011001
b. 1011001 – 10000111
c. 0.1001 – 0.0101
d. 0.0101 – 0.1001
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Assignments Problems…
 5. Convert the following two decimal numbers to binary, octal, and
hexadecimal numbers.
i) 174.25 ii) 250.8
 6. Convert the following two unsigned binary numbers to octal,
hexadecimal, and decimal numbers.
i) 10101.11 ii) 10110110.001
 7. Show how a 16‐bit computer using a two’s complement number
system would perform the following computations.
(i) (16850)10 + (2925)10 = (?)10
ii) (16850)10 ‐ (2925)10 = (?)10
iii) (2925)10 ‐ (16850)10 = (?)10
iv) ‐(16850)10 ‐ (2925)10 = (?)10
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Assignments Problems…
 8. Do the following non-textbook problems:
a. Obtain the 1’s and 2’s complements of the following unsigned
binary numbers: 10001000, 10011001, 10101100, 00000000, and
10000000.
b. Perform the indicated subtraction with the following unsigned
binary numbers by taking the 2’s complement of the subtrahend:
a) 11011 – 10000
b) 10110 – 1011
c) 100 – 101000
d) 1011100 – 1011100
Note: You must choose a size for your 2’s complement numbers.
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Assignments Problems…
c. The following binary numbers are 6-bit
2’s complement numbers. Perform the
indicated arithmetic operations and
verify the answers.
a) 101111 + 111011
b) 001011 + 100010
c) 110001 – 001110
d) 101010 – 110111
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Assignments Problems…
For more problems :
Refer : M. Morris Mano, "Digital Design",
3rd Edition, Prentice Hall of India Pvt.
Ltd., Chapter Pages()
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