Transcript binary calculators

Digital Systems and Binary
Numbers
Mano & Ciletti
Chapter 1
By Suleyman TOSUN Ankara University
Outline
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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
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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
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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
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A digital computer is an interconnection of
digital modules
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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
7 × 103 + 3 × 102 + 9 × 101 + 2 × 100
 Decimal number system is of base (or radix)
10
 In binary system, possible values are 0 and 1
and each digit is multiplied by 2𝑖
 E.g. 11010.11 is
1 × 24 + 1 × 23 + 0 × 22 + 1 × 21 + 0 × 20 + 1
× 2−1 + 1 × 2−2 = 26.75
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Binary Numbers
Hexadecimal (base 16) numbers use digits 09 and letters A, B, C, D, E, F to represent
values 10-15
𝐵65𝐹 16 = 11 × 163 + 6 × 162 + 5 × 16 + 15
= 46687 10
 Operations work similarly in all bases
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Number-Base Conversions
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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
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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
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Conversion of a fraction is similar but the
number is multiplied by to instead of dividing
Octal and Hexadecimal Numbers
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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
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Simplifies
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the subtraction operation
Logical operations
Two types exist
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The radix complement (r’s complement)
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10’s complement, 2’s complement
The diminished radix complement ((r-1)’s
complement)
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9’s complement, 1’s complement
Diminished Radix (r-1) complement
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Given a number N in base r having n digits:
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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)
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(r-1)’s complement of N is (rn-1)-N
999999-546700=453299
9’s complement of 012398 is (n=6)
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999999-012398=987601
1’s complement
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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!!!
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Example:
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If N= 1011000, 1’s comp.= 0100111
If N= 010110, 1’s comp.= 101001
Radix (r’s) complement
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Given a number N in base r having n digits:
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When r=10, r’s complement is called 10’s
complement.
10’s complement of 546700 is (n=6)
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r’s complement of N is rn-N
1000000-546700=453300
10’s complement of 012398 is (n=6)
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1000000-012398=987602
2’s complement
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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:
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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
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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<N, the sum does not produce an end carry and is equal to
rn-(N-M) . To obtain the answer in a familiar form, take the r’s
complement of the sum and place a negative sign in front.
Example
Example
Example
Signed Binary Numbers
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Negative numbers is shown with a minus sign
in math.
In digital systems, the first bit decides the
sign of the number.
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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
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To represent negative number, 1’s
complement and 2’s complements are also
used.
Example
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Represent +9 and -9 in eight bit system
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+9 is same for all systems: 00001001
-9
To determine negative number
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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
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Take the 2’s complement of subtrahend.
Add it to the minuend.
Discard cary if there is any.
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Examples: 10-5 (8 bits), -3-5, 18-(-9)
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Binary Codes – BCD Codes
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n bit can code
upto 2n
combinations.
BCD Addition
Example
Other Decimal Codes
Gray Codes
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Only one bit changes when going from one
number to the next.
How to determine the gray code equivalent of
a number:
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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
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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