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COMPUTER ARITHMETIC. Binary Coded Decimal. Presented By Chung Wai Chow. Binary Coded Decimal. Introduction: Although binary data is the most efficient storage scheme; every bit pattern represents a unique, valid value. However, some applications may not be desirable to work with binary data. - PowerPoint PPT Presentation
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Binary Coded Decimal
Presented By Chung Wai Chow
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Binary Coded Decimal
Introduction:
Although binary data is the most efficient storage scheme; every bit pattern represents a unique, valid value. However, some applications may not be desirable to work with binary data.
For instance, the internal components of digital clocks keep track of the time in binary. The binary value must be converted to decimal before it can be displayed.
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Binary Coded Decimal
Because a digital clock is preferable to store the value as a series of decimal digits, where each digit is separately represented as its binary equivalent, the most common format used to represent decimal data is called binary coded decimal, or BCD.
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1. The BCD format
2. Algorithms for addition
3. Algorithms for subtraction
4. Algorithms for multiplication
5. Algorithms for division
Explanation of Binary Coded Decimal (BCD):
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1) BCD Numeric Format
Every four bits represent one decimal digit.
Use decimal values
from 0 to 9
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4-bit values above 9 are not used in BCD.
1) BCD Numeric Format
The unused 4-bit values are:
BCD Decimal
1010 10
1011 11
1100 12
1101 13
1110 14
1111 15
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1) BCD Numeric Format
Multi-digit decimal numbers are stored as multiple groups of 4 bits per digit.
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1) BCD Numeric Format
BCD is a signed notation
positive or negative.
For example, +27 as 0(sign) 0010 0111.
-27 as 1(sign) 0010 0111.
BCD does not store negative numbers in two’s complement.
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1) BCD Numeric Format
Values represented
b3b2b1b0 Sign and magnitude 1’s complement 2’s complement
0111 +7 +7 +7
0110 +6 +6 +6
0101 +5 +5 +5
0100 +4 +4 +4
0011 +3 +3 +3
0010 +2 +2 +2
0001 +1 +1 +1
0000 +0 +0 +0
1000 -0 -7 -8
1001 -1 -6 -7
1010 -2 -5 -6
1011 -3 -4 -5
1100 -4 -3 -4
1101 -5 -2 -3
1110 -6 -1 -2
1111 -7 -0 -1
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2) Algorithms for Addition
1100 is not used in BCD.
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2) Algorithms for Addition
Two errors will occurs in a standard binary adder.
1) The result is not a valid BCD digit.
2) A valid BCD digit, but not the correct result.
Solution: You need to add 6 to the result generated by a binary adder.
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2) Algorithms for Addition
A simple example of addition in BCD.
0101
+ 1001
1110
+ 0110
1 0100
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+ 9
Incorrect BCD digit
Add 6
Correct answer
1 4
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2) Algorithms for AdditionA BCD adder
10010101
0001 = 1
0100 = 4
If the result,
S3 S2 S1 S0, is not a valid BCD digit,
the multiplexer causes 6 to be added to the result.
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A simple example of subtraction
3) Algorithms for Subtraction
0111
+ 1101
0100
(+7)
(- 3)
(+4)
0011 is 3, the one’s complement is 1100.
Each of the computations adds 1 to the one’s complement to produce the two’s complement of the number.
1100 + 1 = 1101
The two’s complement of 3 is 1101
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3) Algorithms for Subtraction
The second change has to do with complements.
The nine’s complement in BCD, generated by subtracting the value to be complemented from another value that has all 9S as its digits. Adding one to this value produces the ten’s complement, the negative of the original value.
e.g, the nine’s complement of 631 is
999 – 631 = 368.
368 + 1 = 369 is the ten’s complement
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The ten’s complement plays the subtraction and negation for BCD numbers.
3) Algorithms for Subtraction
Hareware generates the nine’s complement of a single BCD digit.
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Conclusion for addition and subtraction
Using a BCD adder and Nine’s complement generation hardware to compute the addition and the subtraction for signed-magnitude binary numbers
The algorithm for adding and subtracting as below:
PM’1: US XS, CU X + Y
PM1: CU X + Y’ + 1, OVERFLOW 0
PM’2: OVERFLOW C
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The algorithm for adding and subtracting
CZ’PM2: US XS
CZPM2: US 0 C’PM2: US X’S, U U’ + 1
2: FINISH 1
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Example of addition of BCD numbers
USU = XSX + YSY
XSX = +33 = 0 0011 0011
YSY = +25 = 0 0010 0101
PM’1: US 0, CU 0 0101 1000
PM’2: OVERFLOW 0
Result: USU = 0 0101 1000 = +58
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Example of subtraction of BCD numbers
USU = XSX + YSY
XSX = +27 = 0 0010 0111
YSY = -13 = 1 0001 0011
PM1: CU 1 0001 0100, OVERFLOW 0
CZ’PM2: US 0
Result: USU = 0 0001 0100 = +14
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4) Algorithms for Multiplication
1101 Multiplicand M
X 1011 Multiplier Q
1101
1101
0000
1101____
10001111 Product P
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4) Algorithms for Multiplication
Multiplicand
Multiplier
Product
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4) Algorithms for Multiplication
Required to use the BCD adder and nine’s complement circuitry.
In BCD, each digit of the multiplicand may have any value from 0 to 9; each iteration of the loop may have to perform up to nine additions. We must incorporate an inner loop in the algorithm for these multiple additions.
In addition, use decimal shifts right operation (dshr), which shift one BCD digit, or four bits at a time.
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The BCD multiplication algorithm
1: US XS+YS, VS XS+YS, U 0, i n, Cd 0
ZY0’2: CSU CdU + X, Yd0 Yd0 – 1, GOTO 2
ZY02: i i - 1
3: dshr (CdUV), dshr (Y)
Z’3: GOTO 2
ZT3: US 0, VS 0
Z3: FINISH 1
4) Algorithms for Multiplication
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4) Algorithms for Multiplication
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Division can be implemented using either a restoring or a non-restoring algorithm. An inner loop to perform multiple subtractions must be incorporated into the algorithm.
5) Algorithms for Division
10
11 ) 1000
11_
10
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5) Algorithms for Division
A logic circuit arrangement implements the restoring-division technique
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A restoring-division example
Initially 0 0 0 0 0 1 0 0 0
0 0 0 1 1
Shift 0 0 0 0 1 0 0 0
Subtract 1 1 1 0 1
Set q0 1 1 1 1 0
Restore 1 1
0 0 0 0 1 0 0 0 0
Shift 0 0 0 1 0 0 0 0
Subtract 1 1 1 0 1
Set q0 1 1 1 1 1
Restore 1 1
0 0 0 1 0 0 0 0 0
Shift 0 0 1 0 0 0 0 0
Subtract 1 1 1 0 1
Set q0 0 0 0 1 0 0 0 0 1
Shift 0 0 0 1 0 0 0 1
Subtract 1 1 1 0 1
Set q0 1 1 1 1 1
Restore 1 1
0 0 0 1 0 0 0 1 0
remainderQuotient
First cycle
Second cycle
Third cycle
Fourth cycle
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5) Algorithms for Division
The restoring-division algorithm:
S1: DO n times
Shift A and Q left one binary position.
Subtract M from A, placing the answer back in A.
If the sign of A is 1, set q0 to 0 and add M back to A (restore A); otherwise, set q0 to 1.
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5) Algorithms for Division
The non-restoring division algorithm:
S1: Do n times
If the sign of A is 0, shift A and Q left one binary position and subtract M from A; otherwise, shift A and Q left and add M to A.
S2: If the sign of A is 1, add M to A.
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References:
Computer Systems Organization & Architecture, Addison Wesley Longman, Inc., 2001
Introduction to Computer Organization 4th Edition. V.Carl hamacher. 1998
http:// www.sfxavier.ac.uk/computing/bcd/bcd1.htm
http:// www.awl.com/carpinelli
Thank you