Transcript Representation of numbers

Representation
of
Numbers and Arithmetic
and Logical Operation
on them
Sahar Mosleh
California State University San Marcos
Page
1
Representation of Numbers
• In general, a number expressed in base r is written as
(….. a3, a2, a1, a0)r
• Where the ai are the symbols of the number system.
• The base r tells us how many different symbols the ai can have
• For example:
• r = 10
•r=2
•r=8
• r = 16
Sahar Mosleh
 ai
 ai
 ai
 ai
e {0, 1, 2,3,4, 5,6 7, 8,9}
e {0, 1}
e {0, 1, 2,3,4, 5,6 7}
e {0, 1, 2,3,4, 5,6 7, 8,9, A, B, C, D, E, F}
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• The right most symbol of a0 is called the least significant digit
(LSD) While the left most symbol of an-1 is called most significant
digit (MSD)
MSD
• Example:
LSD
1000110010
Sahar Mosleh
California State University San Marcos
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3
Value of Numbers
• In digital systems, the determination of value of signed numbers
(positive and negative numbers) is different than unsigned
numbers (just positive numbers)
Value of Unsigned Numbers:
• The value of unsigned number
• Value {(….a3a2a1a0)r} = …. + a3r3 + a2r2+a1r1+a0r0
• In which ai is the ith coefficient and
• r is the base
• Example:
• (15)10 = 1*101 + 5*100 = 10 + 5
• (1010)2 = 1*23 + 0*22 + 1*21 + 0*20 = 8+2=(10)10
• (B7)16 = B*161 + 7*160 = 11*16 + 7 = (183)10
Sahar Mosleh
California State University San Marcos
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4
Binary to Decimal Conversion (Unsigned
Numbers)
Example:
(0001 1001)2 = (….b2b1b0)10 ?
0001 1001 = 1*24 + 1* 23 + 0*22 + 0*21 + 1*20 = (25)10
Example:
(1101 0111 )2 = (….b2b1b0)10 ?
1101 0111 = 1*27 + 1* 26 + 0*25 + 1*24 + 0*23 + 1* 22 + 1*21 + 1*20
= (215)10
Sahar Mosleh
California State University San Marcos
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5
Decimal To Binary Conversion (Unsigned
Numbers)
• To convert from base 10 (decimal to binary) we need to divide the
number in base 10 which is (N)10 by 2 several times until division
by 2 is no longer possible. (dividend becomes 0)
• Example:
• (121) 10 = ( … b3b2b1b0)2
• (121) 10 = (1111001)2
Number Dividend
Remainder
121 / 2
60
1
60 /2
30
0
30 /2
15
0
7
1
3
1
1
1
0
1
15 /2
1) Given the number in base 10
7 /2
(a3a2a1a0)10
3 /2
2) Divide the number by 2 several
times until dividend becomes 0 1 / 2
3) write the remainders from bottom to top
4) This is the representation of (N)10 in (N)2
Sahar Mosleh
California State University San Marcos
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6
Hex To Binary Conversion (Unsigned Numbers)
• A Hex representation of any number can be converted to its
binary representation directly by determining the 4 bit binary
representation of each hex digit and concatenating the 4 bit
binary word in which the hex digit appears
• For (ABC)16 = (….b3b2b1b0)2
(A)16 = (1010) 2
(B)16 = (1011) 2
(C) 16 = (1100) 2
• Therefore, (ABC)16 = (1010 1011 1100)2
Sahar Mosleh
California State University San Marcos
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• The binary representation of any number can be converted to
the hex representation by partitioning the binary number into
groups of 4 digits starting from LSD (the rightmost digit) to
left
• Example:
(01 1011)2 = (…..b3b2b1b0)16
(1B)16
• Example:
• (1 1011 1101 1111 0001 1111 0101)2 = (…..b3b2b1b0)16
1BDF1F5
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= (1BDF1F5) 16
California State University San Marcos
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Reasons for Using Hex Numbers
• Computer and digital systems operate on binary numbers.
• Binary numbers {0,1} constitute the building blocks of computer
machine language
• Hex representation of a binary number is easier and more
convenient for humans to communicate
• Converting between hex and binary representation of the number
is easy
Sahar Mosleh
California State University San Marcos
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9
Bitwise Addition of Binary Numbers (Unsigned
Numbers)
• Example:
• Add binary number
A = 11101100 and
B = 01110011
• Do the addition one column at a time
1 1
A
B
Result
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11101100
+ 01110011
--------------101011111
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Signed Numbers and Signed Arithmetic
• There are only two symbols in the binary number system, digital
systems do not know of the existence of negative and positive
numbers.
• They only know magnitude or positive binary numbers
• In digital world, there are no symbol for ‘+’ or ‘-’
• However, we require positive and negative binary numbers to be
processed by the computer.
• When using the signed number system, we will interpret the string
of binary bits differently
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California State University San Marcos
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• Let us construct a system containing one half positive and other
half negative numbers
• Let the MSB (most significant bit) denote the sign of the binary
number where if the
• MSB = 0  positive number
• MSB = 1  negative number
MSB &
Sign bit
LSB
01111111
• Note that the sign bit does not contribute to the magnitude
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California State University San Marcos
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8-bit Representation of Signed Binary
Number
0000
0000
0000
.
.
.
0111
0111
0111
1000
1000
1000
.
.
.
1111
1111
1111
0000 0
0001
0010
1101
1110
1111 127
0000 -128
0001
0010
1101
1110
1111 -1
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• Range of positive numbers:
• 0 through 2(8-1) -1 = 127
• In general, for n-bit binary words
0 through 2(n-1) -1
• Range of the negative numbers:
• -1 through -2(8-1) = -128
• In general, for n-bit binary words
-1 through -2(n-1)
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1111 1111
1111 1110 -2
-1
0000 0000
0000 0001
0
1
2 0000 0010
3 0000 0011
1111 1101 -3
.
.
[128 negative numbers]
.
[128 positive numbers]
.
Signed
Binary/
Decimal
Number
Wheel
.
.
1000 0011 -125
125 0111 1101
1000 0010 -126
1000 0001
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-127
126 0111 1110
127
-128
0111 1111
1000 0000
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2’s Complement Representation of Signed
Numbers
• We now have two methods for representing and interpreting
strings of binary numbers
1) When it is sufficient to consider only positive numbers, we
use the unsigned interpretation as discussed earlier
2) When we want to consider both positive and negative
numbers, we use the signed number wheel
• The interpretation of the numbers in the signed number wheel is
called the 2’s complement representation.
• This wheel shows us the binary equivalent of signed decimal
numbers, or, the decimal equivalent of signed binary numbers.
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California State University San Marcos
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How to Obtain the Signed Binary Number from
a Signed Small Decimal Number ?
• If the signed decimal number is positive,
• Convert the decimal number to binary using the unsigned
conversion process as discussed earlier
• If the signed decimal number is negative, then
1) Determine the binary number representation of the absolute
value of signed decimal number
2) Take the 1’s complement of the number (flipping every
single bit: 0 to 1 and 1 to 0)
3) Add 1 to it
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California State University San Marcos
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Converting a Positive Decimal Number to the
2’s Complement Representation:
• Example: Represent (+18)10 as a signed 8-bit binary number
• Using unsigned number conversion process
Number Dividend Remainder
18 / 2
9
0
9 /2
4
1
4 /2
2
0
2 /2
1
0
1/2
0
1
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The signed conversion is:
(+18)10 = (00010010 )2
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Converting a Negative Decimal Number to the
2’s Complement Representation:
• Represent (-18)10 as a signed 8-bit binary number
• Solution:
• Abs(-18)10 :
0001 0010
1
• 1’s Complement:
• Add 1:
• Result:
1110 1101
+ 0000 0001
1110 1110
• Therefore, (-18)10 = (1110 1110) 2
Sahar Mosleh
California State University San Marcos
Page 18
How to Obtain the Signed Decimal Number of a
Signed Binary Number
• If the signed binary number is positive (i.e. MSB = 0), convert the
binary number to decimal using the unsigned conversion process
as discussed earlier
• If the signed binary number is negative (i.e., MSB = 1)
1) Take the 1’s complement
2) Add 1
3) Convert to decimal using unsigned process
4) Put a negative sign in front of the decimal number
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California State University San Marcos
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Converting a Positive 2’s Complement Binary
Number to Decimal Representation:
• Example: Represent (0001 1000)2 as a signed decimal number
• Since MSB = 0 then we can use unsigned conversion
(0001 1000)2 = 1*24 + 1*23 + 0*22 + 0*21 + 0*20 = (24)10
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California State University San Marcos
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Converting a Negative 2’s Complement
Number to Decimal Representation:
• Example: Represent (1001 1000)2 as a signed decimal number
• Since MSB = 1 then the number is negative
• Use signed conversion process
• Solution:
The number:
1001 1000
1’s Complement:
0110 0111
Add 1:
+ 0000 0001
Result:
Decimal
-ve sign
• Therefore
Sahar Mosleh
0110 1000
104
-104
(1001 1000)2 = (-104)10
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Signed or 2’s Complement Arithmetic
Addition of 8-bit signed number :
• If both numbers are positive, use unsigned addition process.
(perform bitwise addition)
• Note: If the result of the addition exceed 8-bits, then ignore the
carry over bit. This is because the sum or difference of any two
properly represented 8-bits signed number never exceeds 8-bits
Addition of 8-bit positive signed number
Example: perform the addition of (8)10 + (10)10 in binary
1
+
0000 1000
0000 1010
0001 0010
(0001 0010)2 = 1*24 + 0*23 + 0*22 + 1*21 + 0*20 = (18)10
Sahar Mosleh
California State University San Marcos
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Addition of 8-bit (positive and negative) signed
number:
• Perform the addition:
8 + (-17)
• Find the 2’s complement representation
(8)10
(-17)10
=
=
0000 1000
???
(17)2
0001 0001
1’s complement: 1110 1110
Add 1
+ 0000 0001
Result
1110 1111
• Therefore (-17)10 = (1110 1111)2
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California State University San Marcos
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• Now
(8)10
(-17)10
Result
0000 1000
+ 1110 1111
(1111 0111)2 = (?) 10
• Convert the result to decimal representation:
1111 0111
take 1’s complement
0000 1000
Add 1
+ 0000 0001
Result
(0000 1001)2 = (+9)10
Therefore, (1111 0111)2 is -9 in decimal
Sahar Mosleh
California State University San Marcos
Page 24
• Example: Perform 65 – 68 in 8 bits signed numbering system.
• 65 – 68 is the same as (65)10 + (-68)10.
• The example is the same as the previous one. Therefore, you
can follow the same step as explained to solve this problem
• Example: Perform -26 – 35 in 8 bits signed numbering system.
• -26 – 35 is the same as (-26)10 + (-35)10
• The example is the same as the previous one. Therefore, you
can follow the same step as explained to solve this problem
Sahar Mosleh
California State University San Marcos
Page 25
Logical Operation
• The logical operators And, Or, Xor, Not are used to perform
logical operation on 2 bits. The next few slides shows the
performance of logical operators on all possible combination of
two bits.
• To perform, any logical operation on two 8-bits binary numbers:
• Write two binary numbers and perform bit-wise logical
operation on each bit of the two binary numbers.
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California State University San Marcos
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Logical OR:
• A or B
A
B
A or B
1
1
1
1
0
1
0
1
1
1111 1111
0
0
0
Binary 8-bit
A
B
A and B
1
1
1
1
0
0
0
1
0
0
0
0
Number
Binary 8-bit
A
B
1110 1100
0111 0011
Result
Logical And:
• A and B
Number
A
B
Result
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1110 1100
0111 0011
0110 0000
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Logical XOR:
• A xor B
A
B
A xor B
Number
Binary 8-bit
1
1
0
A
B
1110 1100
0111 0011
1
0
1
0
1
1
Result
1001 1111
0
0
0
Logical NOT:
• Not A
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Number
Binary 8-bit
A
Not A
A
1110 1100
1
0
Result
0001 0011
0
1
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