Post on 19-Jan-2016
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Lecture 3Lecture 3
Representing Integer DataRepresenting Integer Data
ITEC 1000 “Introduction to Information Technology”
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Lecture Template:Lecture Template:
Number RepresentationNumber Representation Unsigned IntegerUnsigned Integer Signed IntegerSigned Integer Complementary RepresentationComplementary Representation 9’s Decimal Representation (skip)9’s Decimal Representation (skip) 1’s Binary Representation (skip)1’s Binary Representation (skip) 10’s Decimal Representation10’s Decimal Representation 2’s Binary Representation2’s Binary Representation
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Number RepresentationNumber Representation
An integer is a number which has no fractional part.
Numbers can be represented as a combination of Sign (plus or minus) Value or magnitude
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Unsigned IntegerUnsigned Integer(Natural Number)(Natural Number)
8-bit storage location28 different values between 0 and 255
16-bit storage location216 different values between 0 and 65535
multiple storage locations4 consecutive 1-byte storage locationsprovide 32 bits of range 232, or 4,294,967,296 different valuesdifficult to calculate and manipulate
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32-bit multiple storage location32-bit multiple storage location
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Signed-Integer RepresentationSigned-Integer Representation
No obvious direct way to represent the sign in binary notation
Options:Sign-and-magnitude representation1’s complement (skip – confusing)2’s complement (most common)
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Sign-and-MagnitudeSign-and-Magnitude
Use left-most bit for sign0 = plus; 1 = minus
Total range of integers the sameHalf of integers positive; half negativeMagnitude of largest integer half as large
Example using 8 bits:Unsigned: 1111 1111 = +255Signed: 0111 1111 = +127
1111 1111 = -127Note: 2 values for 0:
+0 (0000 0000) and -0 (1000 0000)
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Calculation AlgorithmsCalculation Algorithms
Sign-and-magnitude algorithms complex and difficult to implement in hardware
Must test for 2 values of 0Useful with BCD (see slides at end)Order of signed number and carry/borrow makes a difference
Example: Decimal addition algorithm
Addition: 2 Positive Numbers
Addition: 1 Signed Number
4+2
6
4- 2
2
2- 4-2
12- 4
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RangesRanges
No. of bits
Binary
Unsigned Sign-magnitude
Min Max Min Max
1 0 1
2 0 3 -1 1
3 0 7 -3 3
4 0 15 -7 7
5 0 31 -15 15
6 0 63 -31 31
Etc.
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Ranges: General RuleRanges: General Rule
No. of bits
Binary
Unsigned Sign-magnitude
Min Max Min Max
n 0 2n - 1 -(2
n-1 - 1) 2n-1 - 1
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Complementary RepresentationComplementary Representation
Sign of the number does not have to be handled separately
Consistent for all different signed combinations of input numbers
Two methodsRadix: value used is the base numberDiminished radix: value used is the base number minus 1 (skip – confusing)
9’s complement: base 10 diminished radix 1’s complement: base 2 diminished radix
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10’s Decimal Complement10’s Decimal Complement
Complement representation: (1) positive number: remains itself; (2) negative number: subtracting its absolute value from a standard basis value
Decimal (base 10) system: radix complement Radix = 10 10 as the basis 3-digit example: base value = 1000Range of possible values 0 to 999 arbitrarily split at 500
Numbers Negative Positive
Representation method Complement Number itself
Range of decimal numbers -500 -1 +0 499
Calculation 1000 minus abs(number) none
Representation example 500 999 0 499
– Increasing value +1000 – 500
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10’s Decimal Complement10’s Decimal Complement
Necessary to specify number of digits or word size
10’s complement representation in “1000” base
Example: representation of 3-digit number
First digit = 0 through 4 positive numberFirst digit = 5 through 9 negative number
1- 500- if ,)( -
499 0 if ,
valuevalueabsbase
valuevaluecomprep
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Examples with 3-Digit NumbersExamples with 3-Digit Numbers
Example 1:10’s complement representation of 247
247 (positive number)
Example 2:10’s complement representation of -247
1000 – 247 = 753 (negative number)
Example 3:Find a value with 10’s complement representation 777
Negative number because first digit is 71000 – 777 = 223Signed value = -223
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10’s Decimal Complement10’s Decimal Complement
Conversion to sign-and-magnitude value for 9’s complement representation(3 digits)
321 remains +321521: take the difference: base minus complement (999 – 521) with negative sign: – 479
999 500 if ), (
499 0 if ,
comprepcomprepbase
comprepcomprepvalue
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Addition: Counting Upwards Addition: Counting Upwards
Counting upward on scale corresponds to addition Example in 10’s complement: does not cross the
modulus
+250 +250
Representation
500 650 900 999 0 170 420 499
Numberrepresented
-500 -350 -100 -1 +0 170 420 499
+250 +250
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Addition with WraparoundAddition with Wraparound
Count to the right to add a negative number to the representation (since the complement 700 represents the value of -300)
Wraparound scale used to extend the range for the negative result
+700
Representation 500 999 0 200 499 500 900 999
Numberrepresented
-500 -1 0 200 499 -500 -100 -1
-300
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Addition with End-around CarryAddition with End-around Carry
Count to the right crosses the modulus
+300
Representation 500 800 999 0 100 499
Numberrepresented
-500 -200 -1 0 100 499
+300
800300
1100
100
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OverflowOverflow
Fixed word size has a fixed range size Overflow: combination of numbers that
adds to result outside the range End-around carry in modular arithmetic
avoids problem Complementary arithmetic: numbers out
of range have the opposite sign Test: If both inputs to an addition have the
same sign and the output sign is different, an overflow occurred
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2’s Binary Complement2’s Binary Complement
Binary system complementBinary (base 2) system radix complement Radix = 2 2 as the basis
Inversion: change 1’s to 0’s and 0’s to 1s (add 1)Numbers beginning with 0 are positiveNumbers beginning with 1 are negative
Example with 8-bit binary numbers
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2’s Complement2’s Complement
2’s complement representation:Positive value represents itselfNegative value: invert and add “1”
Numbers Negative Positive
Representation method Complement Number itself
Range of decimal numbers -12810 -110 +010 12710
Calculation Inversion None
Representation example 10000000 11111111 00000000 01111111
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Example: 2’s ComplementExample: 2’s Complement
Represent –5 in binary using 2’s complement notation
1. Decide on the number of bits:
2. Find the binary representation of the +ve value in 6 bits
3. Invert all the bits:
4. Add 1 111010+ 1 111011
-5
000101+5
6 (for example)
111010
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Sign Bit in 2’s ComplementSign Bit in 2’s Complement
In 2’s complement representation, the MSB is the sign bit (as with sign-magnitude notation)
0 = positive value1 = negative value
-5: 1 1 1 0 1 1
-ve
+5: 0 0 0 1 0 1
+ve 5
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Estimating Integer Value of 2’s Estimating Integer Value of 2’s Complement RepresentationComplement Representation
Positive numbers begin with 0 Small negative numbers (close to 0)
begin with multiple 1’s1111 1110 = -2 in 8-bit 2’s complements1000 0000 = -128, largest negative 2’s complementsInvert all 1’s and 0’s, add “1” and approximate the value
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Exercise: 2’s complement Exercise: 2’s complement conversionsconversions
What is -20 expressed as an 8-bit binary number in 2’s complement representation?
Answer:
1100011 is a 7-bit binary number in 2’s complement representation. What is the decimal sign-and-magnitude value?
Answer:
Skip answer Answer
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Exercise: 2’s complement Exercise: 2’s complement conversionsconversions
What is -20 expressed as an 8-bit binary number in 2’s complement notation? Answer: 11101100
1100011 is a 7-bit binary number in 2’s complement notation. What is the decimal value? Answer: -29
Answer
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Detail for -20 -> 11101100Detail for -20 -> 11101100
-2010: Positive Value = 00010100
Invert: 11101011
Add 1: + 1
11101100
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Detail for 1100011 -> - 29Detail for 1100011 -> - 29
2’s Complement Rep: 1100011
Invert: 0011100
Add One: + 1
0011101
Converts to: = - 29
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Arithmetic in 2’s ComplementArithmetic in 2’s Complement
Add 2 positive 8-bit numbers
Add 2 8-bit numbers with different signs Take the 1’s
complement of 58 (i.e. invert, add 1)0011 10101100 0110
0010 1101 = 45
1100 0110 = –58
1111 0011 = –13
0010 1101 = 45
0011 1010 = 58
0110 0111 = 103
0000 1101
8 + 4 + 1 = 13Invert to get magnitude
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Addition with CarryAddition with Carry in 2’s in 2’s ComplementComplement
8-bit number Invert (add 1)0000 0010 (210) 1111 1110Adddrop final carry out
0110 1010 = 106
1111 1110 = –2
10110 1000 = 104
(drop 1)
0110 1000
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SubtractionSubtraction
8-bit number Invert (add 1)0101 1010 (9010) 1010 0110
Adddrop final carry out
0110 1010 = 106
-0101 1010 = 90
0110 1010 = 106
–1010 0110 = 90
10001 0000
(drop 1)
0001 0000 = 16
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OverflowOverflow
8-bit number 256 different numbersPositive numbers: 0 to 127
AddTest for overflow 2 positive inputs produced negative result overflow!Wrong answer!
Programmers beware: some high-level languages, e.g., some versions of BASIC, do not check for overflow adequately
0100 0000 = 64
0100 0001 = 65
1000 0001 -127
0111 1110
12710Invert to get magnitude
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Overflow and Carry ConditionsOverflow and Carry Conditions
Carry flag: set when the result of an addition or subtraction exceeds fixed number of bits allocated
Overflow: result of addition or subtraction overflows into the sign bit
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Overflow/Carry ExamplesOverflow/Carry Examples
Example 1: Correct resultNo overflow, no carry
Example 2:Incorrect resultOverflow, no carry
0100 = (+ 4)
0010 = + (+ 2)
0110 = (+ 6)
0100 = (+ 4)
0110 = + (+ 6)
1010 = (– 6)
0101+ 10110
Invert, then add 1 to get magnitude
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Overflow/Carry ExamplesOverflow/Carry Examples
Example 3: Result correct ignoring the carryCarry but no overflow
Example 4:Incorrect resultOverflow, carry ignored
1100 = (– 4)
1110 = + (– 2)
11010 = (– 6)
1100 = (– 4)
1010 = + (– 6)
10110 = (+ 3)
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2’s Complement Subtraction2’s Complement Subtraction
Just add the opposite value!
A – B = A + (-B)
add 2’s complement rep. of -B
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Not quite finished => BCDNot quite finished => BCD
Reading: Lecture slides and Chapter 5Reading: Lecture slides and Chapter 5
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Unsigned IntegersUnsigned Integers
Binary equivalentconversion as discussed in Lecture 2 4 bits can hold 16 different values 0 – 158 bits can hold 28 different values between 0 and 255
BCD: Binary-Coded Decimaldigit-by-digit individual conversion to binary4 bits per decimal digit, i.e. 10 different values 0 - 98-bit storage location can hold 2 BCD digits, i.e. 100 different values 00 - 99
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Binary vs. BCDBinary vs. BCD
Decimal Binary BCD
68 = 0100 0100 = 0110 1000
= 26 + 22 = 64 + 4 = 68 = 22 + 21 = 6 23 = 8
99(largest 8-bit
BCD)
= 0110 0011 = 1001 1001
= 26 + 25 + 21 + 20 == 64 + 32 + 2 + 1 = 99
= 23 + 20 23 + 20 = 9 9
255 (largest 8-bit
binary)
= 1111 1111 = 0010 0101 0101
= 28 – 1 = 2558 bits
= 21 22 + 20 22 + 20
= 2 5 512 bits
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Value Range: Binary vs. BCDValue Range: Binary vs. BCD
No. of Bits BCD Range Binary Range
4 0-9 1 digit 0-15 1+ digit
8 0-99 2 digits 0-255 2+ digits
12 0-999 3 digits 0-4,095 3+ digits
16 0-9,999 4 digits 0-65,535 4+ digits
20 0-99,999 5 digits 0-1 million 6 digits
24 0-999,999 6 digits 0-16 million 7+ digits
32 0-99,999,999 8 digits 0-4 billion 9+ digits
64 0-(1016-1) 16 digits 0-16 quintillion 19+ digits
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Simple BCD MultiplicationSimple BCD Multiplication
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Conventional Binary vs. BCDConventional Binary vs. BCD
Binary representation generally preferredgreater range of value for given number of bitscalculations easier
BCD is still usedin business applications to maintain decimal rounding and decimal precisionin applications with a lot of input and output, but limited calculations