ECSE2610 Computer Components

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ECSE-2610Computer Components Operations
(CoCO)Fall 2008
Number Systems Representation Binary Arithmetic
Read Sections 2.1-2.6 except starred subsections
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Decimal Representation 10 digits 0,1,2,...,9
Positional Number Notation Weight of each digit
determined by its position.
Example 246 200 40
6 2 x 102 4 x 101 6 x 100
In general, for the Base 10 representation N
? Ni x 10i where each Ni ? 0, 1, 2, . . . ,
9 is the weight on the base raised to the
exponent i. Note Decimal fractions occur when i
is negative 0.35 3 x 10-1 5 x 10-2
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Binary Representation 2 digits 0, 1
Positional Number Notation for the Binary
Representation
N ? Ni x 2i with each Ni ? 0,1
Example Converting 1110012 from binary to
decimal 1110012 1000002 100002 10002
0002 002 12 1 x 25 1 x 24
1 x 23 0 x 22 0 x 21 1 x 20 1 x
32 1 x 16 1 x 8 0 x 4 0 x 2 1
x 1 5710
Easy!
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Quote of the Day

• There are 10 kinds of people in this world
• Those who understand binary
• and those who dont.

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Hexadecimal (Hex) Representation 16 digits
0,1,2,...,9,A,B,C,D,E,F
Positional Number Notation for the Hexadecimal
Representation
Note A - F represent the decimal values 10 -
15, respectively.
N ? Ni x 16i with each Ni ? 0,1,2,. .
.,9,A,B,C,D,E,F
Example Converting 8A9B16 from Hex to Decimal
representation 8A9B16 800016 A0016
9016 B16 8 x 163 A x 162 9 x
161 B x 160 8 x 4096 10 x 256 9 x 16
11 x 1 3548310
Still pretty easy! (if you remember 163 4096)
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Converting Decimal to Binary Successive
Division
Idea Use remainders from dividing the decimal
number by powers of 2.
Example Convert 5710 to Binary 57 / 2 28,
remainder 1 (binary number will end with 1) 28
/ 2 14, remainder 0 14 / 2 7, remainder
0 7 / 2 3, remainder 1 3 / 2 1,
remainder 1 1 / 2 0, remainder 1
(binary number will start with 1) Collecting the
remainders (from the bottom up), 5710 1110012
Comes from (((((0x2 1)x2 1)x2 1)x2
0)x2 0)x2 1
1x32 1x16 1x8 0x4 0x2 1x1 57
quotient 0. Time to stop.
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Converting Between Hexadecimal and Binary
Example Convert 1110110001110001012 to
hexadecimal. 1110110001110001012
Hex Binary Decimal 0 0000 0 1 0001
1 2 0010 2 3 0011 3 4 0100 4 5
0101 5 6 0110 6 7 0111 7 8 1000
8 9 1001 9 A 1010 10 B 1011 11 C 1100
12 D 1101 13 E 1110 14 F 1111 15
So 1110110001110001012 3B1C516
Memorize this table!It makes conversions between
hexadecimal and binary easy.
Even easier to go the other way (Hex to Bin)
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BINARY ARITHMETIC

• Computers have circuits that do binary
  arithmetic.
• You already know the rules for decimal addition
  and subtraction (how to handle sums, carries,
  differences, and borrows).
• Analogously, we develop the rules for binary
  addition and subtraction.

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Decimal Addition Refresher
1 x 102 (31) x 101 3 x 100
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Binary Addition of 2 Bits
This table calculates the sum for pairs of binary
numbers
0 0 0 0 1 1 1 0 1 1 1 0 with a
carry of 1
Also known as the Half Adder Table
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Binary Addition of 3 Bits (including Carry)
Table shows all the possible sums for binary
numbers with carries
carry addend augend
sum 0 0 0 0 0 0 1 1 0 1 0
1 0 1 1 0 with a carry of
1 1 0 0 1 1 0 1 0 with a carry
of 1 1 1 0 0 with a carry of
1 1 1 1 1 with a carry of 1
Also known as the Full Adder Table
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Binary Addition Similar to the Decimal Case
Example Add 5 and 3 in binary
(carries) 1 0 12 510 1 12
310 810
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Decimal Subtraction Refresher
9510 1610 7910
Note Borrows are shown as explicit subtractions.
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Binary Subtraction for 2 Bits
This table calculates the difference for pairs of
binary numbers
0 - 0 0 0 - 1 1 with a borrow of 1 1 - 0
1 1 - 1 0
Also known as the Half Subtractor Table
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Binary Subtraction for 3 Bits (including Borrow)
This shows all the possible differences for
binary numbers with borrows
minuend subtrahend borrow difference 0 - 0
- 0 0 0 - 0 - 1 1 with a borrow of 1
0 - 1 - 0 1 with a borrow of
1 0 - 1 - 1 0 with a borrow of
1 1 - 0 - 0 1 1 - 0 - 1 0
1 - 1 - 0 0 1 - 1 - 1 1 with a borrow
of 1
Also known as the Full Subtractor Table
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Binary Subtraction Similar to the Decimal Case
Hanging borrow tells us that result is negative
Example 5 - 3
Example 3 - 5
1 0 12 510 - 1 12 310 1 02
210
1
10
0 1 12 310 - 1 0 12 510 1 02
-210
- 1 0 (borrows)
0
1
Next, we will learn about ways to represent
negative numbers.
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Number Systems Representation of Integers

• Positive numbers are same in most systems
• Major difference is in how negative numbers are
  represented
• Two major schemes
• signed-magnitude twos-complement
• For example, assume a 4 bit machine word
• 16 different values can be represented
• roughly half are positive, half are negative

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Signed-Magnitude Representation 4 Bit Example

• Sign is high order bit 0 positive or zero
  (non-negative)
• 1 negative
• Magnitude is three low order bits 0 000 thru 7
  111
• Number range ?7 for 4 bit numbers for n bits,
  ?(2n-1 -1)

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Signed-Magnitude Operations in Computers

• Addition is performed by
• Binary addition when signs of operand the same
• Binary subtraction when signs of operands differ
• Must compare magnitudes to determine sign of
  result
• Subtraction is performed by
• Binary subtraction when signs of operand the same
• Binary addition when signs of operands differ
• Must compare magnitudes to determine sign of
  result
• Complicated implementation for
• Adder unit
• Subtractor unit
• Comparator unit

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Twos-Complement Representation 4 Bit Example
Most commonly used system in arithmetic logic

• Only one representation for 0 (0000)
• One more negative number than positive number
• The MSB (most significant bit) indicates the sign
• Positive numbers are the same as in
  Signed-Magnitude
• But how do we get 1101 -3? . . .

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Negative Numbers in Twos-Complement
Mathematical Method
Let -N denote negative of N in 2s complement.
-N can be calculated by -N 2n - N
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Negative Numbers in Twos-Complement
Simple Method
-N can be calculated by -N 2n N (2n 1)
N) 1 But (2n 1) 111 A string of n
1s So ((2n 1) N)) is the bit by bit
complement of N
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Twos-Complement Operations

• Addition
• Addition not dependent on the signs of operand
• No need to compare magnitudes to determine sign
  of result
• Subtraction
• Subtraction is treated as an addition
• Add the negative of the subtrahend to the minuend
• Simple implementation
• Adder unit
• Negation circuit unit

Simpler addition/subtraction scheme makes
twos-complement the most common choice for
integer number systems within digital systems