Tutorial: ITI1100

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Tutorial ITI1100

• Dewan Tanvir Ahmed
• SITE, UofO
• Email dahmed_at_site.uottawa.ca

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Binary Numbers

• Base (or radix)
• 2
• example 0110
• Number base conversion
• example 41 101001
• Complements
• 1's complements ( 2n- 1 ) - N
• 2's complements 2n - N
• Subtraction addition with the 2's complement
• Signed binary numbers
• signed-magnitude, 10001001
• signed 1's complement, 11110110
• signed 2's complement. 11110111

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Binary Number System
Base 22 Digits 0, 1 Examples 1001b 1 23
0 22 0 21 1 20 8
1 9  1010 1101b 1 27 1
25 1 23 1 22 1
128 32 8 4 1
173 Note it is common to put
binary digits in groups of 4 to make it easier to
read them.
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Ranges for Data Formats
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In General (binary)
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Signed Integers

• unsigned integers positive values only
• Must also have a mechanism to represent signed
  integers (positive and negative values!)
• -1010 ?2
• Two common schemes
• sign-magnitude and
• twos complement

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Sign-Magnitude

• Extra bit on left to represent sign
• 0 positive value
• 1 negative value
• 6-bit sign-magnitude representation of 5 and 5

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Ranges (revisited)
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In General
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Difficulties with Sign-Magnitude

• Two representations of zero
• Using 6-bit sign-magnitude
• 0 000000
• 0 100000
• Arithmetic is awkward!

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Complementary Representations

• 1s complement
• 2s complement
• 9s complement
• 10s complement

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Complementary Notations

• What is the 3-digit 10s complement of 207?
• Answer
• What is the 4-digit 10s complement of 15?
• Answer
• 111 is a 10s complement representation of what
  decimal value?
• Answer

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Exercises Complementary Notations

• What is the 3-digit 10s complement of 207?
• Answer 793
• What is the 4-digit 10s complement of 15?
• Answer 9985
• 111 is a 10s complement representation of what
  decimal value?
• Answer 889

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2s Complement

• Most common scheme of representing negative
  numbers
• natural arithmetic - no special rules!
• Rule to represent a negative number in 2s C
• Decide upon the number of bits (n)
• Find the binary representation of the ve value
  in n-bits
• Flip all the bits
• Add 1

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2s Complement Example

• Represent -5 in binary using 2s complement
  notation
• Decide on the number of bits
• Find the binary representation of the ve value
  in 6 bits
• Flip all the bits
• Add 1

6 (for example)
111010
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Sign Bit

• In 2s complement notation, the MSB is the sign
  bit (as with sign-magnitude notation)
• 0 positive value
• 1 negative value

2s complement
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Complementary Notation

• Conversions between positive and negative numbers
  are easy
• For binary (base 2)

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Example
2s C
ve
-ve
2s C
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Range for 2s Complement

• For example, 6-bit 2s complement notation

000000
111111
000001
011111
100000
100001
-32 -31 ... -1 0 1 ... 31
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Ranges
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In General (revisited)
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What is -6 plus 6?

• Zero, but lets see

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2s Complement Subtraction

• Easy, no special rules
• Subtract??
• Actually addition!

A B A (-B)
add
2s complement of B
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What is 10 subtract 3?

• 7, but
• Lets do it (well use 6-bit values)

10 3 10 (-3) 7
001010111101 000111
3 000011 -3 111101
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What is 10 subtract -3?

• 13, but
• Lets do it (well use 6-bit values)

10 (-3) 10 (-(-3)) 13
-3 111101 3 000011
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M - N

• M 2s complement of N
• M (2n - N) M - N 2n
• If M ? N
• Produce an carry, which is discarded
• If M lt N
• results in 2n - (N - M), which is the 2s
  complement of (N-M)

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Overflow

• Carry out of the leading digit
• If we add two positive numbers and we get a carry
  into the sign bit we have a problem
• If we add two negative numbers and we get a carry
  into the sign bit we have a problem
• If we add a positive and a negative number we
  won't have a problem
• Assume 4 bit numbers (7 -8)

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N 4
Number Represented
Unsigned 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
Binary 0000 0001 0010 0011 0100 0101 0110 0111 1
000 1001 1010 1011 1100 1101 1110 1111
Signed Mag 0 1 2 3 4 5 6 7 -0 -1 -2 -3 -4 -5 -6 -
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1's Comp 0 1 2 3 4 5 6 7 -7 -6 -5 -4 -3 -2 -1 -0
2's Comp 0 1 2 3 4 5 6 7 -8 -7 -6 -5 -4 -3 -2 -1
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Overflow

• If we add two positive numbers and we get a carry
  into the sign bit we have a problem

• 3 0011 4 0100
• 3 0011 4 0100
• 6 0110 8 1000

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Overflow

• -5 1011 -4 1100
• -3 1101 -5 1011
• -8 11000 -9 10111

• If we add two negative numbers and we get a carry
  into the sign bit we have a problem

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Overflow

• If we add a positive and a negative number we
  won't have a problem
• 5 0101 -4 1100
• -3 1101 5 0101
• 2 10010 1 10001

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Overflow

• If we add two positive numbers and we get a carry
  into the sign bit we have a problem
• 3 0011 4 0100
• 3 0011 4 0100
• 6 0110 8 1000

carry in 0 carry out 0
carry in 1 carry out 0
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Overflow

• If we add two negative numbers and we get a carry
  into the sign bit we have a problem
• -5 1011 -4 1100
• -3 1101 -5 1011
• -8 11000 -9 10111

carry in 1 carry out 1
carry in 0 carry out 1
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Overflow

• If we add a positive and a negative number we
  won't have a problem
• 5 0101 -4 1100
• -3 1101 5 0101
• 2 10010 1 10001

carry in 1 carry out 1
carry in 1 carry out 1
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Binary Codes

• n-bit binary code
• 2n distinct combinations
• BCD Binary Coded Decimal (4-bits)
• 0 0000
• 1 0001
• 9 1001
• BCD addition
• Get the binary sum
• If the sum gt 9, add 6 to the sum
• Obtain the correct BCD digit sum and a carry

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Binary Codes

• ASCII code
• American Standard Code for Information
  Interchange
• alphanumeric characters, printable characters
  (symbol), control characters
• Error-detection code
• one parity bit - an even numbered error is
  undetected
• A 411000001 - - gt
• 01000001 (even),
• 11000001 (odd)

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Binary Logic

• Boolean algebra
• Binary variables X, Y
• two discrete values (true or false)
• Logical operations
• AND, OR, NOT
• Truth tables

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Logic Gates

• Logic circuits
• circuits logical manipulation paths
• Computations and controls
• combinations of logic circuits
• Logic Gates

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Timing diagram
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Thank You!