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Tutorial: ITI1100
Dewan Tanvir Ahmed
SITE, UofO
Email: [email protected]
1
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 = 2
2 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
No. of bits
Binary
BCD
1
0–1
2
0–3
3
0–7
4
0 – 15
7
0 – 127
8
0 – 255
0 – 99
16
0 - 65,535
0 – 9999
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0 – 16,777,215
0 – 999999
0–9
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In General (binary)
Binary
No. of bits
Min
Max
n
0
2n – 1
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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:
+5:
+ve
0 0 0 1 0 1
5
-5:
-ve
1 0 0 1 0 1
5
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Ranges (revisited)
Binary
No. of bits
Unsigned
Min
Max
1
0
1
2
0
3
Sign-magnitude
Min
Max
3
-1
1
0
7
-3
3
4
0
15
-7
7
5
0
31
-15
15
6
0
63
-31
31
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In General …
Binary
No. of bits
Unsigned
Min
n
0
Sign-magnitude
Max
Min
Max
n
n-1
n-1
2 - 1 -(2
- 1) 2
-1
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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
•
•
•
•
1’s complement
2’s complement
9’s complement
10’s complement
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Complementary Notations
• What is the 3-digit 10’s complement of 207?
– Answer:
• What is the 4-digit 10’s complement of 15?
– Answer:
• 111 is a 10’s complement representation of what
decimal value?
– Answer:
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Exercises – Complementary Notations
• What is the 3-digit 10’s complement of 207?
– Answer: 793
• What is the 4-digit 10’s complement of 15?
– Answer: 9985
• 111 is a 10’s complement representation of what
decimal value?
– Answer: 889
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2’s Complement
•
•
•
Most common scheme of representing negative
numbers
natural arithmetic - no special rules!
Rule to represent a negative number in 2’s C
1.
2.
3.
4.
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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2’s Complement Example
•
Represent -5 in binary using 2’s complement
notation
1. Decide on the number of bits
6 (for example)
2. Find the binary representation of the +ve value in 6 bits
3. Flip all the bits
4. Add 1
000101
+5
111010
111010
+
1
111011
-5
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Sign Bit
• In 2’s complement notation, the MSB is the
sign bit (as with sign-magnitude notation)
– 0 = positive value
– 1 = negative value
+5:
0 0 0 1 0 1
+ve
5
-5:
1
-ve
1 1 0 1 1
2’s complement
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“Complementary” Notation
• Conversions between positive and negative
numbers are easy
• For binary (base 2)…
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Example
+5
0 0 0 1 0 1
2’s C
1 1 1 0 1 0
+
1
-5
1 1 1 0 1 1
2’s C
0 0 0 1 0 0
+
1
+5
0 0 0 1 0 1
2’s C
+ve
-ve
2’s C
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Range for 2’s Complement
• For example, 6-bit 2’s complement notation
100000
100001
-32
-31
111111
...
Negative, sign bit = 1
-1
000000
0
000001
1
011111
...
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Zero or positive, sign bit = 0
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Ranges
Binary
No. of bits
Unsigned
Min
Max
1
0
1
2
0
3
Sign-magnitude
2’s complement
Min
Max
Min
Max
3
-1
1
-2
1
0
7
-3
3
-4
3
4
0
15
-7
7
-8
7
5
0
31
-15
15
-16
15
6
0
63
-31
31
-32
31
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In General (revisited)
No. of
bits
Binary
Unsigned
Min
n
0
Sign-magnitude
Max
Min
Max
n
n-1
n-1
2 - 1 -(2
- 1) 2
2’s complement
Min
n-1
-1 -2
Max
2
n-1
-1
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What is -6 plus +6?
• Zero, but let’s see
Sign-magnitude
-6:
+6:
10000110
+00000110
10001100
2’s complement
-6:
11111010
+6: +00000110
00000000
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2’s Complement Subtraction
• Easy, no special rules
• Subtract??
• Actually … addition!
A – B = A + (-B)
add
2’s complement of B
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What is 10 subtract 3?
• 7, but…
• Let’s do it (we’ll use 6-bit values)
10 – 3 = 10 + (-3) = 7
+3: 000011
-3: 111101
001010
+111101
000111
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What is 10 subtract -3?
• 13, but…
• Let’s do it (we’ll use 6-bit values)
10 – (-3) = 10 + (-(-3)) = 13
-3: 111101
+3: 000011
001010
+000011
001101
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M-N
• M + 2’s complement of N
– M + (2n - N) = M - N + 2n
• If M  N
– Produce an carry, which is discarded
• If M < N
– results in 2n - (N - M), which is the 2’s 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
Binary
Unsigned
0000
0001
0010
0011
0100
0101
0110
0111
1000
1001
1010
1011
1100
1101
1110
1111
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
Signed
Mag
1's
Comp
2's
Comp
0
1
2
3
4
5
6
7
-0
-1
-2
-3
-4
-5
-6
-7
0
1
2
3
4
5
6
7
-7
-6
-5
-4
-3
-2
-1
-0
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
3
6
0011
0011
0110
4
4
8
0100
0100
1000
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Overflow
-5
-3
-8
1011
1101
11000
-4
-5
-9
1100
1011
10111
• If we add two negative numbers and we get a carry
into the sign bit we have a problem
30
Overflow
• If we add a positive and a negative number
we won't have a problem
5
-3
2
0101
1101
10010
-4
5
1
1100
0101
10001
31
Overflow
• If we add two positive numbers and we get a
carry into the sign bit we have a problem
3
3
6
0011
0011
0110
carry in 0
carry out 0
4
4
8
0100
0100
1000
carry in 1
carry out 0
32
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
33
Overflow
• If we add a positive and a negative number
we won't have a problem
5
-3
2
0101
1101
10010
carry in 1
carry out 1
-4
5
1
1100
0101
10001
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
1
…
9
0000
0001
…
1001
• BCD addition
• Get the binary sum
• If the sum > 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” 41:100|0001 - - >
– 0100|0001 (even),
– 1100|0001 (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
39
Thank You!
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