computer engineering (logic circuits) lec. # 2 dr. tamer samy gaafar dept. of computer & systems...
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Computer Engineering(Logic Circuits)
Lec. # 2
Dr. Tamer Samy Gaafar
Dept. of Computer & Systems EngineeringFaculty of Engineering
Zagazig University
Sections will start this week (today)
A quiz will be held on the next week lecture (23/2/2015) at the 1st 20 mins.
The quiz will be on ch.1 as a whole.
No cheating is allowed (zero for both cheaters).
Announcements
Lecture 2
Number Systems (cont.)
5
Binary Numbers Again•Recall that N binary digits (N bits) can represent unsigned integers from 0 to 2N-1.
4 bits = 0 to 158 bits = 0 to 25516 bits = 0 to 65535
•Besides simply representation, we would like to also do arithmetic operations on numbers in binary form.
•Principal operations are addition and subtraction.
6
Binary Arithmetic, SubtractionThe rules for binary arithmetic are:
0 + 0 = 0, carry = 0
1 + 0 = 1, carry = 0
0 + 1 = 1, carry = 0
1 + 1 = 0, carry = 1
The rules for binary subtraction are:
0 - 0 = 0, borrow = 0
1 - 0 = 1, borrow = 0
0 - 1 = 1, borrow = 1
1 - 1 = 0, borrow = 0
Borrows, Carries from/to digits to left of current of digit.
Binary subtraction, addition works just the same as decimal addition, subtraction.
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Binary, Decimal addition
34
+ 17------ 51
from LSD to MSD:7+4 = 1; with carry out of 1 to next column
1 (carry) + 3 + 1 = 5.answer = 51.
Decimal
101011
+ 000001--------------- 101100
From LSB to MSB:1+1 = 0, carry of 11 (carry)+1+0 = 0, carry of 11 (carry)+0 + 0 = 1, no carry1 +0 = 10 + 0 = 01 + 0 = 1 answer = 101100
Binary
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SubtractionDecimal
900
- 001------- 899
0-1 = 9; with borrow of 1 from next column0 -1 (borrow) - 0 = 9, with borrow of 1 9 - 1 (borrow) - 0 = 8.Answer = 899.
Binary 100
- 001 ------- 011
0-1 = 1; with borrow of 1 from next column0 -1 (borrow) - 0 = 1, with borrow of 1 1 - 1 (borrow) - 0 = 0.Answer = 011.
9
Number systems
◦ Sign and magnitude
◦ Ones-complement ( 9’s Complement )
◦ Twos-complement (10’s Complement )
Signed Numbers
Decimal
Binary
10
Arithmetic
1 1 1011+ 1010
10101
1011– 0110 0101
11 1234+ 5678
6912
5274– 1638 3636
How do we write negative binary numbers?◦ Prefix numbers with minus symbol?
3 approaches:
◦ Sign and magnitude◦ Ones-complement◦ Twos-complement
All 3 approaches represent positive numbers in the same way
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Negative numbers
12
Most significant bit (MSB) is the sign bit◦ 0 ≡ positive◦ 1 ≡ negative
Remaining bits are the number's magnitude
Sign and magnitude
0000
0001
0011
1111
1110
1100
1011
1010
1000 0111
0110
0100
0010
0101
1001
1101
+ 0
+ 1
+ 2
+ 3
+ 4
+ 5
+ 6
+ 7– 0
– 1
– 2
– 3
– 4
– 5
– 6
– 7
Problem 1: Two representations of for zero◦ +0 = 0000 and also –0 = 1000
Problem 2: Arithmetic is cumbersome
◦ 4 – 3 != 4 + (-3)
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Sign and magnitude
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Negative number: Bitwise complement of positive number◦ 0111 ≡ 710
◦ 1000 ≡ –710
Ones(1’s)-complement
0000
0001
0011
1111
1110
1100
1011
1010
1000 0111
0110
0100
0010
0101
1001
1101
+ 0
+ 1
+ 2
+ 3
+ 4
+ 5
+ 6
+ 7– 7
– 6
– 5
– 4
– 3
– 2
– 1
– 0
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Solves the arithmetic problem
1’s complement
end-around carry
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The ones-complement of an 4-bit positive number y is 11112 – y◦ 0111 ≡ 710
◦ 11112 – 01112 = 10002 ≡ –710
What is 11112?◦ 1 less than 100002 = 24 – 1◦ –y is represented by (24 – 1) – y
Why 1’s complement works
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Still have two representations for zero!
◦ +0 = 0000 and also –0 = 1111
So what's wrong?
Negative number: Bitwise complement plus one◦ 0111 ≡ 710
◦ 1001 ≡ –710
Benefits:◦ Simplifies
arithmetic◦ Only one zero!
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Twos (2’s)-complement
0000
0001
0011
1111
1110
1100
1011
1010
1000 0111
0110
0100
0010
0101
1001
1101
0
+ 1
+ 2
+ 3
+ 4
+ 5
+ 6
+ 7– 8
– 7
– 6
– 5
– 4
– 3
– 2
– 1
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2’s complement
Recall: The ones-complement of a b-bit positive number y is (2b – 1) – y
Twos-complement adds one to the bitwise complement, thus, –y is 2b – y
Or Leaving all zeros and the first one from the
right as it is then, complement each 0 and 1 after the first 1 from the right
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Obtaining 2’s complement
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Adding representations of x and –y where x, y are positive numbers, we get x + (2b – y) = 2b + (x – y)◦ If there is a carry, that means that x y and
dropping the carry yields x – y
◦ If there is no carry, then x < y, then we can think of it as 2b – (y – x), which is the twos-complement representation of the negative number resulting from x – y.
Why 2’s complement works
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Sign-extension
◦ Write +6 and –6 as twos-complement 0110 and 1010
◦ Sign-extend to 8-bit bytes
00000110 and 11111010
Can't infer a representation from a number
◦ 11001 is 25 (unsigned)◦ 11001 is -9 (sign and magnitude)◦ 11001 is -6 (ones complement)◦ 11001 is -7 (twos complement)
Miscellaneous
10’s Complement Process The 10’s Complement process uses base-10 (decimal) numbers. Later, when we’re working with base-2 (binary) numbers, you will see that the 2’s Complement process works in the same way.
First, complement all of the digits in a number.
◦ A digit’s complement is the number you add to the digit to make it equal to the largest digit in the base (i.e., 9 for decimal). The complement of 0 is 9, 1 is 8, 2 is 7, etc.
Second, add 1.
◦ Without this step, our number system would have two zeroes (+0 & -0), which no number system has.
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10’s Complement Examples
24
-003
+1
996
997
-214
+1
785
786
Example #1
Example #2
Complement Digits
Add 1
Complement Digits
Add 1
8-Bit Binary Number SystemApply what you have learned to the binary number systems. How do you represent negative numbers in this 8-bit binary system?
Cut the number system in half.
Use 00000001 – 01111111 to indicate positive numbers.
Use 10000000 – 11111111 to indicate negative numbers.
Notice that 00000000 is not positive or negative.
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01111111
01111110
01111101
00000001
00000000
11111111
11111110
10000001
10000000
pos(+)
neg(-)
+127
+126
+125
+1
0
-1
-2
-127
-128
Sign Bit What did do you notice about
the most significant bit of the binary numbers?
The MSB is (0) for all positive numbers.
The MSB is (1) for all negative numbers.
The MSB is called the sign bit.
In a signed number system, this allows you to instantly determine whether a number is positive or negative.
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01111111
01111110
01111101
00000001
00000000
11111111
11111110
10000001
10000000
pos(+)
neg(-)
+127
+126
+125
+1
0
-1
-2
-127
-128
2’S Complement Process The steps in the 2’s Complement process are similar to the 10’s Complement process. However, you will now use the base two.
First, complement all of the digits in a number.
◦ A digit’s complement is the number you add to the digit to make it equal to the largest digit in the base (i.e., 1 for binary). In binary language, the complement of 0 is 1, and the complement of 1 is 0.
Second, add 1.
◦ Without this step, our number system would have two zeroes (+0 & -0), which no number system has.
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2’s Complement Examples
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Example #1
Example #2
Complement Digits
Add 1
5 = 00000101
-5 = 11111011
11111010
+1
Complement Digits
Add 1
-13 = 11110011
13 = 00001101
00001100
+1
Using The 2’s Complement Process
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9 + (-5)
4
(-9) + 5
- 4
(-9)+ (-5)
- 14
9 + 5
14
POS + POS
POS
POS + NEG
POS
NEG + POS
NEG
NEG + NEG
NEG
Use the 2’s complement process to add together the following numbers.
POS + POS → POS Answer
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If no 2’s complement is needed, use regular binary addition.
00001001 9 + 5
14
00001110
00000101 +
POS + NEG → POS Answer
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Take the 2’s complement of the negative number and use regular binary addition.
00001001 9 + (-5)
4
11111011+
00000101
11111010+1
11111011
2’s Complement
Process
1]000001008th Bit = 0: Answer is Positive
Disregard 9th Bit
POS + NEG → NEG Answer
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Take the 2’s complement of the negative number and use regular binary addition.
11110111 (-9) + 5
-4
00000101+
00001001
11110110+1
11110111
2’s Complement
Process
111111008th Bit = 1: Answer is Negative
11111100
00000011+1
00000100
To Check:Perform 2’s ComplementOn Answer
NEG + NEG → NEG Answer
33
Take the 2’s complement of both negative numbers and use regular binary addition.
11110111 (-9) + (-5)
-14
11111011 +
2’s ComplementNumbers, See Conversion ProcessIn Previous Slides
1]111100108th Bit = 1: Answer is Negative
Disregard 9th Bit
11110010
00001101+1
00001110
To Check:Perform 2’s ComplementOn Answer
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◦ Summing two positive numbers can give a negative result
◦ Summing two negative numbers can give a positive result
Overflow
00000001
0011
11111110
1100
10111010
1000 01110110
0100
0010
01011001
1101
0+ 1
+ 2
+ 3
+ 4
+ 5
+ 6+ 7– 8
– 7
– 6
– 5
– 4
– 3– 2
– 10000
0001
0011
11111110
1100
10111010
1000 01110110
0100
0010
01011001
1101
0+ 1
+ 2
+ 3
+ 4
+ 5
+ 6+ 7– 8
– 7
– 6
– 5
– 4
– 3– 2
– 1
6 + 4 ⇒ –6 –7 – 3 ⇒ +6
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Addition Examples (8 bit numbers)
Add 7 and 4 (both positive)
Add 15 and -6 (positive > negative)
Add 16 and -24 (negative > positive)
Add -5 and -9 (both negative)
00000111 7+00000100 +4 00001011 11
00001111 15 +11111010 +(-6)1 00001001 9Discard carry
00010000 16+11101000 +(-24) 11111000 -8Sign bit is negative so negative
number in 2’s complement form
11111011 -5 +11110111 +-9 1 11110010 -14Discard carry
36
Subtraction Examples Find 8 minus 3.
Find 12 minus -9.
Find -25 minus 19.
Find -120 minus -30.
00001000 8 +11111101 -31 00000101 5Discard carry
MinuendSubtrahendDifference
11100111 -25 +11101101 - 191 11010100 -44Discard carry
00001100 12 +00001001 - -9 00010101 21
10001000 -120 +00011110 - -30 10100110 -90
BCD Addition
Result must be corrected if
1. Result > 9 (further during all the addition steps)
2. A carry occurs in the first addition step only from one digit ( 4bit binary ) to another.
BCD Addition Rules
BCD example
Example #1
52 +
46----- ----------------
98
No correction is needed
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0101 0010
0100 0110
1001 1000 nc
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BCD example Example #2
17 0 0001 0111 +
56 0 0101 0110----- ---------------------
73 0 nc 0110 nc 1101 Correction 0 0000 0110
--------------------0 0111 0011
9/15/09 - L2Copyright 2009 - Joanne DeGroat,
ECE, OSU 41
BCD example Example #3
39 0 0011 1001 +
28 0 0010 1000----- ---------------------
67 0 nc 0110 c 0001
Correction 0 0000 0110--------------------0 0110 0111
42
BCD example Example #4
71 0 0111 0001 +
65 0 0110 0101----- --------------------- 136 0 nc 1101 nc 0110 Correction 0 0110 0000
-------------------- 0 0001 0011 0110
43
BCD example Example #5
87 0 1000 0111 +
15 0 0001 0101----- ---------------------
102 0 nc 1001 nc 1100
Correction 0 0000 0110--------------------0 1010 0010
0 0110 0000 -------------------------
0 0001 0000 0010
