1. Computer Engineering (Logic Circuits) Dr. Tamer Samy Gaafar Lec. # 2
Dept. of Computer & Systems Engineering
Faculty of Engineering
Zagazig University

2. Course Web Page http://www.tsgaafar.faculty.zu.edu.eg

3. Announcements 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).

4. Lecture 2 Number Systems (cont.)

5. Binary Numbers Again
Recall that N binary digits (N bits) can represent unsigned integers from 0 to 2N bits = 0 to 15 8 bits = 0 to 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, Subtraction
The rules for binary arithmetic are:
The rules for binary subtraction are:
0 + 0 = 0, carry = 0
0 - 0 = 0, borrow = 0
1 + 0 = 1, carry = 0
1 - 0 = 1, borrow = 0
0 + 1 = 1, carry = 0
0 - 1 = 1, borrow = 1
1 + 1 = 0, carry = 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.

7. Binary, Decimal addition
101011
From LSB to MSB: 1+1 = 0, carry of 1 1 (carry)+1+0 = 0, carry of 1 1 (carry) = 1, no carry = = = 1 answer = 34
from LSD to MSD: 7+4 = 1; with carry out of 1 to next column
1 (carry) = 5. answer = 51.

8. Subtraction Decimal Binary 900 100
= 9; with borrow of 1 from next column 0 -1 (borrow) - 0 = 9, with borrow of (borrow) - 0 = 8. Answer = 899.
100
= 1; with borrow of 1 from next column 0 -1 (borrow) - 0 = 1, with borrow of (borrow) - 0 = 0. Answer = 011.

9. Signed Numbers Number systems Sign and magnitude
Ones-complement ( 9’s Complement )
Twos-complement (10’s Complement )

10. Arithmetic 6912 10101 Decimal Binary 11 1234 5274 + 5678 – 1638 3636
1011
+ 1010
10101
1011
– 0110
0101

11. Negative numbers Sign and magnitude Ones-complement Twos-complement
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

12. Sign and magnitude Most significant bit (MSB) is the sign bit
0 ≡ positive
1 ≡ negative
Remaining bits are the number's 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

13. Sign and magnitude Problem 1: Two representations of for zero
+0 = 0000 and also –0 = 1000
Problem 2: Arithmetic is cumbersome
4 – 3 != 4 + (-3)

14. Ones(1’s)-complement
Negative number: Bitwise complement of positive number
0111 ≡ 710
1000 ≡ –710
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

15. 1’s complement
Solves the arithmetic problem
end-around carry

16. Why 1’s complement works
The ones-complement of an 4-bit positive number y is – y
0111 ≡ 710
11112 – = ≡ –710
What is 11112?
1 less than = 24 – 1
–y is represented by (24 – 1) – y

17. So what's wrong? Still have two representations for zero!
+0 = 0000 and also –0 = 1111

18. Twos (2’s)-complement Negative number: Bitwise complement plus one
0111 ≡ 710
1001 ≡ –710
Benefits:
Simplifies arithmetic
Only one zero!
0000
0001
0011
1111
1110
1100
1011
1010
1000
0111
0110
0100
0010
0101
1001
1101
+ 1
+ 2
+ 3
+ 4
+ 5
+ 6
+ 7
– 8
– 7
– 6
– 5
– 4
– 3
– 2
– 1

19. 2’s complement

20. Obtaining 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

21. Why 2’s complement works
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.

22. Miscellaneous Sign-extension
Write +6 and –6 as twos-complement
0110 and 1010
Sign-extend to 8-bit bytes
and
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)

23. 2's Complement Arithmetic
Digital Electronics 
Lesson 2.4 – Specific Comb Circuit & Misc Topics
2's Complement Arithmetic
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.
This slide describes the 10’s complement conversion process.
Project Lead The Way, Inc.
Copyright 2009

24. 10’s Complement Examples
2's Complement Arithmetic
Digital Electronics 
Lesson 2.4 – Specific Comb Circuit & Misc Topics
10’s Complement Examples
Example #1
-003
Complement Digits

996
Add 1 +1
997
Example #2
Examples of the 10’s Complement Process.
-214
Complement Digits

785
Add 1 +1 24
786
Project Lead The Way, Inc.
Copyright 2009 24

25. 8-Bit Binary Number System
Digital Electronics 
Lesson 2.4 – Specific Comb Circuit & Misc Topics
2's Complement Arithmetic
8-Bit Binary Number System
Apply 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 – to indicate positive numbers. Use – to indicate negative numbers. Notice that is not positive or negative.
+127
pos(+)
+126
+125 +1 -1
Introduction to the 8-Bit Binary Number system and how negative numbers are represented. -2
-127
neg(-)
-128
Project Lead The Way, Inc.
Copyright 2009

26. 2's Complement Arithmetic
Digital Electronics 
Lesson 2.4 – Specific Comb Circuit & Misc Topics
2's Complement Arithmetic
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.
+127
pos(+)
+126
+125 +1 -1
Explanation of the sign bit. -2
-127
neg(-)
-128
Project Lead The Way, Inc.
Copyright 2009

27. 2's Complement Arithmetic
Digital Electronics 
Lesson 2.4 – Specific Comb Circuit & Misc Topics
2's Complement Arithmetic
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.
This slide describes the 2’s complement conversion process.
Project Lead The Way, Inc.
Copyright 2009

28. 2’s Complement Examples
Digital Electronics 
Lesson 2.4 – Specific Comb Circuit & Misc Topics
2's Complement Arithmetic
2’s Complement Examples
Example #1
5 =
Complement Digits

Add 1 +1
-5 =
Example #2
Examples of the 2’s Complement Process.
-13 =
Complement Digits

Add 1 +1
13 =
Project Lead The Way, Inc.
Copyright 2009

29. Using The 2’s Complement Process
Digital Electronics 
Lesson 2.4 – Specific Comb Circuit & Misc Topics
2's Complement Arithmetic
Using The 2’s Complement Process
Use the 2’s complement process to add together the following numbers.
POS
+ POS  9
+ 5 14
NEG
+ POS 
(-9)
+ 5
- 4
This slide show that there are only four possible combinations for adding together two signed numbers. The next four slides demonstrate each of these examples.
POS
+ NEG  9
+ (-5) 4
NEG
+ NEG 
(-9)
+ (-5)
- 14
Project Lead The Way, Inc.
Copyright 2009

30. 2's Complement Arithmetic
Digital Electronics 
Lesson 2.4 – Specific Comb Circuit & Misc Topics
2's Complement Arithmetic
POS + POS → POS Answer
If no 2’s complement is needed, use regular binary addition. 9
+ 5 14  +  
Addition of two Positive numbers.
Project Lead The Way, Inc.
Copyright 2009

31. 2's Complement Arithmetic
Digital Electronics 
Lesson 2.4 – Specific Comb Circuit & Misc Topics
2's Complement Arithmetic
POS + NEG → POS Answer
Take the 2’s complement of the negative number and use regular binary addition. 9
+ (-5) 4  +  1]
8th Bit = 0: Answer is Positive
Disregard 9th Bit
This example shows the addition of one positive and one negative numbers. Note that this is done in the same way as subtracting a positive number from a positive number. In this case, the answer is positive.
 +1
2’s
Complement
Process
Project Lead The Way, Inc.
Copyright 2009

32. 2's Complement Arithmetic
Digital Electronics 
Lesson 2.4 – Specific Comb Circuit & Misc Topics
2's Complement Arithmetic
POS + NEG → NEG Answer
Take the 2’s complement of the negative number and use regular binary addition.
(-9)
+ 5 -4  + 
8th Bit = 1: Answer is Negative
This slide demonstrates the addition of one positive and one negative number. Again, this is is the same a subtracting a positive number from a positive number. In this case the answer happens to be negative.
 +1
To Check:
Perform 2’s
Complement
On Answer
 +1
2’s
Complement
Process
Project Lead The Way, Inc.
Copyright 2009

33. 2's Complement Arithmetic
Digital Electronics 
Lesson 2.4 – Specific Comb Circuit & Misc Topics
2's Complement Arithmetic
NEG + NEG → NEG Answer
Take the 2’s complement of both negative numbers and use regular binary addition.
(-9)
+ (-5)
-14 
2’s Complement
Numbers, See
Conversion Process
In Previous Slides  +  1]
8th Bit = 1: Answer is Negative
Disregard 9th Bit
This slide demonstrates the addition of two negative numbers.
 +1
To Check:
Perform 2’s
Complement
On Answer
Project Lead The Way, Inc.
Copyright 2009

34. Overflow Summing two positive numbers can give a negative result
Summing two negative numbers can give a positive result
0000
0001
0011
1111
1110
1100
1011
1010
1000
0111
0110
0100
0010
0101
1001
1101
+ 1
+ 2
+ 3
+ 4
+ 5
+ 6
+ 7
– 8
– 7
– 6
– 5
– 4
– 3
– 2
– 1
6 + 4 ⇒ –6
–7 – 3 ⇒ +6

35. 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)
(-6)
Discard carry
(-24)
Sign bit is negative so negative number in 2’s complement form
Discard carry

36. Subtraction Examples Find 8 minus 3. Find 12 minus -9.
Discard carry
Minuend Subtrahend Difference
Discard carry

37. BCD Addition

38. BCD Addition Rules 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.

39. BCD example 0101 0010 0100 0110 1001 1000 nc Example #1 52 + 46 98
No correction is needed
1001
1000 nc

40. BCD example
Example #2 +
nc nc
Correction

41. BCD example
Example #3 +
nc c
Correction
Copyright Joanne DeGroat, ECE, OSU
9/15/09 - L2

42. BCD example
Example #4 +
nc nc
Correction

43. BCD example
Example #5 +
nc nc
Correction