negative Representing numbers
A negative number is usually indicated by its complement.
2s complement is the most common.
ECE 15A
Fundamentals of Logic Design
Lecture 2
Example:
+11
-11
Malgorzata Marek-Sadowska
:
:
00001011
10001011 (signed magnitude)
11110100 (signed ones complement)
11110101 (signed twos complement)
Signed 2s complement has only one representation for 0 (+)
Electrical and Computer Engineering Department
UCSB
2
One's complement format - 8 bit arithmetic
One's Complement to Decimal
Change the number N to binary, ignoring the sign.
Add 0s to the left of the binary number to make a
total of 8 bits
If the sign is positive, do nothing.
If the sign is negative, complement every bit (i.e.
change from 0 to 1 or from 1 to 0)
In this way we compute (28 -1) - N
Convert the following 1's complement
representation to decimal:
a) 11110001
Since the sign bit is 1, complement the number:
00001110
Convert to decimal: 000011102 = 1410
Put a negative sign in front: -14
b) 00011010 -> 26
3
Two's complement format - 8 bit arithmetic
Example
Most computers today use 2's complement
representation for negative numbers.
The 2's complement of a negative number N is
obtained by adding 1 to the 1's complement.
Write 25 in one's complement
00011001
25
Write -25 in one's complement
8
Since the number is negative, complement each bit
11100110
4
-25
5
This is the same as computing 2 - N
For -13
00001101 base integer
11110010 1's complement
+1
11110011 2's complement
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1
Example
Two's Complement to Decimal
If the sign bit is 0, convert the binary number
to decimal.
If the sign bit of N is 1
Write -25 in two's complement format.
00011001
11100110
11100111
25
one's complement
two's complement
Compute 2s complement of N
convert the binary number to decimal
put a minus sign in front
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Complements
Example
Convert the following 2's complement
representation to decimal: 11100011
Compute 2s complement:
11100011 -> (1s complement: 00011100) ->
(Add 1: 00011101)
(change to decimal) -> 29 -> (put in front)
-> -29
Used to simplify the subtraction
Arithmetic subtraction
(+-A) - (+B) = (+-A) + (-B)
(+-A) - (-B) =(+-A) + (+B)
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Example (1-s complement)
Application:
11000000
-00100111
The carry out of the MSB is added to LSB
(end around carry)
10
Example (1-s complement)
<=> 1 1 0 0 0 0 0 0
+ 1 1 0 1 1 0 0 0(ones complement)
1 1 0 0 11 0 0 0
+
1 (end around carry)
100 1 1 0 0 1
The difference
Application:
10110011
-01101101
The carry out of the MSB is added to LSB
(end around carry)
<=>
101 10011
[phone]
1 0 1 0 0 0 1 0 1(ones complement)
1 (end around carry)
???
01 0 0 0 1 1 0
The difference
11
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2
Suppose
Example
M = 10110011 (negative)
N = 00001011 (positive)
Example
Suppose
Compute M-N
M = 10110011 (negative)
N = 00001011 (positive)
Compute M-N
1s complement
If M is in 1s complement,
then the actual value of M
is -1001100
2s complement
Subtraction in 1s complement:
10110011
11110100
(original M in 1s complement)
(1s complement of N)
If M is in 2s complement,
then the actual value of M
is -01001101
10110011
11110101
(original M in 1s complement)
(2s complement of N)
So, the result of M-N is:
So, the result of M-N is:
110100111
- 1001100
- 0001011
- 1010111
Subtraction in 2s complement:
110101000
- 1001101
- 0001011
- 1011000
+1
10101000
Changing to 2s
complement
Changing to 1s
complement
10101000
10101000
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14
Overflow
Today: The Algebra of Sets
0
The Algebra of Sets is
Carry into sign bit
Carry out of the sign bit
11
0 1010011
+ 0 1011100
1 0101111
carry
1
Sign bit
an example of Boolean Algebra
useful in manipulating digital circuits
01
1 1010011
+ 1 0011100
0 1101111
We will investigate the nature of sets and the
way in which they may be combined
Sign bit
Adding two positive numbers results
in a negative number!
Adding two negative numbers results
in a positive number!
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Definitions
Definitions
Elements are basic objects
Collections of objects constitute sets
Example:
Basic objects
S3
Universal set, denoted by 1:
Consists of all elements
under consideration
S1
S4
Sets
The algebra will be developed as an algebra for sets, not for elements of sets
Every set is a subset of 1; 0 is a subset of every other set.
0 and 1 are not numbers!
m
Example, continues:
Basic objects
Universal set 1
S5
S2
S4
m is a member of S4
m
p
S3 ={p}
S3
S4
S5
Complement
of a set S6
Null denoted set, by 0,
Contains no elements
S6
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S6
(also denoted as S6)
S2
S2
r
k
u
S1
S2 = {r,k,u}
S1
S2 S1
S4 = S5
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3
Example: red, black and yellow books
The red (R) books and some of the
black books (B) are in English (E);
The reminder of the black books (B) are in
German (G); Yellow books (Y) are in French (F).
B
R
Rules of forming new sets
The union of sets X and Y is a set X+Y consisting of
all elements which are either in X or in Y.
Example:
Y
E
R
B
G
B
R
E
Y
R+Y = B
E
G
F
R+E=E
G
Y=F
F
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20
Rules of forming new sets
Theorem
The intersection of sets X and Y is a set X Y consisting
of all elements which are both in X and in Y.
Example:
If m 1 and X,Y are arbitrary sets, then m is a member
of only one of the sets: XY, XY, XY, XY.
E
m X or m X, but not to both.
m Y or m Y, but not to both.
If m X then mXY or m XY, but not to both.
If mX then mXY or m XY, but not to both.
G
F
BY
R
E B is the set of black bound books which are in English
R Y is a null set
R E=R
For any set X: X + X = 1; X X = 0
E
G
F
21
Symbols in common use
Meaning
22
Venn diagrams
Symbolic notation
X Y , X Y , X Y
XY , X Y , X Y
Union of sets X and Y
Intersections of X and Y
Complement of X
X ',
X
Y
The set of points interior to the rectangle
is taken as the universal set.
X, ~ X
XY
Example
B
R
G
Y
Say in words, what are: Y+G; RB, G(B+R); B+BR
F
Show that (G+R) = GR; (GB)=G+B
XY
X+Y
E
(X+Y) (all blue)
(X+Y)=XY
X
Y
X
X (all blue)
Y
Y (all blue)
XY
XY (all blue)
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24
4
Venn diagrams for X+YZ=(X+Y)(X+Z)
Y
X
Y
X
Z
Z
YZ
Y
X
X+YZ
Z
Each of the laws given here may be justified
intuitively using Venn diagrams
Y
X
Z
X+Y
Commutative laws
(1a) XY = YX
X+Z
(1b) X+Y = Y+X
Associative laws
Y
X
Fundamental laws of the algebra of sets
(Also valid in any Boolean algebra)
(2a) X(YZ) = (XY)Z
Z
(2b) X+(Y+Z) = (X+Y)+Z
Distributive laws
(3a) X(Y+Z) = XY+XZ
(X+Y)(X+Z)
(3b) X+YZ= (X+Y)(X+Z)
25
Fundamental laws of the algebra of sets
(Also valid in any Boolean algebra)
26
Fundamental laws of the algebra of sets
(Also valid in any Boolean algebra)
Laws of Tautology
(4a) XX = X
Laws of De Morgan
(8a) (XY) = X+Y
(4b) X+X = X
Laws of Absorbtion
(5a) X(X+Y) = X
(9a) 0X=0
(10a) 1X=X
(11a) 0=1
(5b) X+XY = X
Laws of Complementation
(6a) XX=0
(8b) (X+Y) = XY
Operations with 0 and 1
(6b) X+X=1
(9b) 1+X=1
(10b)0+X=X
(11b) 1=0
Expressions such as 2X or X2 will never appear in the algebra of sets.
Double Laws of Complementation
Principle of duality: If in any identity, each union is replaced by intersection,
each intersection by union, 0 by 1, and 1 by 0, then the resulting equation is
also an identity.
(7) (X)=X
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Monomials and polynomials
Factors
A monomial is either a single letter representing a set
(with or without a prime), or a product of two or
more of such symbols representing the intersection
of sets.
In any expression representing an intersection
of sets, each such set is called a factor of the
set of intersection.
X, Y, XYZ are examples of monomials
The factors of X(Y+Z) are X and Y+Z.
A factor is linear if it is a letter (possibly with
a prime) or a sum of such letters.
The polynomial represents the union of the sets
corresponding to separate terms
X+Z is linear
(X+Y), X+YZ are not linear
X+Y+XYZ is an example of a polynomial
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5
Expanding expressions into polynomials
Theorem
Expand (X+Y)(Z+W) into a polynomial
(X+Y)(Z+W)=(X+Y)Z+(X+Y)W by (3a) [x(y+z)=xy+xz]
=Z(X+Y)+W(X+Y) by(1a) [xy=yx]
=ZX+ZY+WX+WY (by 3a)
Factor AC+AD+BC+BD into linear factors
AC+AD+BC+BD=A(C+D)+B(C+D) (by 3a)
=(A+B)(C+D) (by 1a and 3a)
It can be shown, that any expression can be factored
into linear factors by repeated application of the
(3b). [x+yz=(x+y)(x+z)]
For any sets X and Y, X+XY=X+Y
Proof:
(X+XY) = (X+X)(X+Y)
(by 3b) [x+yz=(x+y)(x+z)]
= 1(X+Y)
(6b) [x+x=1]
= X+Y
31
X
Y
X
Y
XY
32
At home
Get ready for discussion #1:
Do the following problems from CHR: 2.1- 2.3; 2.5; 2.6; 2.11.
Read before lecture #3: CHR, pp.62-68.
Homework #1 is posted, due Monday, January 23 at noon.
You can read more on the algebra of sets in
R. Serrel, Elements of Boolean Algebra for the Study of
Information-Handling Systems, in Proc. of the IRE, vol. 41, issue
10, Oct. 1953, pp.1366-1380.
This publication can be found on IEEE Xplore:
http://www.library.ucsb.edu/eresources/databases/
Scroll down to I and find IEEE Xplore
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