Introduction to Computing

1
Binary Numbers (contd)

• Addition
• 1 0 1
• 1 1 1
• _________________

2
Binary Numbers (contd)

• Addition Our basic addition table is really
  easy now
• 0 0 0
• 0 1 1
• 1 0 1
• 1 1 10
• What about 1 1 1?
• 1 1 1 11

3
Binary Numbers (contd)

• Addition just like usual
• 1 0 1
• 1 1 1
• _______________

0 0 0 0 1 1 1 0 1 1 1 10 1
1 1 11
4
Binary Numbers (contd)

• Addition just like usual
• 1
• 1 0 1
• 1 1 1
• _______________
• 0

0 0 0 0 1 1 1 0 1 1 1 10 1
1 1 11
5
Binary Numbers (contd)

• Addition just like usual
• 1 1
• 1 0 1
• 1 1 1
• _______________
• 0 0

0 0 0 0 1 1 1 0 1 1 1 10 1
1 1 11
6
Binary Numbers (contd)

• Addition just like usual
• 1 1
• 1 0 1
• 1 1 1
• _______________
• 1 1 0 0

0 0 0 0 1 1 1 0 1 1 1 10 1
1 1 11
7
Binary Numbers (contd)

• Can also do
• Subtraction, Multiplication, etc
• Negative Numbers
• Fractions
• Great, but where do Logic Circuits fit in?

8
Binary Numbers (contd)

• And you thought we were done with truth tables

• For Addition on small numbers, can make a truth
  table

Problem Output canbe 2 bitssometimes
9
Binary Numbers (contd)

• Addition Truth Table with Multiple Outputs

X Y A B 0 0 0 0 0 1 0 1 1 0
0 1 1 1 1 0
10
Binary Numbers (contd)

• But we know how to deal with multiple outputs
• Now we can convert into 2 circuits!

11
Binary Numbers (contd)
X
Y
B
X
Y
X
A

Y
12
Binary Numbers (contd)
X
Y
B
X
Y
This is often called the eXclusive OR (XOR)
circuit We say that B is the exclusive OR of X
and Y. We write B X Y.

13
A Logic Puzzle

• Bob will go to the party if Ed goes OR
  Dan goes.
• Dan will go if Xena does NOT go AND
  Yanni goes.
• Ed will go if Xena goes AND
  Yanni does NOT go.
• Turns out this is addition without the carry
• Bob goes if the XOR of Xena and Yanni go.

14
A Simple Breakthrough

• We can represent information by bits.(So we
  interpret bits to mean things like numbers.)
• Then we re-interpret those bits as Logical
  True/False values and construct truth tables for
  operations on information (e.g. addition)
• Finally we use Universal Method to construct
  circuits for these operations

15
Intermission

• Questions??
• How do we actually build a circuit for addition?

16
Addition
X1 X2 Y1 Y2 Z1 Z2 C
0 0 0 0 0 0 0
0 0 0 1 0 1 0
0 0 1 0 1 0 0
0 0 1 1 1 1 0
0 1 0 0 0 1 0
0 1 0 1 1 0 0
0 1 1 0 1 1 0
0 1 1 1 0 0 1
1 0 0 0 1 0 0
1 0 0 1 1 1 0
1 0 1 0 0 0 1
1 0 1 1 0 1 1
1 1 0 0 1 1 0
1 1 0 1 0 0 1
1 1 1 0 0 1 1
1 1 1 1 1 0 1
X1 X2 Y1 Y2 C Z1
Z2 C is the carry bit
17
Addition
X1 X2 Y1 Y2 Z1 Z2 C
0 0 0 0 0 0 0
0 0 0 1 0 1 0
0 0 1 0 1 0 0
0 0 1 1 1 1 0
0 1 0 0 0 1 0
0 1 0 1 1 0 0
0 1 1 0 1 1 0
0 1 1 1 0 0 1
1 0 0 0 1 0 0
1 0 0 1 1 1 0
1 0 1 0 0 0 1
1 0 1 1 0 1 1
1 1 0 0 1 1 0
1 1 0 1 0 0 1
1 1 1 0 0 1 1
1 1 1 1 1 0 1
X1 X2 Y1 Y2 C Z1
Z2 C is the carry bit Could have 3 circuits (for
Z1, Z2, C) 16 inputs each 2 w/ 8 ANDs, 1 OR 1 w/
6 ANDs, 1 OR 187 25 gates
18
Simplifying the Addition Circuit
X1 X2 Y1 Y2 C Z1
Z2
Rewrite as X2 X1 Y2 Y1 C2 C2
Z2 C Z1
19
Addition
X1 Y1 C2 C Z1
X2 Y2 C2 Z2
X1 Y1 C2 C Z1
0 0 0 0 0
0 0 1 0 1
0 1 0 0 1
0 1 1 1 0
1 0 0 0 1
1 0 1 1 0
1 1 0 1 0
1 1 1 1 1
2 circuits (C, Z1) 10 gates
2 circuits (C2, Z2) 4 gates
20
Addition
X2
Z2
Circuit For Z2 and carry
Y2
C2
Z1
Circuit For Z1 and carry
X1
Carry
Y1

• Can be cascaded for adding longer numbers
• We can use 2 building blocks to add numbers of
  any length.

21
Adding Binary Numbers of Any Length

• Actually we need only 1 building block
• lowest-bit circuit also has 3 inputs, but carry
  in is always 0

ith bit of X
ith bit of Z
ith bit of Y
carry bit out
carry bit in
Abstraction in action -- This is a piece of a
carry-ripple adder
22
Adding Binary Numbers of Any Length

• Write the numbers as
• X4 X3 X2 X1 X0
• Y4 Y3 Y2 Y1 Y0
• Represent the sum as
• C Z4 Z3 Z2 Z1 Z0

23
Adding Binary Numbers of Any Length

• Write the numbers as
• X4 X3 X2 X1 X0
• Y4 Y3 Y2 Y1 Y0
• Represent the sum as
• C Z4 Z3 Z2 Z1 Z0
• Start with X0, Y0, 0 in, Z0 and C0 out
• Then X1,Y1,C0 in and Z1 and C1 out.
• And so on

24
Adding Binary Numbers of Any Length
At the ith stage
ith bit of X
Addition Circuit
ith bit of Z
ith bit of Y
carry bit out
carry bit in
Abstraction in action -- This is a piece of a
carry-ripple adder
25
Adding Binary Numbers of Any Length
At each stage, take 2 addends and a bit telling
whether there is a carry in, and produce a sum
bit and a bit telling if there is a carry out
We can build a black box to do this
(abstraction)
addend
Sum bit
addend
carry bit out
carry bit in
26
Carry-Ripple Adder
Z1
Z2
Black Box Circuit
X2
Z0
Black Box Circuit
X1
Black Box Circuit
X0
C2
Y2
Y0
Y1
C1
C0
0
Fixed at 0
X2 X1 X0 Y2 Y1
Y0 C2 Z2 Z1 Z0
27
Whats inside the box?
Zi
Black Box Circuit
Xi
Yi
Cout
Cin
28
Whats inside the box?
Zi
Universal Circuit
Xi
Yi
Cout
Cin
Abstraction in Action Use Truth Tables and the
Universal Method
29
Carry ripple adders

• Useful for some applications
• Too slow for other
• Delay while waiting for the carry to ripple
• One solution add larger blocks
• Details in the homework assignment

30
Arithmetic Logical Unit (ALU)

• This is the part of the CPU that does arithmetic,
  logical and comparison operations
• Weve built a piece of the ALU
• Using abstraction, we could build the other parts
• Multiplication/subtraction/division
• Comparison
• Then we would write a language to let the user
  talk to the ALU (programming)

31
Pause

• Questions??
• Back to representing information

32
Representing information

• How do we represent characters?
• How many characters might we want to represent?
• What characters might we want to represent?

33
Representing information

• How do we represent characters?
• How many characters might we want to represent?
• What characters might we want to represent?
• A-Z 26
• A-Z and a-z 52
• All the keys on my keyboard 104
• Maybe use a power of 2? 128
• Maybe use an even power of 2? 256
• Maybe an even bigger power of 2? 65536

34
Representing characters

• ASCII is the American Standard Code for
  Information Interchange. It is a 7-bit code.
• Many 8-bit codes contain ASCII as the lower half
  of their range of values
• The ASCII standard was published by the United
  States of America Standards Institute (USASI) in
  1968
• There are other character coding standards
  EBCDIC, and more recently Unicode

35
Unicode

• Universal Character Set (UCS) contains all
  characters of all other character set
  standards. It also guarantees round-trip
  compatibility, i.e., conversion tables can be
  built such that no information is lost when a
  string is converted from any other encoding to
  UCS and back.
• UCS contains the characters required to
  represent almost all known languages. In
  addition to the many languages which use
  extensions of the Latin script, this also
  includes the following scripts and languages
  Greek, Cyrillic, Hebrew, Arabic, Armenian,
  Gregorian, Japanese, Chinese, Hiragana,
  Katakana, Korean, Hangul, Devangari, Bengali,
  Gurmukhi, Gujarati, Oriya, Tamil, Telugu,
  Kannada, Malayam, Thai, Lao, Bopomofo, and a
  number of others. Work is going on to include
  further scripts like Tibetian, Khmer, Runic,
  Ethiopian, Hieroglyphics, various Indo-European
  languages, and many others.
• It uses 31 bits (32768 possible characters)

36
What do we do in practice?

• Problems
• Bits represent too little information too many
  are needed
• Decimal numbers dont translate well into
  hardware
• So
• Group into blocks of 4 and 8 bits
• 8 bits 256 characters, holds ASCII
• 8 bits make 1 byte things are organized into
  bytes
• 4 bits make 1 nibble
• Hexadecimal numbers use 1 nibble per slot or hex
  digit
• Digits go from 0-9 and then A-F, for 16 possible
  values

37
Shorthand Hexadecimal
Dec Hex Binary
Dec Hex Binary
8 9 10 11 12 13 14 15
0 1 2 3 4 5 6 7
0 0 0 0 0 0 0 1 0 0 1 0 0 0 1 1 0 1 0 0 0 1
0 1 0 1 1 0 0 1 1 1
8 9 A B C D E F
1 0 0 0 1 0 0 1 1 0 1 0 1 0 1 1 1 1 0 0 1 1
0 1 1 1 1 0 1 1 1 1
0 1 2 3 4 5 6 7
38
Hexadecimal

• We can add hex numbers
• 112, 224, 448, 48C, 28A,
• We can combine 2 hexadecimal numbers to make a
  byte (2C)
• Hex is easier to read than binary
• Decimal 22 is 16 in hex (1161 6160) rather
  than 10110 in binary
• In ASCII
• Hex 41 through 5A represent A to Z
• Hex 61 through 7A represent a to z

39
Summary

• Logic gates and the Universal Method
• Binary representation of numbers and arithmetic
• Universal method can lead to very big circuits
• Solve by reducing to smaller circuits
• Demonstrated with cascading smaller circuits
• Carry-ripple adder
• Represented characters
• Hexadecimal
• Bytes, nibbles

40
Next Time Memory
Now that we can represent it, how do we store
it??