1. COMPSCI 210S1C 2014 Computer Systems 1 Introduction

2. James Goodman (revised by Robert Sheehan)
Computer Science 210 s1c Computer Systems Semester 1 Lecture Notes
Introduction & Layers
Lecture 1
James Goodman (revised by Robert Sheehan)

3. Lecturers Tutors Class Representative Robert Sheehan (week 1-6)
Office
Office hours: by appointment, after class, or whenever the office door is open
Xinfeng Ye (week 7-12)
Office
Office hours:
Tutors
Ahmad Obidat
Office hours: TBA
Class Representative
TBD (volunteers?)
Arash Heidarian
Office hours: TBA

4. Forum Tutors We are using the Cecil discussion list.
Check it as often as you want.
The tutor/s will be regularly participating.
I follow the discussion area of Cecil irregularly and participate as appropriate but likely not as quickly as the tutors.
Tutors
Are available to help 4

5. Tutorials
Tutorials are not compulsory but are strongly recommended. All tutorials will start from week.
9:00-10:00, Monday, 303S-191 (N.B. different room)
10:00-11:00, Monday, 303S-G75
13:00-14:00, Monday, 303S-G75
16:00-17:00, Monday, 303S-G75
12:00-13:00, Tuesday, 303S-G75
17:00-18:00, Tuesday, 303S-G75
13:00-14:00, Wednesday, 303S-G75
Tutor's office hours ?
Tutorials will be available online
Lecture recordings in the Knowledge Map area of Cecil

6. Miscellaneous Come to class . It is very dangerous to fall behind.
Each new lecture will require you to know and understand the content of the previous lectures.
Lectures are tied to the content of the textbook. It is important to read the textbook – exam questions may require more detail than is covered in the lectures.
Tutorials are a great way to supplement lectures
Outside office hours and tutorials, tutors are not expected to be at hand

7. Course Outline Bits and Bytes Digital Logic
How do we represent information using electrical signals?
Digital Logic
How do we build circuits to process information?
Computing Engines, Processors and Instruction Sets
How do we build a processor out of logic elements?
What operations (instructions) will we implement?
Assembly Language Programming
How do we use processor instructions to implement algorithms?
How do we write modular, reusable code? (subroutines)
I/O, Traps, and Interrupts
How does a processor communicate with the outside world?
C Programming
How do we write programs in C?
How do we implement high-level programming constructs?
Redundant?

8. Textbook LC-3 simulator
LC-3 simulator

9. Data Representation Data Performing Arithmetic
Binary
Octal
Decimal
Signed Numbers
Performing Arithmetic
Addition
Subtraction
Shifting (Mul/Div)
Types and Representation
Integer
Floating point -- IEEE format
Alpha-numeric representation
IEEE Format is so widely adopted it is often not mentioned.
Note that P&P don’t mention octal except in Appendix D!!!

10. Low-level Processes
Introduction
Digital logic structures
Finite state machine
ISA/Memory organisation
Opcodes
Operate instructions, data movement operations
Control instructions (loop, if-then-else control)
The Assembly process
Input & Output
Sub-routines / Stacks
Coding examples
Note that tutorials will closely follow the course materials progression offering you a chance to apply new knowledge immediately. This will be very important for both assembly and C.

11. C programming Basic components Data representation Binary fraction
Floating point representation
Introduction to C
Operators
Control structure
Functions
Pointers, arrays, string
I/O
Advanced programming

12. Assignments
There will be three assignments. The assignments count 20% of your grade.
Don’t pass your work to someone else; don’t copy some one else’s work. Do not copy other sources.
If you are caught you will receive zero for the entire assignment and your previous and future submissions will be scrutinised.
For assembly and C, an assignment not compiling will receive 0 marks
Submissions Deadline: an assignment due date means the assignment should be turned in by the time specified in the assignment description. The submission dropboxes normally stay open for late assignments but penalties apply.
3 or 4???
New system this semester – bugs still being worked out!

13. Test Tentative: During class on 30 April, Week 7 (28 April – 2 May)
Multiple-choice questions (MCQ)
Material through week 6
See example from previous semesters
Worth 20% towards course grade
SURVEY: How many would prefer MCQ ONLY?

14. Exam Assignments are important!!! Multiple-choice questions (MCQ)
See examples from previous semesters
Worth 60% towards course grade
Assignments are important!!!

15. How to Do Well in CompSci 210
1. Read the lecture notes after each lecture
2. Read the relevant textbook sections
a. To learn more
b. To complement lectures
3. If you have questions or do not understand something
Attend the tutorials
Check the forum
Discuss with other 210 students
Ask a tutor during office hours
or see me
4. How to prepare for exams
Do 1, 2 & 3 above
Do exercises of the course/tutorials/exercises/textbook
Study previous years’ exams: You can get these online from the library website
Point 3 is Patrice’s point, not mine!
The internet is a wonderful tool, especially, Wikipedia

16. Assignments Assignment Subject Approx date out Date due
% of the final mark
# 1
Data representation/Digital logic/simple machines
10/03/14
27/03/14 5
# 2
Low-level programming
31/03/14
01/05/14
# 3
C code
05/05/14
29/05/14 10

17. Yale N. Patt Sanjay J. Patel
Introduction to Computing Systems: From Bits and Gates to C and Beyond 2nd Edition
Yale N. Patt Sanjay J. Patel
Based on slides originally prepared by Gregory T. Byrd, North Carolina State University

18. Chapter 1 Welcome Aboard

19. Introduction to the World of Computing
Computer: electronic genius?
NO! Electronic idiot!
Does exactly what we tell it to, nothing more.
Goal of the course:
You will be able to write programs in C and understand what’s going on underneath – no magic!
Approach:
Build understanding from the bottom up.
Bits  Gates  Processor  Instructions  C Programming

20. Two Recurring Themes Abstraction Hardware vs. Software
Productivity enhancer – don’t need to worry about details…
Can drive a car without knowing how the internal combustion engine works.
…until something goes wrong!
Where’s the dipstick? What’s a spark plug?
Important to understand the components and how they work together.
Hardware vs. Software
It’s not either/or – both are components of a computer system.
Even if you specialize in one, it is important to understand capabilities and limitations of both.
Many aspects can be either S/W or H/W, or a combination. Driving choice: cost and speed (flexibility).

21. Big Idea #1: Universal Computing Device
All computers, given enough time and memory, are capable of computing exactly the same things. = =
Smart phone
Desktop
Supercomputer

22. Alan Turing (12 Jun 1912 – 7 Jun 1954)
Added

23. Tadd Tmul Turing Machine
Mathematical model of a device that can perform any computation – Alan Turing (1937)
ability to read/write symbols on an infinite “tape”
state transitions, based on current state and symbol
Every computation can be performed by some Turing machine. (Turing’s thesis)
Previously, a “computer” was built to solve a problem: ordinance calculations, controlling a textile loom (Jacquard), computing logarithms, code breaking, tabulating results.
Tadd
a,b
a+b
Turing machine that adds
Tmul
a,b ab
Turing machine that multiplies
For more info about Turing machines, see

24. Universal Turing Machine
A machine that can implement all Turing machines -- this is also a Turing machine!
inputs: data, plus a description of computation (other TMs) U
a,b,c
c(a+b)
Universal Turing Machine
Tadd, Tmul
U is programmable – so is a computer!
instructions are part of the input data
a computer can emulate a Universal Turing Machine
A computer is a universal computing device.

25. From Theory to Practice
In theory, a computer can compute anything
that’s possible to compute
given enough memory and time
In practice, solving problems involves computing under constraints.
time
weather forecast, next frame of animation, ...
cost
cell phone, automotive engine controller, ...
power
cell phone, handheld video game, ...
Further Caveats:
1. A machine may be deliberately restricted for size or function (needs access to unbounded memory).
2. Accuracy may be limited.

26. Big Idea #2: Transformations Between Layers
Problems
Algorithms
Language
Instruction Set Architecture
Microarchitecture
Circuits
Devices

27. How do we solve a problem using a computer?
A systematic sequence of transformations between layers of abstraction.
Problem
Software Design:
choose algorithms and data structures
Algorithm
Programming:
use language to express design
Program
Compiling/Interpreting:
convert language to machine instructions
Instr Set
Architecture

28. Deeper and Deeper… Processor Design:
Instr Set
Architecture
Processor Design:
choose structures to implement ISA
Microarch
Logic/Circuit Design:
gates and low-level circuits to implement components
Circuits
Process Engineering & Fabrication:
develop and manufacture lowest-level components
Devices

29. Descriptions of Each Level
Problem Statement
stated using "natural language"
may be ambiguous, imprecise
Algorithm
step-by-step procedure, guaranteed to finish
definiteness, effective computability, finiteness
Program
express the algorithm using a computer language
high-level language, low-level language
Instruction Set Architecture (ISA)
specifies the set of instructions the computer can perform
data types, addressing mode

30. Descriptions of Each Level (cont.)
Microarchitecture
detailed organization of a processor implementation
different implementations of a single ISA
Logic Circuits
combine basic operations to realize microarchitecture
many different ways to implement a single function (e.g., addition)
Devices
properties of materials, manufacturability

31. Many Choices at Each Level
Solve a system of equations
Gaussian
elimination
Jacobi
iteration
Red-black SOR
Multigrid
FORTRAN C
C++
Java
Intel x86
ARM
NVIDIA
Haswell
Nehalem
Atom
Ripple-carry adder
Carry-lookahead adder
CMOS
Bipolar
GaAs
Tradeoffs:
cost
performance
power
(etc.)
SOR = Successive Over Relaxation
Cloud? Multiple processors?
Sun and Java are trademarks of Sun Microsystems, Inc. Intel, Pentium, Centrino, and Xeon are trademarks of Intel Corporation. AMD and Athlon and trademarks of Advanced Micro Devices, Inc. Atmel and AVR are registered trademarks of Atmel Corporation.

32. Course Outline Bits and Bytes Digital Logic
How do we represent information using electrical signals?
Digital Logic
How do we build circuits to process information?
Processor and Instruction Set
How do we build a processor out of logic elements?
What operations (instructions) will we implement?
Assembly Language Programming
How do we use processor instructions to implement algorithms?
How do we write modular, reusable code? (subroutines)
I/O, Traps, and Interrupts
How does processor communicate with outside world?
C Programming
How do we write programs in C?
How do we implement high-level programming constructs?
Review

33. James Goodman (revised by Robert Sheehan)
Computer Science 210 s1c Computer Systems Semester 1 Lecture Notes
Data Representation
Lecture 2
James Goodman (revised by Robert Sheehan)
Credits: Adapted from slides prepared by Gregory T. Byrd, North Carolina State University

34. Chapter 2 Bits, Data Types, and Operations
Reading is critical – keep current

35. How do we represent data in a computer?
At the lowest level, a computer is an electronic machine.
works by controlling the flow of electrons
Easy to recognize two conditions:
presence of a voltage – we’ll call this state “1”
absence of a voltage – we’ll call this state “0”
Could base state on value of voltage, but control and detection circuits more complex.
compare turning on a light switch to measuring or regulating voltage
Early work on analog computation.

36. Computer is a Binary Digital System.
finite number of symbols
Binary (base two) system:
has two states: 0 and 1
Basic unit of information is the binary digit, or bit.
Values with more than two states require multiple bits.
A collection of two bits has four possible states: 00, 01, 10, 11
A collection of three bits has eight possible states: 000, 001, 010, 011, 100, 101, 110, 111
A collection of n bits has 2n possible states.

37. What kinds of data do we need to represent?
Numbers – signed, unsigned, integers, floating point, complex, rational, irrational, …
Text – characters, strings, …
Images – pixels, colors, shapes, …
Sound
Logical – true, false
Instructions …
Data type:
representation and operations within the computer
We’ll start with numbers…
Others are important, but numbers and text occur in nearly every application.

38. 101 329 Unsigned Integers 22 21 20 102 101 100 Non-positional notation
could represent a number (“5”) with a string of ones (“11111”)
problems?
Weighted positional notation
like decimal numbers: “329”
“3” is worth 300, because of its position, while “9” is only worth 9
101 22 21 20
most
significant
least
329
102
101
100
3x x10 + 9x1 = 329
1x4 + 0x2 + 1x1 = 5

39. Unsigned Integers (cont.)
An n-bit unsigned integer represents any of 2n (integer) values: from 0 to 2n-1. 22 21 20
Value 1 2 3 4 5 6 7
All eight possible combinations

40. Unsigned Binary Arithmetic
Base-2 addition – just like base-10!
add from right to left, propagating carry
carry
10111
+ 111
Subtraction, multiplication, division,…

41. “There are 10 kinds of people in the world: those who understand binary, and those who don’t.”
“Funny” vs. “Ambiguous”

42. Signed Integers With n bits, we can distinguish 2n unique values
assign about half to positive integers (1 through 2n-1) and about half to negative (-2n-1 through -1)
that leaves two values: one for 0, and one extra
Positive integers
just like unsigned, but zero in most significant (MS) bit = 5
Negative integers
Sign-Magnitude (or Signed-Magnitude) – set MS bit to show negative, other bits are the same as unsigned = -5
One’s complement – flip every bit to represent negative = -5
In either case, MS bit indicates sign: 0=positive, 1=negative
For sign bit, we arbitrarily set 0 for positive, 1 for negative (for good reason)
Problem: two representations of zero!

43. Two’s Complement + 11011 (-5) + (-9) 00000 (0) 00000 (0)
Problems with sign-magnitude and 1’s complement
two representations of zero (+0 and –0)
arithmetic circuits are complex
How to add two sign-magnitude numbers?
e.g., try 2 + (-3)
How to add two one’s complement numbers?
e.g., try 4 + (-3)
Two’s complement representation developed to make circuits easy for arithmetic.
for each positive number (X), assign value to its negative (-X), such that X + (-X) = 0 with “normal” addition, ignoring carry out
Answer: = 10111
00101 (5) (9)
(-5) + (-9)
00000 (0) (0)

44. Two’s Complement Representation
If number is positive or zero,
normal binary representation, zeroes in upper bit(s)
If number is negative,
start with positive number
flip every bit (i.e., take the one’s complement)
then add one
2’s complement negation: 1’s complement + 1
Works for both positive and negative numbers!
01001(9) -> = = -9
00101 (5) (9)
11010 (1’s comp) (1’s comp)
11011 (-5) (-9)

45. Two’s Complement Signed Integers
MS bit is sign bit – it has weight –2n-1.
Range of an n-bit number: -2n-1 through 2n-1 – 1.
The most negative number (-2n-1) has no positive counterpart.
-23 22 21 20 1 2 3 4 5 6 7
-23 22 21 20 1 -8 -7 -6 -5 -4 -3 -2 -1

46. “Biased” Representation of Signed Integers
All integers (positive & negative) are represented as an unsigned integer supplemented with a “bias” to be subtracted out.
Range of an n-bit number: (0 - bias) through (2n-1 - bias).
Bias 8: 23 22 21 20
Bias-8 -8 1 -7 -6 -5 -4 -3 -2 -1 23 22 21 20
Bias-8 1 2 3 4 5 6 7

47. “Biased” Representation of Signed Integers
All integers (positive & negative) are represented as an unsigned integer supplemented with a “bias” to be subtracted out.
Range of an n-bit number: (0 - bias) through (2n-1 - bias).
Bias 7: 23 22 21 20
Bias-7 -7 1 -6 -5 -4 -3 -2 -1 23 22 21 20
Bias-7 1 2 3 4 5 6 7 8
Why do this?
Represent more positive numbers than negative numbers
But means that zero is not represented by all zeros

48. Converting Binary (2’s C) to Decimal
If leading bit is one, take two’s complement to get a positive number.
Add powers of 2 that have “1” in the corresponding bit positions.
If original number was negative, add a minus sign. n 2n 1 2 4 3 8 16 5 32 6 64 7
128
256 9
512 10
1024
X = two
= =
= 104ten
Assuming 8-bit 2’s complement numbers.

49. More Examples X = 00100111two = 25+22+21+20 = 32+4+2+1 = 39ten
= =
= 39ten n 2n 1 2 4 3 8 16 5 32 6 64 7
128
256 9
512 10
1024
X = two
-X =
= =
= 26ten
X = -26ten
Assuming 8-bit 2’s complement numbers.

50. Converting Decimal to Binary (2’s C)
First Method: Division
Find magnitude of decimal number. (Always positive.)
Divide by two – remainder is least significant bit.
Keep dividing by two until answer is zero, writing remainders from right to left.
Append a zero as the MS bit; if original number was negative, take two’s complement.
Key observation: LSB for a binary representation tells you if the number is even or odd.
X = 104ten 104/2 = 52 r0 bit 0
52/2 = 26 r0 bit 1
26/2 = 13 r0 bit 2
13/2 = 6 r1 bit 3
6/2 = 3 r0 bit 4
3/2 = 1 r1 bit 5
X = two 1/2 = 0 r1 bit 6

51. Converting Decimal to Binary (2’s C)1 2 4 3 8 16 5 32 6 64 7
128
256 9
512 10
1024
Second Method: Subtract Powers of Two
Find magnitude of decimal number.
Subtract largest power of two less than or equal to number.
Put a one in the corresponding bit position.
Keep subtracting until result is zero.
Append a zero as MS bit; if original was negative, take two’s complement.
X = 104ten = 40 bit 6
= 8 bit 5
8 - 8 = 0 bit 3
X = two

52. Operations: Arithmetic and Logical
Recall: a data type includes representation and operations.
We now have a good representation for signed integers, so let’s look at some arithmetic operations:
Addition
Subtraction
Sign Extension
We’ll also look at overflow conditions for addition.
Multiplication, division, etc., can be built from these basic operations.
Logical operations are also useful:
AND OR
NOT

53. Addition + 11110000 (-16) + 11110111 (-9) (1)01011000 (88) (1)11101101
As we’ve discussed, 2’s comp addition is just binary addition.
assume all integers have the same number of bits
ignore carry out
for now, assume that sum fits in n-bit 2’s comp. representation
+9 :
-9 : 1
(104) (-10)
(-16) (-9)
(1) (88)
(1)
(-19)
Assuming 8-bit 2’s complement numbers.

54. Subtraction
Negate subtrahend (2nd no.) and add.
assume all integers have the same number of bits
ignore carry out
for now, assume that difference fits in n-bit 2’s comp representation
(104) (-10)
(16) (-9)
(-16) (9)
(88) (-1)
Work out on blackboard? Don’t have to compute -9! Just use +9
Assuming 8-bit 2’s complement numbers.

55. Sign Extension
To add two numbers, we must represent them with the same number of bits.
If we just pad with zeroes on the left:
Instead,
propagate the MS bit (the sign bit):
4-bit 8-bit
0100 (4) (still 4)
1100 (-4) (12, not -4)
Replicate the sign bit:
4-bit 8-bit
0100 (4) (still 4)
1100 (-4) (still -4)

56. Logic and more data types
Computer Science 210 s1c Computer Systems Semester 1 Lecture Notes
Logic and more data types
Lecture 3
James Goodman (revised by Robert Sheehan)
Credits: Adapted from slides prepared by Gregory T. Byrd, North Carolina State University

57. Overflow + 01001 (9) + 10111 (-9) 10001 (-15) 01111 (+15)
If operands are too big, their sum cannot be represented as an n-bit 2’s comp number.
We have overflow if:
signs of both operands are the same, and
sign of sum is different.
Another test -- easy for hardware:
carry into MS bit does not equal carry out
01000 (8) (-8)
(9) (-9)
10001 (-15) (+15)

58. Overflow 1010 10000 carries 0110 10011(4-bit)= 0011 + 0111
Example: 4-bit
Two’s complement - 8 <= x <= 7
Examples: -6 -7
= -13
(outside the range)
6 + 7 = 13
(outside the range)
1010
+ 1001
10000 carries
10011(4-bit)= 0011 6 7
0110
01100 carries
01101 (4-bit) => 1101
6 + 7 : two positive numbers gives a negative result (4 bits)
: two negative numbers gives a positive result
: CAN’T overflow!
2 + 3 : two positive numbers give s positive result
Answer = 3 Invalid answer
Answer = -3 Invalid answer 2 3
2 + 3 = 5 -2 3
= 1
0010
+0011
0100 carries
0101
1110
11100 carries
10001 (4-bit)=0001
Answer = 1 valid answer
Answer = 5 Valid answer

59. Addition/Subtraction with 2’s Complement
Two’s complement representation allows addition and subtraction from a single simple adder. Circuit to add : S = A + B To subtract A – B : invert B and enable carry in
From chapter 3 of textbook (coming soon!)
Easy to invert B inputs
Easy to add carry in

60. Logical Operations Operations on logical TRUE or FALSE
two states -- takes one bit to represent: TRUE=1, FALSE=0
View n-bit number as a collection of n logical values
operation applied to each bit independently A B
A AND B 1 A B
A OR B 1 A
NOT A 1
Truth table
Also EXCLUSIVE OR

61. Examples of Logical Operations
AND
AND
useful for clearing bits
AND with zero = 0
AND with one = no change OR
useful for setting bits
OR with zero = no change
OR with one = 1
NOT
unary operation -- one argument
flips every bit OR
Note that NOT is also the 1’s complement operation!
Also XOR (as opposed to inclusive OR)
NOT

62. Hexadecimal Notation (not Representation)
It is often convenient to write binary (base-2) numbers using hexadecimal (base-16) notation instead.
fewer digits -- four bits per hex digit
less error prone -- easy to corrupt long string of 1’s and 0’s
Binary
Hex
Decimal
0000
0001 1
0010 2
0011 3
0100 4
0101 5
0110 6
0111 7
Binary
Hex
Decimal
1000 8
1001 9
1010 A 10
1011 B 11
1100 C 12
1101 D 13
1110 E 14
1111 F 15
Hexadecimal is a Shorthand NOTATION, not a DATA TYPE
Can talk about “unsigned hexadecimal” or hexadecimal “2’s complement” representation.
We are NOT talking about 15’s complement and 16’s complement!
Can perform BINARY arithmetic operation on hexadecimal notation.

63. Converting from Binary Notation to Hexadecimal Notation
Every four bits is a hex digit.
start grouping from right-hand side 7 D 4 F 8 A 3
This is not a new machine representation, just a convenient way to write the number.

64. Representing Text
American Standard Code for Information Interchange (ASCII)
Developed from telegraph codes, alternative to IBM’s Extended Binary Coded Decimal Interchange Code (EBCDIC) in 1960s
Printable and non-printable (ESC, DEL, …) characters (127)
Limited set of characters – many character missing, especially language-specific
Many national “standards” developed
Unicode – more than 110,000 characters covering 100 scripts
UTF-8 is the form of Unicode which preserves the ASCII encoding

65. Text: ASCII Characters
ASCII: Maps 128 characters to 7-bit code.
both printable and non-printable (ESC, DEL, …) characters
“ASCIIbetical” order 00
nul 10
dle 20 sp 30 40 @ 50 P 60 ` 70 p 01
soh 11
dc1 21 ! 31 1 41 A 51 Q 61 a 71 q 02
stx 12
dc2 22
" 32 2 42 B 52 R 62 b 72 r 03
etx 13
dc3 23 # 33 3 43 C 53 S 63 c 73 s 04
eot 14
dc4 24 $ 34 4 44 D 54 T 64 d 74 t 05
enq 15
nak 25 % 35 5 45 E 55 U 65 e 75 u 06
ack 16
syn 26
& 36 6 46 F 56 V 66 f 76 v 07
bel 17
etb 27 ' 37 7 47 G 57 W 67 g 77 w 08 bs 18
can 28 ( 38 8 48 H 58 X 68 h 78 x 09 ht 19 em 29 ) 39 9 49 I 59 Y 69 i 79 y 0a nl 1a
sub 2a * 3a : 4a J 5a Z 6a j 7a z 0b vt 1b
esc 2b + 3b ; 4b K 5b [ 6b k 7b { 0c np 1c fs 2c , 3c
< 4c L 5c \ 6c l 7c | 0d cr 1d gs 2d - 3d = 4d M 5d ] 6d m 7d } 0e so 1e rs 2e . 3e
> 4e N 5e ^ 6e n 7e ~ 0f si 1f us 2f / 3f ? 4f O 5f _ 6f o 7f
del
Note definition did not define usage, simply name
Missing: different kinds of quotes
accent marks, diphthong,
swung dash: NOT tilda (which is missing)
bs: backspace
nl: newline
ht: (horizontal tab)
esc: escape

66. Interesting Properties of ASCII Code
What is relationship between a decimal digit (‘0’, ‘1’, …) and its ASCII code?
What is the difference between an upper-case letter (‘A’, ‘B’, …) and its lower-case equivalent (‘a’, ‘b’, …)?
Given two ASCII characters, how do we tell which comes first in alphabetical order?
Is 128 characters enough? (
“ASCIIbetical order” : why “B” comes before “a” (Note that Patt/Patel index is ASCIIbetical!)
Other properties: integers are consecutive (can do limited arithmetic by subtracting ‘0’)
No new operations -- integer arithmetic and logic.

67. Representation of non-Integers
Text, strings
Fractions
Scientific notation/Floating point representation

68. Other Data Types Text strings Image Sound
sequence of characters, terminated with NULL (0)
typically, no hardware support
Image
array of pixels
monochrome: one bit (1/0 = black/white)
color: red, green, blue (RGB) components (e.g., 8 bits each)
other properties: transparency
hardware support:
typically none, in general-purpose processors
MMX -- multiple 8-bit operations on 32- or 64-bit word
GPUs
Sound
sequence of fixed-point numbers

69. LC-3 Data Types
Some data types are supported directly by the instruction set architecture.
For LC-3, there is only one hardware-supported data type:
16-bit 2’s complement signed integer
Operations: ADD, AND, NOT
Other data types are supported by interpreting 16-bit values as logical, text, fixed-point, etc., in the software that we write.
ASCII characters appear to be supported in I/O operations

70. Fractions: Fixed-Point
How can we represent fractions?
Use a “binary point” to separate positive from negative powers of two -- just like “decimal point.”
2’s comp addition and subtraction still work
only if binary points are aligned
2-1 = 0.5
2-2 = 0.25
2-3 = 0.125
Note that alignment is required (“de-normalizing”)
Note that such computations can be done in software, but multiplication requires additional shifting
(40.625)
(-1.25)
(39.375)
No new operations -- same as integer arithmetic.

71. Scientific Notation
We can only represent 232 (~ 4 billion) or maybe 264 (~ 18 quintillion) or even 2128 (~ 3.4 x 1038 or 340,000 decillion) unique values, but there are
Infinitely many numbers between any two integers
Infinitely many numbers between any two real numbers!
We can only represent a (small) finite number of values.
These values are not spread uniformly along number line
Many numbers between zero and one
Not many numbers between 1,000,000,000,000 and 1,000,000,000,001
Use board to illustrate number line and non-representation of zero.

72. Scientific Notation
Conventional (decimal) notation: ± mantissa x 10exponent 1 ≤ mantissa < 10 exponent is signed integer Binary notation: ± mantissa x 2exponent 1 ≤ mantissa < 2
Mantissa – also called “significand” is the part of a floating point number containing the significant digits
Note that the 1 (most significant digit) can be implied, because it’s (almost) always 1.

73. Significant Digits
Accuracy of measurement leads to notion of Significant Digits
For most purposes, we don’t need high precision
Accuracy of calculations is generally limited by least precise numbers
Can represent numbers with a few significant digits
* 1023 Avogadro constant (approximately)
299,792,458 meters/sec -- Speed of Light (exactly!)
By definition, a meter is the distance light travels through a vacuum in exactly 1/ seconds …
Computable to arbitrary accuracy, but
More digits probably won’t improve result.
Use board to illustrate number line and non-representation of zero.
SPECIAL CASES: speed of light, Pi

74. Representation of Floating Point Numbers (Reals)
As with integers and chars, we ask
Which reals? There is an infinite number between two adjacent integers.
In fact, there are an infinite number between any two reals!!!!!!!
Which bit patterns for selected reals?
Answer for both strongly related to scientific notation.

75. Very Large and Very Small: Floating-Point
Large values: x requires 79 bits
Small values: x requires >110 bits
Use equivalent of “scientific notation”: F x 2E
Need to represent F (fraction), E (exponent), and sign.
IEEE 754 Floating-Point Standard (32-bits):
Exponent uses “biased” representation
Fraction has implicit 1
Exponent = 255 used for special values:
If Fraction is non-zero, NaN (not a number).
If Fraction is zero and sign is 0, positive infinity.
If Fraction is zero and sign is 1, negative infinity. 1b 8b
23b S
Exponent
Fraction

76. Floating Point Example
Single-precision IEEE floating point number:
Sign is 1 – number is negative.
Exponent field is = 126 (decimal).
Fraction is … = 0.5 (decimal).
Value = -1.5 x 2( ) = -1.5 x 2-1 =
sign
exponent
fraction
Why subtract 127???
Biased exponent is a variant of 2’s complement (basically reversing sense of the sign bit)

77. IEEE Floating-Point Standard (32-bit)
Exponent
Fraction 1b 8b
23b
Exponent = 255 used for special values:
If Fraction is non-zero, NaN (not a number).
If Fraction is zero and sign is 0, positive infinity.
If Fraction is zero and sign is 1, negative infinity.

78. Example: Show FP Representation for 40.625ten
Exponent
Fraction 1b 8b
23b
From earlier slide ten = two
Put the binary rep. into normal form (make it look like scientific notation): x 20 = x 25
5 is the true exponent; with bias: = 132ten = two
Mantissa/Fraction occupies 23 bits: (1.)
Note that “Biased” means unsigned, but with 127 added.
Fixedslide!!! Note that animated fraction has different number! 1b 8b
23b

79. Floating-Point Operations
Will regular 2’s complement arithmetic work for Floating Point numbers?
(Hint: In decimal, how do we compute 3.07 x x 108?)
Short answer: NO! (Need to “de-normalize” numbers!)

80. Floating-Point Arithmetic
Floating point operations may overflow but, more importantly, floating point operations are inherently inexact
Some numbers (e.g. “repeating decimal”) cannot be represented exactly.
Introduces the “Rounding” problem
Every inexact result creates a difference between the mathematical value and the computed value.
Errors accumulate, often benignly by cancelling out.
Worst-case accumulation of error can be enormous.
Unlike integers which use operations defined to give a correct value within a range, every two floating point numbers have infinitely many numbers between them. Only a few can be represented.
No irrational numbers can be represented exactly!