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CPSC 121: Models of Computation
2010 Winter Term 2
Number Representation

Steve Wolfman, based on notes by
Patrice Belleville and others

1
Outline
• Prereqs, Learning Goals, and Quiz Notes
• Problems and Discussion
– Clock Arithmetic and Two’s Complement
– 1/3 Scottish and Fractions in Binary
– Programs and Numbers
– Programs as Numbers
• Next Lecture Notes
2
Learning Goals: Pre-Class
By the start of class, you should be able to:
– Convert positive numbers from decimal to binary
and back.
– Convert positive numbers from hexadecimal to
binary and back.
– Take the two’s complement of a binary number.
– Convert signed (either positive or negative)
numbers to binary and back.

3
Learning Goals: In-Class
By the end of this unit, you should be able to:
– Critique the choice of a digital representation
scheme—including describing its strengths,
weaknesses, and flaws (such as imprecise
representation or overflow) —for a given type of
data and purpose, such as (1) fixed-width binary
numbers using a two’s complement scheme for
signed integer arithmetic in computers or (2)
hexadecimal for human inspection of raw binary
data.

4
Where We Are in
The Big Stories
Theory                     Hardware
How do we model            How do we build devices to
computational systems?     compute?

Now: showing that our        Now: enabling our
logical models can         hardware to work with
connect smoothly to             data that’s more
models of number                meaningful to
systems.                        humans. (And once
we have numbers,
we can represent
pictures, words, sounds,   5
and everything else!)
Outline
• Prereqs, Learning Goals, and Quiz Notes
• Problems and Discussion
– Clock Arithmetic and Two’s Complement
– 1/3 Scottish and Fractions in Binary
– Programs and Numbers
– Programs as Numbers
• Next Lecture Notes
6
Prelude: Unsigned Integers
We can choose any
arrangement of Ts        #   p   q   r
and Fs to represent      0   F   F   F
numbers...               1   F   F   T
2   F   T   F
But, we might as well       3   F   T   T
choose something         4   T   F   F
convenient.              5   T   F   T

If we let F correspond to   6   T   T   F
7   T   T   T
0 and T to 1, then our
representation is...
7
Prelude: Unsigned Integers
...base 2 numbers.
When we represent                  #    p     q    r
negative numbers, the           0    0     0    0
choice is also arbitrary,       1    0     0    1
but may as well be              2    0     1    0
convenient:                     3    0     1    1
• Just one representation       4    1     0    0
for zero
5    1     0    1
• Easy to tell negative
from positive (or non-        6    1     1    0
negative?)                    7    1     1    1
• Basic operations easy.

8
But... What does it mean for basic operations to be ―easy‖?
The ―additive inverse‖ of a number x is
another number y such that x + y = 0.

What is the additive inverse of 3?
What is the additive inverse of -7?

We want to be able to add signed binary numbers. We need
x + -x to be 0. And, we want addition to be easy to implement.
9
Outline
• Prereqs, Learning Goals, and Quiz Notes
• Problems and Discussion
– Clock Arithmetic and Two’s Complement
– 1/3 Scottish and Fractions in Binary
– Programs and Numbers
– Programs as Numbers
• Next Lecture Notes
14
Problem: Clock Arithmetic
Problem: It’s 0500h.
How many hours until
midnight? Give an
algorithm that
requires a 24-hour
clock, a level, and no
arithmetic.

A level is a carpentry tool, essentially a straightedge
15
that indicates when it is either horizontal or vertical.
Clock Arithmetic
0500 is five hours from midnight.
1900 is five hours to midnight.

5 and 19 are ―additive inverses‖
in clock arithmetic: 5 + 19 = 0.

So are any other numbers that
are ―across the clock‖ from
each other.
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That’s even true for 12. Its additive inverse is itself!
Clock Arithmetic Problem
It’s 18 hundred. Without using numbers
larger than 24 in your calculations, what
time will it be 22*7 hours from now?
(Don’t multiply 22 by 7!)
a. 0 hundred (midnight)
b. 4 hundred
c. 8 hundred
d. 14 hundred
e. None of these           (Clock arithmetic is also known as 17
modular arithmetic in mathematics.)
Clock Arithmetic:
Food for Thought
If we wanted negative
numbers on the clock,
we’d probably put them                 0
―across the clock‖ from          -3
the positives.
-6
After all, if 3 + 21 is
already 0, why not put -3        -9
where 21 usually goes?                -12

18
Unsigned Binary Clock

Here’s a 3-bit unsigned
000            binary clock,
111          001
numbered from 0
(000) to 7 (111).
110              010

101          011
100
19
Crossing the Clock

To ―cross the clock‖, go
000             as many ticks left
111         001
from the top as you
previously went right
110             010     from the top.
Here’s a clock labelled
101         011       with 0 (000) to 3 (011)
100             and -1 (111) to -4
(100).
20
Reminder:
Two’s Complement
Taking two’s complement of B = b1b2b3...bn:

Flip  1 1 1 ...1
the - b1b2b3...bn
bits
----------
x1x2x3...xn
one +          1
----------
-B

21
A Different View of
Two’s Complement
Taking two’s complement of B = b1b2b3...bn:

Flip  1 1 1 ...1                   1 1 1 ...1
the - b1b2b3...bn                +          1
bits
----------                   ----------
x1x2x3...xn                1 0 0 0 ...0
one +          1                 - b1b2b3...bn
----------
----------
-B
-B
Or... Just subtract from 100...0   22
Two’s Complement vs.
Crossing the Clock
Two’s complement with k        ―Crossing the clock‖ with k
bits:                          bits:
1 1 1 ...1
+          1
000
----------                     111         001
1 0 0 0 ...0
110            010
- b1b2b3...bn
----------                     101         011
100
-B

Equivalent to subtracting      Equivalent to subtracting
from 100...000 with k 0s.      from 100...000 with k 0s.
Two’s complement turns numbers into their
23
Problem: Why Two’s
complement?
Why make the negation of 010 be 110?
a. 010 + 110 already equals 0.
b. 010 + 110 equals 1000.           000
111       001
c. 110 is the easiest
negation to calculate.
110            010
d. 110 isn’t being used
for any other purpose.
101       011
e. If you invert the bits in        100
110 and add 1, you get 010.             24
Summary Questions (1 of 2)
• With n bits, how many distinct values can we represent?

• What’s the smallest/largest n-bit unsigned binary integer?

• What’s the smallest/largest n-bit signed (using two’s complement)
binary integer?

• Why the ―extra‖ negative number?

• How many representations for each number do we have with
unsigned/signed binary integers?
25
Summary Questions (2 of 2)
• How do we tell if an unsigned binary integer is: negative, positive,
zero?

• How do we tell if a signed binary integer is: negative, positive, zero?

• How do we negate a signed binary integer?

• What one value does that negation fail (or at least do something
weird) on for signed binary integers?

• How do we calculate the decimal value of a positive signed binary
integer?

• How do we calculate the decimal value of a negative signed binary
integer?

26
Outline
• Prereqs, Learning Goals, and Quiz Notes
• Problems and Discussion
– Clock Arithmetic and Two’s Complement
– 1/3 Scottish and Fractions in Binary
– Programs and Numbers
– Programs as Numbers
• Next Lecture Notes
27
Problem: 1/3 Scottish
Problem: Can you be 1/3 Scottish?

28
Problem: 1/3 Scottish
Problem: Can you be 1/3 Scottish?
To build a model, we must clearly specify
multiple models that lead to fundamentally
different results.

We’re going to use the model of
parentage ―endowing‖ 50% of each
parent’s ―ish-ness‖. That’s a coarse, even
silly model. (By that model, none of us
are Canadian, since humans did not

Our model is handy for us, but it’s not
necessarily what people’s identity is
Can you be one-third Scottish?

Focus on Mom (and Mom’s Mom and so on).
We’ll just make Dad “Scot” or “Not” as needed at each step.

??                                             ??
Can you be one-third Scottish?

??            ??

2/3          Not
Can you be one-third Scottish?

??            ??

1/3             Scot

2/3          Not
And so on...

Mom:
1/3
Can you be one-third Scottish?
Scot

2/3          Not

1/3             Scot
What’s
happening
here?
2/3          Not
0 . 0 1 0 1..
We can represent fractions in
binary by making “Scottish family trees”:
1/3          Scot

2/3          Not

1/3                    Scot

2/3                             Not
Here’s 0.375 in binary...

0.0 1 1
0.0          Scot

0.5             Scot

0.75          Not
Problem: 1/3 Scottish
Which of the following numbers can be
precisely represented with a finite number of
digits/bits using a ―decimal point‖-style
representation in base 10 but not base 2?
a. 1/9
b. 1/8
c. 1/7
d. 1/6
e. None of these.
36
So... Computers Can’t
Represent 1/3?
No! Using a different scheme (e.g., a
rational number with a separate integer
numerator and denominator), computers
can perfectly represent 1/3!

The point is: Representations that use a
finite number of bits (all of them) have
weaknesses. Know those weaknesses
and their impact on your computations!
37
Outline
• Prereqs, Learning Goals, and Quiz Notes
• Problems and Discussion
– Clock Arithmetic and Two’s Complement
– 1/3 Scottish and Fractions in Binary
– Programs and Numbers
– Programs as Numbers
• Next Lecture Notes
38
What Doesn’t Work is
Not Always Obvious (1 of 2)
Class Main {
public static void main(String[] args) {
// Let's add up 4 quarters.
System.out.println("4 quarters gives us:");
System.out.println(0.25 + 0.25 + 0.25 + 0.25);

// Let's do something a hundred times.
int i = 100;
do {
// Make i one smaller.
i--;
} while (i > 0);

System.out.println("Done!");
System.out.println("i ended up with the value: " + i);
System.out.println("It went down by: " + (100 - i));
}
}                                                            39
What Doesn’t Work is
Not Always Obvious (2 of 2)
Class Main {
public static void main(String[] args) {
// Let's add up 10 dimes.
System.out.println("10 dimes gives us:");
System.out.println(0.1 + 0.1 + 0.1 + 0.1 + 0.1 +
0.1 + 0.1 + 0.1 + 0.1 + 0.1);
// Let's try do something a hundred times..
// but accidentally go forever
int i = 100;
do {
// Make i one LARGER. Oops!
i++;
} while (i > 0);

System.out.println("Done!");
System.out.println("i ended up with the value: " + i);
System.out.println("It went down by: " + (100 - i));
}                                                        40
}
Number Representation Prediction
// Let's add up 10 dimes.
System.out.println("10 dimes gives us:");
System.out.println(0.1 + 0.1 + 0.1 + 0.1 + 0.1 +
0.1 + 0.1 + 0.1 + 0.1 + 0.1);
// Let's try do something a hundred times..
// but accidentally go forever
int i = 100;
do {
// Make i one LARGER. Oops!
i++;
} while (i > 0);
What will this print?
a. First 1.0 and then nothing because it runs forever.
b. First 1.0 and then some other value, because it won’t run forever.
c. First something OTHER than 1.0 and then nothing because it runs forever.
d. First something OTHER than 1.0 and then some other value,
because it won’t forever.                                          41
e. None of these
Outline
• Prereqs, Learning Goals, and Quiz Notes
• Problems and Discussion
– Clock Arithmetic and Two’s Complement
– 1/3 Scottish and Fractions in Binary
– Programs and Numbers
– Programs as Numbers
• Next Lecture Notes
42
Preface: Java Byte Code
Java programs are compiled to a language called ―byte
code‖ that is then interpreted on a particular computer.
Why? Byte code is hard for humans to write, read, and
understand... but it’s easy to write a program that reads
and executes it (compared to writing a program to
directly read and execute Java source code).
So, if you create a brand new type of computer tomorrow,
and I want to run my Java code on it, I don’t have to
write a program that works on your computer and knows
how to execute Java; I just need to write a program that
knows how to execute byte code.
Java byte code is also designed to be compact so it’s
cheap to transmit across the internet.
Problem: Java Byte Code
Problem: When compiled to bytecode, i = 100 might be
―push 100; store in variable 1‖. The ―opcode‖ for
bipush (push a byte) is 1610. The opcode for
istore_1 is 6010. Here’s a typical ―hex‖ view of ~1/5th
of the previous program’s byte code. Where is i =
100?                     a

b
b
d
c
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e: None of these.
Problem:
Binary Byte Code
Why would the same task (finding a particular snippet of
code in a bytecode file) be much more difficult if the
file were represented in binary?
a. Because we would have to translate all the opcodes
and values to binary.
b. Because many bytecode files would have no binary
representation.
c. Because the binary representation of the file would
be much longer.
d. Because data like 1100100 (100 in base 2) might not
show up as the sequence of numbers 1 1 0 0 1 0 0.
e. It wouldn’t be much more difficult.

45
Problem:
Decimal Byte Code
Why would the same task (finding a particular
snippet of code in a bytecode file) be much more
difficult if the file were represented in decimal?
a. Because we would have to translate all the
opcodes and values to decimal.
b. Because many bytecode files would have no
decimal representation.
c. Because the decimal representation of the file
would be much longer.
d. Because data like 100 might not show up as the
sequence of numbers 1 0 0.
e. It wouldn’t be much more difficult.
46
Outline
• Prereqs, Learning Goals, and Quiz Notes
• Problems and Discussion
– Clock Arithmetic and Two’s Complement
– 1/3 Scottish and Fractions in Binary
– Programs and Numbers
– Programs as Numbers
• Next Lecture Notes
47
Learning Goals: In-Class
By the end of this unit, you should be able to:
– Critique the choice of a digital representation
scheme—including describing its strengths,
weaknesses, and flaws (such as imprecise
representation or overflow) —for a given type of
data and purpose, such as (1) fixed-width binary
numbers using a two’s complement scheme for
signed integer arithmetic in computers or (2)
hexadecimal for human inspection of raw binary
data.

48
Next Lecture Learning Goals:
Pre-Class
By the start of class, you should be able to:
– Use truth tables to establish or refute the
validity of a rule of inference.
– Given a rule of inference and propositional
logic statements that correspond to the rule’s
premises, apply the rule to infer a new
statement implied by the original statements.

49
Next Lecture Prerequisites
Read Section 1.3 (Epp 3rd ed) or 2.3 (Epp 4th
ed).
Complete the open-book, untimed quiz on
Vista that is due before next class.

50
snick

snack

Some Things to Try...

(on your own if you have time,
not required)

51
Problem: Weighty Numbers
Problem: You have a balance scale and
four weights. You may choose the mass
of the weights, as long as they’re in whole
units of grams. What’s the largest number
n such that you can exactly measure
every weight 0…n?

?g        ?g
?g        ?g
52
Problem: Representing Data
Problem: Devise two different ways to
represent each of the following with bits:
• black-and-white images
• text
• the shape of your face

53
Representing Characters

54
Problem:
256-hour Clock Arithmetic
Problem: Imagine you’ve built a computer
that uses 256-hour clock faces, each with
a single dial, as storage units. How would
you store, add, subtract, and negate
integers?

55
Concept Q: 256-hour Clock
Arithmetic
Java has a type called ―byte‖ that is an 8-bit signed
integer. What will the following code print?
byte b = 70;
b = b + 64;
b = b + 64;
b = b + 64;
a. A positive number, greater than 70.
b. 70.
c. A positive number, less than 70.
d. 0.
e. A negative number.
56
Problem: Number Rep
Breakdown
Problem: Explain what’s happening in each
of these…

57
Program for Introductions:
Testing
Java version with 100: 9900
Java version with 8675309: 265642364
Java version with 1526097757: -645820308

Reminder: this is to calculate n*(n-1), the number of
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introductions for a group of size n.
Program for Introductions:
Testing
Racket version with 100: 9900
Racket version with 8675309:
75260977570172
Racket version with 1526097757:
2328974362394333292

Reminder: this is to calculate (* n (- n 1)), the
59
number of introductions for a group of size n.

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