Mod odule 3: Representing Values in a Com omputer Du Due Da - - PowerPoint PPT Presentation

Mod

• dule 3: Representing

Values in a Com

• mputer

2

Du Due Da Dates

• Assignment #1 is due Thursday January 19th at

4pm

• Pre-class quiz #4 is due Thursday January 19th at

7pm.

• Assigned reading for the quiz:
• Epp, 4th edition: 2.3
• Epp, 3rd edition: 1.3
• Come to my office hours (ICCS X563)!
• Wednesday 4-5pm, Thursday 2-3pm, and

Friday 12-1pm

3

Le Learn rning goals: : pre-cl class

• By the start of this class you should be able to
• Convert unsigned integers from decimal to binary

and back.

• Take two's complement of a binary integer.
• Convert signed integers from decimal to binary

and back.

• Convert integers from binary to hexadecimal and

back.

• Add two binary integers.

4

Le Learn rning goals: : in-cl class

• By the end of this module, 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

• fixed-width binary numbers using a two’s

complement scheme for signed integer arithmetic in computers

• hexadecimal for human inspection of raw binary

data.

5

CP CPSC SC 121: 121: the BI BIG ques questions ns:

• We will make progress on two of them:
• How does the computer (e.g. Dr. Racket) decide if

the characters of your program represent a name, a number, or something else? How does it figure

• ut if you have mismatched " " or ( )?
• How can we build a computer that is able to

execute a user-defined program?

8

Mo Module 3 3 outline

• Unsigned and signed binary integers.
• Characters.
• Real numbers.
• Hexadecimal.

Re Review question

• What does it mean for a binary integer to be

unsigned or signed?

10

Re Recall the 7-se segm gment disp splay y problem

• Mapping unsigned integers between decimal and

binary

Number X1 X2 X3 X4 1 1 2 1 3 1 1 4 1 5 1 1 6 1 1 7 1 1 1 8 1 9 1 1

11

Un Unsig igned ed bin inar ary à dec decimal

• The binary value
• Represents the decimal value
• Examples:
• 010101 = 24 + 22 + 20 = 16 + 4 + 1 = 21
• 110010 = 25 + 24 + 21 = 32 + 16 + 2 = 50

De Decim cimal al à uns unsigned gned bi bina nary

• Divide x by 2 and write down the remainder
• The remainder is 0 if x is even, and 1 if x is odd.
• Repeat this until the quotient is 0.
• Write down the remainders from right (first one)

to left (last one).

De Decim cimal al à uns unsigned gned bi bina nary

• Example: Convert 50 to a 8-bit binary integer
• 50 / 2 = 25with remainder 0
• 25 / 2 = 12 with remainder 1
• 12 / 2 = 6 with remainder 0
• 6 / 2 = 3 with remainder
• 3 / 2 = 1 with remainder

1

• 1 / 2 = 0 with remainder

1

• Answer is 00110010.

14

To To negate a (signed) binary integer

The algorithm (also called taking two’s complement

• f the binary number):
• Flip all bits: replace every 0 bit by a 1, and every 1

bit by a 0.

• Add 1 to the result.

Why does it make sense to negate a binary integer by taking its two’s complement?

Addi Additive In Inver erse

• Two numbers x and y are “additive inverses” of

each other if they sum to 0.

• Examples: 3 + (-3) = 0. (-7) + 7 = 0.

15

16

Cl Clock k ari rithme metic

• Pre-class quiz #3b:
• It is currently 18:00, that is 6pm.

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!)

Cl Clock k ari rithme metic

• 3:00 is 3 hour from midnight.
• 21:00 is 3 hours to midnight.
• 21 and 3 are additive inverses in

clock arithmetic: 21 + 3 = 0.

• Any two numbers across the clock

are additive inverses of each other.

• 21 is equivalent to -3 in any

calculation.

18

Cl Clock k ari rithme metic

• Pre-class quiz #3b:
• It is currently 18:00, that is 6pm.

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. 0:00 (midnight)
• b. 4:00 (4am)
• c. 8:00 (8am)
• d. 14:00 (2pm)
• e. None of the above

Ne Negative n numbers i in cl clock ck a arithmetic

• Put negative numbers

across the clock from the positive ones (where the additive inverses are).

• Since 21 + 3 = 0, we

could put -3 where 21 is.

• 3
• 6
• 9
• 12

Bi Binary y clock k ari rithme metic

• Suppose that we have 3 bits to work with.
• If we are only representing positive values or zero:

000 001 010 011 100 101 110 111

1 2 3 4 5 6 7

Bi Binary y clock k ari rithme metic

• If we want to represent negative values...
• We can put them across the clock from the

positive ones (where the additive inverses are).

000 001 010 011 100 101 110 111

1 2 3

• 4
• 3
• 2
• 1

Tw Two’s Complement

Taking two’s complement of B = b1b2b3...bn: 1 1 1 ...1

• b1b2b3...bn
• x1x2x3...xn

+ 1

• B

22

Flip the bits Add

• ne

A Different Vi View of Tw Two’s Complement

Taking two’s complement of B = b1b2b3...bk:

1 1 1 ...1

• b1b2b3...bk
• x1x2x3...xk

+ 1

• B

Equivalent to subtracting from 100...000 with k 0s.

23

Flip the bits Add

• ne

1 1 1 ...1 + 1

• 1 0 0 0 ...0
• b1b2b3...bk
• B

Tw Two’s Complement vs. Cr Crossi ssing the Cl Clock

Two’s complement with k bits: Equivalent to subtracting from 100...000 with k 0s.

24

1 1 1 ...1 + 1

• 1 0 0 0 ...0
• b1b2b3...bn
• B

000 001 010 011 100 101 110 111

1 2 3

• 4
• 3
• 2
• 1

Adv Advantages s of f two’s s compl plement

• Which one(s) are the advantages of the two’s

complement representation scheme?

• a. There is a unique representation of zero.
• b. It is easy to tell a negative integer from

a non-negative integer.

• c. Basic operations are easy. For example,

subtracting a number is equivalent to adding the two’s complement of the number.

• d. All of (a), (b), and (c).
• e. None of (a), (b), and (c).

Adv Advantages s of f two’s s compl plement

• Why did we choose to let 100 represent -4 rather

than 4?

000 001 010 011 100 101 110 111

1 2 3

• 4
• 3
• 2
• 1

000 001 010 011 100 101 110 111

1 2 3 4

• 3
• 2
• 1

Wha What do does s two’s s compl plement mean? n?

• a. Taking two’s complement of a binary integer

means flip all of the bits and then add 1.

• b. Taking two’s complement is a mathematical

procedure to negate a binary integer.

• c. Two’s complement is a representation scheme

we use to map signed binary integers to decimal integers.

• d. 2 of (a), (b), and (c) are true.
• e. All of (a), (b), and (c) are true.

29

Ne Negative b binary à dec decimal

• Example: convert the 6-bit signed binary integer

101110 to a decimal number.

• 1. Convert the binary integer directly to decimal

(2 + 4 + 8 + 32 = 46)

• 2. Subtract 2^6 from it (2^6 because of 6-bit)

(46 – 64 = -18).

• 3. Answer is -18.

30

Ne Negative b binary à dec decimal

• The signed binary value
• represents the integer
• Example: convert the 6-bit signed binary integer

101110 to a decimal number.

• -25 + 23 + 22 + 21 = -32 + 8 + 4 + 2 = -18.
• Answer is -18.

Ne Negative d deci cimal à bi bina nary

• Example: convert -18 to a 6-bit signed binary

number.

• 1. Add 2^6 to it (2^6 because of 6-bit)

(-18 + 2^6 = -18 + 64 = 46).

• 2. Convert the positive decimal integer directly to
• binary. 46 / 2 = 23 … 0, 23 / 2 = 12 ... 1, 12 / 2 = 6

... 0, 6 / 2 = 3 ... 0, 3 / 2 = 1 ... 1, 1 / 2 = 0 ... 1.

• 3. Answer is 110010.

32

Qu Questions to ponder: r:

• With n bits, how many distinct values can we

represent?

• What are the smallest and largest n-bit unsigned

binary integers?

• What are the smallest and largest n-bit signed

binary integers?

33

Mo More questions to ponder: r:

• Why are there more negative n-bit signed integers

than positive ones?

• 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?
• On what value does this “negation” not behave

like negation on integers?

• How do we perform the subtraction x – y?

34

Mo Module 3 3 outline

• Unsigned and signed binary integers.
• Characters.
• Real numbers.
• Hexadecimal.

35

How do computers represent ch charact cters?

• It uses sequences of bits (like for everything else).
• Integers have a “natural” representation of this kind.
• There is no natural representation for characters.
• So people created arbitrary mappings:
• EBCDIC: earliest, now used only for IBM mainframes.
• ASCII: American Standard Code for Information

Interchange7-bit per character, sufficient for upper/lowercase, digits, punctuation and a few special characters.

• UNICODE:16+ bits, extended ASCII for languages other

than English

36

Re Representing characters

What does the 8-bit binary value 11111000 represent? a.-8 b.The character c.248 d.More than one of the above e.None of the above.

ø

Show Answer

38

Mo Module 3 3 outline

• Unsigned and signed binary integers.
• Characters.
• Real numbers.
• Hexadecimal.

39

Ca Can so some meone be 1/ 1/3r 3rd Sc Scottish sh?

• No, but what if I once wore a kilt?
• Not normally, since every person's genetic code is

derived from two parents, branching out in halves going back. However, someone with a chromosome abnormality that gives them three chromosomes (trisomy), a person might be said to be one-third/two-thirds of a particular genetic marker.

40

Ca Can so some meone be 1/ 1/3r 3rd Sc Scottish sh?

• I have come to the conclusion that no, you cannot be one-

third Scottish. I will provide my reasoning with the following two examples. Say there are two progenitors, P and Q. P is Korean and Q is American. If P and Q have a child, PQ, he will be 1/2 Korean and 1/2 American. Now say that PQ grows up to the legal age (no pedophillia in here) and enters a romantic relationship with another progenitor, R, who is Scottish. PQ and R have a child, PQR, and he will be 1/2 Scottish, 1/4 Korean, and 1/4 American. The second example, we can say PQ conceives a child, PQUT, with someone named UT, who is 1/2 Ethiopian and 1/2 African. PQUT would be 1/4 of each ethnicity. We can see that you will only be (1/2)^n - where n is a nonnegative integer - of an ethnicity (n in this context means which generation it is). Notice that the denominator will always be a multiple of 2. Therefore, you can never be 1/3 of any ethnicity.

41

Ca Can so some meone be 1/ 1/3r 3rd Sc Scottish sh?

While debated, Scotland is traditionally said to be founded in 843AD, approximately 45 generations ago. Your mix of Scottish, will therefore be n/2

45; using 2

45/3 (rounded to

the nearest integer) as the numerator gives us 11728124029611/2

45 which give us

approximately 0.333333333333342807236476801 which is no more than 1/10

13 th away from

1/3.

42

Ca Can so some meone be 1/ 1/3r 3rd Sc Scottish sh?

If we assume that two Scots have a child, and that child has a child with a non-Scot, and this continues in the right proportions, then eventually their Scottishness will approach 1/3: This is of course discounting the crazy citizenship laws we have these days, and the effect of wearing a kilt on one's heritage.

43

Ca Can so some meone be 1/ 1/3r 3rd Sc Scottish sh?

In a mathematical sense, you can create 1/3 using infinite sums of inverse powers of 2 1/2 isn't very close 1/4 isn't either 3/8 is getting there... 5/16 is yet closer, so is 11/32, 21/64, 43/128 etc 85/256 is 0.33203125, which is much closer, but which also implies eight generations of very careful romance amongst your elders. 5461/16384 is 0.33331298828125, which is still getting there, but this needs fourteen generations and a heck of a lot of Scots and non-Scots.

Ca Can so some meone be 1/ 1/3rd

rd Sc

Scottish?

• To answer this question, we need to make some

assumptions.

• What does being Scottish mean? Nationality?

Ethic identity?

• How does the Scottish-ness of a parent influence

the Scottish-ness of a child?

• Our model of ancestry: Each parent gives you 50%
• f an ancestry.

45

Ca Can so some meone be 1/ 1/3rd

rd Sc

Scottish?

• Suppose we start with people who are either 0%
• r 100% Scottish.
• After 1 generation, how Scottish can a child be?
• After 2 generations, how Scottish can a grand-

child be?

• What about 3 generations?
• What about n generations?

46

Re Representing real numbers in binary

Which of the following values have a finite binary representation? a.1/4 b.1/6 c.1/9 d.More than one of the above. e.None of the above.

▷

Fr Frac actio tions ns in in bas base e 10 and and bas base e 2

• 5/8 = 0.625 (in base 10) = 0.101 (in base 2)
• Can be represented exactly in both.
• 1/3 = 0.333…(in base 10) = 0.0101...(in base 2)
• Can be represented exactly in neither.
• Could you come up with a number that can be

represented exactly in base 10 but not in base 2?

• Could you come up with a number that can be

represented exactly in base 2 but not in base 10?

48

Ho How w do does es co computer represent va values

• f
• f t

the f for

• rm xxx.

xxx.yyyy?

Java uses scientific notation 1724 = 1.724 x 10

3

But in binary, instead of decimal. 1724 = 1.1010111100 x 2

1010

Only the mantissa and exponent need to be stored. The mantissa has a fixed number of bits (24 for float, 53 for double). Scheme/Racket uses this for inexact numbers.

mantissa exponent

49

Flo Floating ting po poin int t comput putatio tions ns

• Computations involving floating point numbers

are imprecise.

• The computer does not store 1/3, but a number

that's very close to 1/3.

• The more computations we perform, the further

away from the “real” value we are.

50

Flo Floating ting po poin int t comput putatio tions ns

• Predict the output of:

(* #i0.1 0.1 0.1 10 10 10)

• a. 1
• b. Not 1, but a value close to 1.
• c. Not 1, but a value far from 1.
• d. I don’t know.

▷

51

Flo Floating ting po poin int t comput putatio tions ns

Consider the following: What value will (addfractions 0) return? a. 10 b. 11 c. Less than 10 d. More than 11 e. No value will be printed

(define (addfractions x) (if (= x 1.0) (+ 1 (addfractions (+ x #i0.1)))))

▷

52

Mo Module 3 3 outline

Summary Unsigned and signed binary integers. Characters. Real numbers. Hexadecimal.

53

Mo Module 3. 3.4: 4: Hexadecima mal

As you learned in CPSC 110, a program can be Interpreted: another program is reading your code and performing the operations indicated. Example: Scheme/Racket Compiled: the program is translated into machine language. Then the machine language version is executed directly by the computer.

54

Mo Module 3. 3.4: 4: Hexadecima mal

What does a machine language instruction look like? It is a sequence of bits! Y86 example: adding two values. In human-readable form: addl %ebx, %ecx. In binary: 0110000000110001

Arithmetic operation Addition %ebx %ecx

55

Mo Module 3. 3.4: 4: Hexadecima mal

Long sequences of bits are painful to read and write, and it's easy to make mistakes. Should we write this in decimal instead? Decimal version: 24625. Problem: Solution: use hexadecimal 6031

Arithmetic operation Addition %ebx %ecx

We can not tell what operation this is.

56

Mo Module 3. 3.4: 4: Hexadecima mal

Another example: Suppose we make the text in a web page use color 15728778. What color is this?

Red leaning towards purple.

Written in hexadecimal: F00084

Red Green Blue