Lecture 5

Topics

Sec 1.4 Representing Information as Bit Patterns

Representing Text Representing Numeric Values Representing Images

Sec 1.6 Storing Integers Two’s Complement Notation Excess Notation

Representing Text Each character (letter, punctuation,

etc.) is assigned a unique bit pattern.

ASCII: Uses patterns of 7-bits to represent most symbols used in written English text

Unicode: Uses patterns of 16-bits to represent the major symbols used in languages world side

ISO standard: Uses patterns of 32-bits to represent most symbols used in languages world wide

Figure 1.12 The message “Hello.” in ASCII

Representing Numeric Values

Binary notation: Uses bits to represent a number in base two

Limitations of computer representations of numeric values

Overflow – occurs when a value is too big to be represented

Truncation – occurs when a value cannot be represented accurately

Representing Images Images are stored using a

variety of formats and compression techniques

The simplest representation is a bitmap

Bitmaps partition an image into a grid of picture elements, called pixels, and then convert each pixel into a bit patternResolution refers to the

sharpness or clarity of an image bitmaps that are divided into

smaller pixels will yield higher resolution images

the left image is stored using 96 pixels per square inch, and the right image is stored using 48 pixels per square inch

the left image appears sharp, but has twice the storage requirements

What about “negative seventeen”?

Can use “+” and “-”, as usual.

But requires extra symbols. How can we use bits instead?

Decimal Binary

+17 +10001

-17 -10001

Attempt #1: Sign & magnitude Fix the #bits used to represent the magnitude.

Q: What is the range of “n” bits? Replace “+” with “0”, and “-” with “1”.

Decimal Binary(5-bit magnitude)

Binary(7-bit magnitude)

+17 010001 00010001

-17 110001 10010001

Problems• “0000” = “1000” !• How to add and subtract? Can’t just handle each digit at a time…

Attempt #2: One’s complement To negate, invert all bits.

Decimal Binary3-bit mag.

Decimal Binary3-bit mag.

0 0000 -0 1111

1 0001 -1 1110

2 0010 -2 1101

3 0011 -3 1100

4 0100 -4 1011

5 0101 -5 1010

6 0110 -6 1001

7 0111 -7 1000

Still same problems – ambiguous zero, can’t add/subtract one bit at a time.

Attempt #3: Two’s complement Invert all bits, then add 1.

Decimal Binary3-bit mag.

Decimal Binary3-bit mag.

0 0000

1 0001 -1 1111

2 0010 -2 1110

3 0011 -3 1101

4 0100 -4 1100

5 0101 -5 1011

6 0110 -6 1010

7 0111 -7 1001

-8 1000

No ambiguous zero. Adding and subtracting work nicely…

Two’s complement – it just works! 0011 (3 decimal)

+ 0010 (2 decimal)

= (5 decimal ?)

1101 (-3 decimal)

+ 1110 (-2 decimal)

= (-5 decimal ?)

0011 (3 decimal)

+ 1110 (-2 decimal)

= (1 decimal ?)

0101

11011

10001

Problems with binary arithmetic Fixed number of bits can go out of

range.

0011 (3 decimal)

+ 0111 (7 decimal)

= 1010 (-6 decimal???)

Overflow: When result can’t be represented within range of bits.

In addition, if operands have same sign and sum doesn’t.

Summary

Representations Binary Hexadecimal

Values Unsigned Two’s complement

0000 0001

0010

0011

0100

0101

0110

011110001001

1011

1100

1101

1010

1110

1111

01

2

3

4

5

6

789

A

D

C

B

E

F

0 1

2

3

4

5

6789

10

11

12

13

14

15

0 1

-8

3

4

5

67

2

-7-6

-5

-4

-3-2

-1