1. Lecture 5

2. 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

3. 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

5. Figure 1.12 The message “Hello.” in ASCII

6. 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

7. 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 pattern
Resolution 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

8. 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

9. 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)
(7-bit magnitude)
+17
010001
-17
110001
Problems
“0000” = “1000” !
How to add and subtract? Can’t just handle each digit at a time…

10. Attempt #2: One’s complement
To negate, invert all bits.
Decimal
Binary
3-bit mag.
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.

11. Attempt #3: Two’s complement
Invert all bits, then add 1.
Decimal
Binary
3-bit mag.
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…

12. Two’s complement – it just works!
0011 (3 decimal)
(2 decimal)
= (5 decimal ?)
0101
1101 (-3 decimal)
(-2 decimal)
= (-5 decimal ?) 
11011
0011 (3 decimal)
(-2 decimal)
= (1 decimal ?) 
10001

13. Problems with binary arithmetic
Fixed number of bits  can go out of range.
(3 decimal)
(7 decimal)
= (-6 decimal???)
Overflow: When result can’t be represented within range of bits.
In addition, if operands have same sign and sum doesn’t.

14. Summary Representations Values Binary Hexadecimal Unsigned
0000
0001
0010
0011
0100
0101
0110
0111
1000
1001
1011
1100
1101
1010
1110
1111 1 2 3 4 5 6 7 8 9 A D C B E F 10 11 12 13 14 15 -8 -7 -6 -5 -4 -3 -2 -1
Representations
Binary
Hexadecimal
Values
Unsigned
Two’s complement