CIS 115
Lecture 8: Encoding Data
Binary - Natural Numbers
| 128 | 64 | 32 | 16 | 8 | 4 | 2 | 1 |
| 0 | 0 | 1 | 0 | 1 | 0 | 1 | 0 |
| 0*128 | 0*64 | 1*32 | 0*16 | 1*8 | 0*4 | 1*2 | 0*1 |
32 + 8 + 2 = 42
Binary Data Types
- Unsigned Integer (Natural Number)
- Signed Integer
- Float
Negative Numbers
- One's Compliment - Just invert the bits
- Sign Bit: 0 is positive, 1 is negative
| 0 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 42 |
Negative Numbers
- One's Compliment - Just invert the bits
- Sign Bit: 0 is positive, 1 is negative
| 0 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 42 |
| 1 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | -42 |
One's Compliment Addition
| 0 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 42 |
+
| 1 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | -42 |
=
One's Compliment Addition
| 0 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 42 |
+
| 1 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | -42 |
=
| 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | -0 |
Hmm, that's not quite right...
Negative Numbers
- Two's Compliment -
Invert the bits and add 1
| 0 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 42 |
invert
| 1 | 1 | 0 | 1 | 0 | 1 | 0 | 1 |
Negative Numbers
- Two's Compliment -
Invert the bits and add 1
| 0 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 42 |
invert
| 1 | 1 | 0 | 1 | 0 | 1 | 0 | 1 |
plus 1
| 1 | 1 | 0 | 1 | 0 | 1 | 1 | 0 | -42 |
Two's Compliment Addition
| 0 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 42 |
+
| 1 | 1 | 0 | 1 | 0 | 1 | 1 | 0 | -42 |
=
Two's Compliment Addition
| 0 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 42 |
+
| 1 | 1 | 0 | 1 | 0 | 1 | 1 | 0 | -42 |
=
| 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |
That works!
Other Values
| Binary | Unsigned | Signed |
| 00000000 | 0 | 0 |
| 00000001 | 1 | 1 |
| 00000010 | 2 | 2 |
| 01111110 | 126 | 126 |
| 01111111 | 127 | 127 |
| 10000000 | 128 | -128 |
| 10000001 | 129 | -127 |
| 10000010 | 130 | -126 |
| 11111110 | 254 | -2 |
| 11111111 | 255 | -1 |
Range of Values
- 8 Bit numbers
- Unsigned: 0 → 28 - 1
- Signed: -(27) → 27 - 1
- General Numbers - n bits
- Unsigned: 0 → 2n - 1
- Signed: -(2n-1) → 2n-1 - 1
Rational Numbers
The decimal point can "float" around
Floating Point
- IEEE 754 Standard - 16 bits (Half)
- The exponent has a bias of 15
- The leading one of the mantissa is implied
| - | Exponent | Mantissa | |||||||||||||
| 0 | 0 | 1 | 0 | 1 | 0 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 | 1 |
Floating Point Example
| - | Exponent | Mantissa | |||||||||||||
| 0 | 1 | 0 | 1 | 0 | 0 | 0 | 1 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
Mantissa: (1).01010 = 1.3125
Exponent: 10100 - 01111 = 20 - 15 = 5
Floating Point Example
| - | Exponent | Mantissa | |||||||||||||
| 0 | 1 | 0 | 1 | 0 | 0 | 0 | 1 | 0 | 1 | 0 | 0 | 0 | 0 | 0 | 0 |
Mantissa: (1).01010 = 1.3125
Exponent: 10100 - 01111 = 20 - 15 = 5
Value: 1.3125 * 25 = 42
1.01010 * 25 = 101010 = 42
Range of Values
- -65504 → +65504
- 5.96046 x 10-8 : minimum positive
- 0 11111 0000000000 = infinity
- 1 11111 0000000000 = -infinity
- 0 01101 0101010101 ≈ 0.33325 ≈ 1/3
Not exact, but not bad either
Real World
- Integer - 32 bits
- Long Integer - 64 bits
- Half - 16 bits (5 + 10)
- Float (Single) - 32 bits (8 + 23)
- Double - 64 bits (11 + 52)
Text - ASCII
011001100110111101110010011101000111
100100100000011101000111011101101111
Text - ASCII
011001100110111101110010011101000111
100100100000011101000111011101101111
- 01100110 (102)
- 01101111 (111)
- 01110010 (114)
- 01110100 (116)
- 01111001 (121)
- I00100000 (32)
- 01110100 (116)
- 01110111 (119)
- 01101111 (111)
Text - ASCII
011001100110111101110010011101000111
100100100000011101000111011101101111
- 01100110 (102) - f
- 01101111 (111) - o
- 01110010 (114) - r
- 01110100 (116) - t
- 01111001 (121) - y
- I00100000 (32) - sp
- 01110100 (116) - t
- 01110111 (119) - w
- 01101111 (111) - o
Compression
How much wood could
a woodchuck chuck if a
woodchuck could chuck wood?
Compression
How much wood could
a woodchuck chuck if a
woodchuck could chuck wood?
wood = 1 could = 2 chuck = 3
Compression
How much wood could
a woodchuck chuck if a
woodchuck could chuck wood?
wood = 1 could = 2 chuck = 3
How much 1 2 a 13 3 if a 13 2 3 1?
Assignments
- Read and be prepared to discuss:
- Pattern on the Stone Chapter 7: Speed: Parallel Computers
- Blog 4: Computer Systems in Daily Life - Due 2/24 10:00 PM
- Scratch Sorting Project - Due 2/19 10:00 PM
Blog 4: Computer Systems in Daily Life
We interact with a variety of computer systems on a daily basis, but most of the time we don’t take the time to think about where they came from and how they work. Choose a computer system you see in your everyday life and write about it and its history. Tell us how it works and how it affects us in our daily life. A simple example would be the keycard entry systems at the K-State dorms and the engineering labs. Some questions to ask yourself while you are doing your research:
- Who created this system?
- What are the underlying technologies?
- Is this a very common system, or something unique to a particular field or use?
- How long has this system been around?
- How difficult would life be without this system?
- Are there any privacy, security or safety concerns with this system? Are there any concerns if this system didn’t exist?
Binary Worksheet