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THE BINARY NUMBERING SYSTEM
prepared by Burak Galip ASLAN ([email protected])
September, 2006
Information representation
Decimal to binary conversion table
Base 2
positional
numbering
system
Simple examples
111001
?
57
23
10111
?
Same amount of information in more number
of digits in binary form
Arithmetic overflow
integer storing capacity
1 1 1 1 1 1 1 1
27
0-255
20
1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1
215
0
2
0-65535
67890?
Arithmetic overflow!
Should be handled!
Sign-magnitude notation
sign bit
0
positive
1 1 1 1 1 1 1 1
1
negative
27 26
20
1 0 1 0 1 0 1 0
170?
- 86?
?ball?
Interpretation
Negation
0 0 0 0 1 1 0 0
+ 12
Take one’s complement
1 1 1 1 0 0 1 1
Add 1
1 1 1 1 0 1 0 0
- 12
This is two’s complement technique for negation
Binary addition
12 + (- 5)
0 0 0 0 1 1 0 0
+ 12
0 0 0 0 0 1 0 1
+5
1 0
1 0
1 0
1 0
1 1
0 1 1
0
+- 57
Binary addition
5 + (- 12)
0 0 0 0 0 1 0 1
+5
0 0 0 0 1 1 0 0
+ 12
0 0
1 0 1
0
1 1 1 1 1
--12
?
7
0 0 0 0 0 1 1 1
Binary multiplication and division
(- 3) X 2
6/3
http://courses.cs.vt.edu/~cs1104/BuildingBlocks/divide.010.html
Decimal to binary conversion
47
15
7
3
1
0
47
23
11
5
2
1
20 21 22 23 24 25 26
1
0
1
1
1
1
1
0
1
1
1
1
46
22
10
4
0
http://www.math.grinnell.edu/~rebelsky/Courses/CS152/97F/Readings/student-binary.html
Storing text / code mapping
0 0 0 0 0 0 0 1 A
0 0 0 0 0 0 1 0 B
0 0 0 0 0 0 1 1 C
8-bit code
representation
0 0 0 0 0 1 0 0 D
AD
0 0 0 0 0 0 0 1 0 0 0 0 0 1 0 0
A
D
ASCII coding standard
American Standard Code for Information Interchange
ASCII art
Reliability of binary representation
Why not decimal, octal or ternary computers?
Nature of electrical systems: bistability!
Reliability v.s. conversion time
Binary storage devices
Two stable energy states (for 0 and 1)
States separated by large energy barrier
Sense the state without destroying information
Switch states by applying energy
How about a light switch?
bulky and slow, but OK! 
Binary storage devices
Magnetic core memories (1955 – 1975)
128 X 128 bits = 2 KB memory
each ring stores 1 bit
each ring is ~ 1 mm
16 cm X 16 cm
speed up to 1 MHz
non-volatile storage
Binary storage devices
Transistor memories
transistor
extremely small
extremely fast
extremely cheap
Integrated Circuit
Binary storage devices
Attacking the complexity problem
IC: Integrated Circuits (1950’s)
SSI: Small-Scale Integration (1960’s) (10’s)
MSI: Medium-Scale Integration (1960’s) (100’s)
LSI: Large-Scale Integration (1970’s) (1000’s)
VLSI: Very-Large-Scale Integration (1980’s)
(millions of transistors)
ULSI: Ultra-Large-Scale Integration (> 1M)
Binary storage devices
Extreme intregration techniques
WSI: Wafer-Scale Integration (1980’s)
SOC: System-On-Chip Design (today)
FPGA: Field Programmable Gate Arrays (today)
(ten thousands of LSI circuits)
Summary
• What is binary representation?
• Why is it suitable to be used in computers?
• How is it realized? (abstract level)
References
• An Invitation to Computer Science,
1st Edition (1995) (Schneider & Gersting)
(Section 4.2)
• Wikipedia & Google
September, 2006