Numbers and number systems

1
Numbers and number systems

• Lecture 2

2
Some material from the last lecture

• Electronic computers represent information as
  voltage levels.
• To make the computer hardware simple and
  reliable, computers represent information in
  binary form.
• example voltages greater than 3V are interpreted
  as representing one value (called 1), voltages
  less than 2V are interpreted as representing
  another value (called 0).
• In principle, could use more voltage levels.
• example 0 to .75V represents 0, 1 to 1.75V
  represents 1, 2 to 2.75V represents 2, and so
  forth.
• In practice, this is rarely done.
• requires more complex circuits
• circuits are more susceptible to noise, hence
  less reliable

3
Some material from the last lecture

• Computers, like all electronic systems, are
  affected by noise.
• noise has various sources (nearby signal changes,
  thermal vibrations of molecules in semiconductor
  materials, . . . )
• in computers, noise can cause binary
• signals to be misinterpreted
• The noise margin is the amountof noise that a
  system cantolerate and still correctlyidentify
  a logic high or low.

4
Radix number systems

• Some number of positions and some number of
  symbols
• The number of positions varies by context
• The number of symbols is a property of the number
  system
• Decimal -- 10 symbols
• Binary -- 2 symbols
• Octal -- 8 symbols
• Hexadecimal -- 16 symbols

5
Start with whole numbers

• Each position has a value
• Each symbol has a value
• Multiply the value of the symbol by the value of
  the position, then add
• In decimal, 3874 means
• 3 times 1,000
• plus 8 time 100
• plus 7 times 10
• plus 4 times 1

6
Decimal, binary, octal, hex

• In decimal there are 10 symbols (0..9) and the
  value of each position is a power of 10.
• 100 1 value of the units position
• 101 10 value of next position to the left
• etc.
• In binary, there are 2 symbols, 0 and 1, and the
  value of each position is a power of 2.
• In octal, 8 symbols, and powers of 8
• In hexadecimal, 16 symbols, and powers of 16

7
Most Significant digit
Decimal Number 378.4
Least Significant digit
Weight
3 7 8 4
. is called the radix point 3 is the MSD 4 is the
LSD
8
Least Significant bit
Most Significant bit
weight 27 26 25 24 23 22 21 20
1 1 0 0 1 0 0 1
110010012 1?27 1?26 1?23 1?20 201
Least Significant bit
Most Significant bit
weight 25 24 23 22 21 20 2-1 2-2
1 1 0 0 1 0 0 1
110010.012 1?25 1?24 1?21 1?2-2 50.25
9
K, M and G

• 210 is referred as K (kilo)
• 220 is referred as M (mega)
• 230 is referred as G (giga)

10
Trinary Numbers
Trinary number 110010023
weight 37 36 35 34 33 32 31 30
1 1 0 0 1 0 0 2
110010013 1?37 1?36 1?33 2?30 2945
11
Octal Numbers
Octal number 127.48
weight 82 1 81 2 80 7 8-1 4
82 1 81 2 80 7 8-1 4
110010013 1?82 2?81 7?80 4?8-1 87.5
12
Hexadecimal Numbers
Hex Dec
0 0
1 1
2 2
3 3
4 4
5 5
6 6
7 7
8 8
9 9
A 10
B 11
C 12
D 13
E 14
F 15
Hexadecimal number FA9H
weight 162 161 160
F A 9
FA9H 15?162 10?161 9?160 4009
13
Things to do

• Review from your class notes what we discussed
  today.
• Conversions between Number Systems
• Binary, octal and hexadecimal
• Solve 1-6 before coming to the
• class