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9/15/09 - L1 Crs Ovrvw Copyright 2009 - Joanne DeGroat, ECE, OSU 2
Class 1 outline Course Syllabus Information Representation Number Systems
Material from sections 1-1 and 1-2 of text
Personal course website ece.osu.edu/~degroat
Information Representation In the real world
Items that we want to measure are continuous, i.e., they can have any value Weight Temperature Pressure Velocity And many, many others
9/15/09 - L1 Crs Ovrvw Copyright 2009 - Joanne DeGroat, ECE, OSU 3
But in a computer The paradigm of a computer that we are
familiar with is electronic. But there have been others like Babbage’s compute engine and cash registers which were mechanical.
The electronics make it easy to represent two states, on and off, or high voltage level and low voltage level.
9/15/09 - L1 Crs Ovrvw Copyright 2009 - Joanne DeGroat, ECE, OSU 4
Realization Result: representation of real world quantities
by a system that can only represent discrete states, and thus discrete quantities.
Example Use you fingers to represent temperature from
10oF to 100oF. What values did you represent?
9/15/09 - L1 Crs Ovrvw Copyright 2009 - Joanne DeGroat, ECE, OSU 5
Realization Result: representation of real world quantities by
a system that can only represent discrete states, and thus discrete quantities.
Example Use you fingers to represent temperature from 10oF
to 100oF. What values did you represent? Ans: partially depends on whether it was in 8 or 10
steps. So it was either in 10oF steps or 12.5oF steps.9/15/09 - L1 Crs Ovrvw Copyright 2009 - Joanne DeGroat, ECE, OSU 6
Digital Systems Digital electronics have progressed from 12V
logic elements to 1.2V logic elements. Could use these values OR Use a representation of 1 for a HIGH And a representation of 0 for a LOW
In electronics we restrict ourselves to just these two states to provide interpretation of a range of voltages as HIGH or a 1 and another range as LOW or a 0. Why?
9/15/09 - L1 Crs Ovrvw Copyright 2009 - Joanne DeGroat, ECE, OSU 7
The Digital Computer “Virtually every aspect of digital system
design is encompassed in a computer design” (Hill and Peterson)
“Computers are the most important type of digital sytem”
Computers and digital systems have become pervasive in our world From cell phones to MP3 players to ……
9/15/09 - L1 Crs Ovrvw Copyright 2009 - Joanne DeGroat, ECE, OSU 8
Computers & digital systems today There everywhere, there everywhere
PCs and Workstations Cell phones MP3 players Cameras Your automobile – an automobile will have 15 to 40
embedded processors Microwaves Stoves, Dishwashers, fautces (auto temp control) Washers, Dryers, Vacuum cleaners And on and on and on and on and ………….
9/15/09 - L1 Crs Ovrvw Copyright 2009 - Joanne DeGroat, ECE, OSU 9
The computer Major components
The CPU – central processing unit – the datapath combined with the control unit Datapath – performs the actual arithmetic and logic
functions on the data Control Unit – THE BRAIN – controls the flow of data
and instructions, decoding and executing instructions MEMORY – Consists of both registers, main
memory and secondary memory I/O or Input/Output – communication channels to get
information into and out of the computer
9/15/09 - L1 Crs Ovrvw Copyright 2009 - Joanne DeGroat, ECE, OSU 10
Computers and digital systems Each of the components of “The Computer”
is a digital component “The computer” consists of an interconnected
set of digital modules. Most of the components is are designed with
not much more than the techniques of this class.
9/15/09 - L1 Crs Ovrvw Copyright 2009 - Joanne DeGroat, ECE, OSU 11
Number Systems In digital systems you can only represent 2
states. A base 10 number systems is simply not
straightforward.
Will need a different number system. Concepts across number systems are the
same.
9/15/09 - L1 Crs Ovrvw Copyright 2009 - Joanne DeGroat, ECE, OSU 12
General number systems A number system has a base or radix, r A number in base r contains digits 0,1,2,…,r-1 The value represented as a power series in r as
An-1rn-1 + An-2rn-2 + … + A1r1 + A0r0 +
A-1r-1 + A-2r-2 + … + A-mr-m
and is written An-1 An-2 …A1 A0 . A-1 A-2 …A-m
9/15/09 - L1 Crs Ovrvw Copyright 2009 - Joanne DeGroat, ECE, OSU 13
General number systems (2) The “ . “ is called the radix point.
In base 10 it is the decimal point In base 2 it is the binary point
is referred to as the most significant digit Referred to as the msb
is referred to as the least significant digitReferred to as the lsb Usually m=0 so is the lsb
9/15/09 - L1 Crs Ovrvw Copyright 2009 - Joanne DeGroat, ECE, OSU 14
For base 10 or radix 10 Example - book 724.5 421
From prior education may have referred to digits as the units, tens and hundred positions
421 = 4.21 x 102 in scientific notation 421 = 4 x 102 + 2 x 101 + 1 x 100
(Recalling that anything to the 0 power is 1)
9/15/09 - L1 Crs Ovrvw Copyright 2009 - Joanne DeGroat, ECE, OSU 15
Something in base or radix 5 (123)5 What is it’s value?
(Note method of showing base of number) Value
Val (123)5 = 1 x 52 + 2 x 51 + 3 x 50
= 25 + 10 + 3 = (38)10
9/15/09 - L1 Crs Ovrvw Copyright 2009 - Joanne DeGroat, ECE, OSU 16
Binary numbers – base or radix 2 Have binary digits 0 and 1 So all number represented in binary have only
digits 0 and 1 11010 would have value?
= 1 x 24 + 1 x 23 + 0 x 22 + 1 x 21 + 0 x 20 + = 1 x 16 + 1 x 8 + 0 x 4 + 1 x 2 + 0 x 1 = 16+8+2 = (26)10
9/15/09 - L1 Crs Ovrvw Copyright 2009 - Joanne DeGroat, ECE, OSU 17
Know the powers of two The first 16 powers of two are in the
following table.
9/15/09 - L1 Crs Ovrvw Copyright 2009 - Joanne DeGroat, ECE, OSU 18
n 2^n n 2^N
0 1 8 256
1 2 9 512
2 4 10 1024
3 8 11 2048
4 16 12 4096
5 32 13 8192
6 64 14 16384
7 128 15 32768
Some common size terms 210 is commonly referred to as kilo or K 220 is commonly referred to as mega or M 230 is commonly referred to as giga or G 240 is commonly referred to as tera or T
So 4K is = 22 x 210 = 212 = 409610
9/15/09 - L1 Crs Ovrvw Copyright 2009 - Joanne DeGroat, ECE, OSU 19
Base 10 to binary One method
Subtract the largest power of two from the value and repeat until done.
Text example 652 146
Largest if 128 or 27 leaving18 Then have 16 or 24 leaving just 2 Which is 21 giving value 1 0 0 1 0 0 1 0
9/15/09 - L1 Crs Ovrvw Copyright 2009 - Joanne DeGroat, ECE, OSU 20
Fractional Parts In base 10
Have xx.12 which is 1 x 10-1 + 2 x 10-2 In base 2
Have binary point xxx.1 which is 1 x 2-1 or 0.5 (1/2) And xxx.01 which is 1 x 2-2 or 0.25 (1/4) And xxx.001 which is 1 x 2-3 or 0.125 (1/8) And xxx.0001 which is 1 x 2-4 or 0.0625 (1/16)
9/15/09 - L1 Crs Ovrvw Copyright 2009 - Joanne DeGroat, ECE, OSU 21
In digital systems and computer Have two other radix systems that are related
to binary or base 2 as they are powers of it. Octal or base 8
Can get an octal representation by grouping a binary number into groups of 3 digits.
Octal numbers use 8 distinct digits 0 through 7 110010 = 110 010 = (6 2)8 for example And can use same general number systems
expansion shown earlier.
9/15/09 - L1 Crs Ovrvw Copyright 2009 - Joanne DeGroat, ECE, OSU 22
Hex Hex or Hexadecimal is more common than
Octal representation today Hex is base or radix 16 Thus group 4 binary digits 1111 0100 1011 would be?
F 4 B For representation need 16 symbols
Use 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F
9/15/09 - L1 Crs Ovrvw Copyright 2009 - Joanne DeGroat, ECE, OSU 23