Binary number, Bits and Byte

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Binary number, Bits and Byte

• Sen Zhang

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• Number systems
• Decimal
• Binary
• Bits
• bytes
• Hexadecimal
• Octal
• Numbers conversion among different systems
• Ascii code

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Bits Bytes?

• Have you ever heard of words bits and bytes? Have
  you heard of an advertisement that says, "This
  computer has a 64-bit Pentium IV processor with
  256 mega-bytes of RAM and 100 giga-bytes of hard
  disk space."
• Probably the answer is yes, even for a normal
  computer user, not to mention you, a C
  programmer.
• As a computer programmer, you should know what
  bits and bytes are and how to work with numbers
  expressed in binary and hexadecimal notations.

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• In this lecture, we will discuss bits and bytes,
  binary and decimal numbers in detail so that you
  will gain a fundamental understanding about their
  meanings, what these systems are and how they
  work.
• To help you understand, let's first review the
  well known decimal number system.

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The Decimal Number System

• The decimal system is the base-10 system that we
  use every day.
• A number, say 6357, represented in the base-10
  system consists of multiple ordered digits. (In
  other words, digits are normally combined
  together in groups to create larger numbers.)
• A digit is a single place that can hold numerical
  values between 0 and 9 (10 different values).

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Let us start from an arbitrary decimal number

• For example, 6,357 has four digits.
• It is understood that in the number 6,357,
• the 7 is filling the "1s place,"
• while the 5 is filling the 10s place,
• the 3 is filling the 100s place
• and the 6 is filling the 1,000s place.
• So you could express 6,357 this way if you want
  to be explicit
• (6 1000) (3 100) (5 10) (7 1)
• 6000 300 50 7
• 6357

103
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Continue ..

• Another way to express it would be to use the
  concept of powers of 10.
• A specific digit is associated with a specific
  weight expressed as powers of 10. The first digit
  (counting from the right) gives 10 to the 0
  power, the second digit gives 10 to the 1 power,
  and so on.

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• Exponents are a shorthand way to show how many
  times a number, called the base, is multiplied
  times itself. A number with an exponent is said
  to be "raised to the power" of that exponent.
• Assuming that we are going to represent the
  concept of "raised to the power of" with the ""
  symbol.
• "10 squared or 10 to the power of 2 is written
  as "102"
• 10 to the fourth power is denoted 104

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• Thus, another way to express the previous number
  is like this
• (6 103) (3 102) (5 101) (7
  100)
• 6000 300 50 7
• 6357

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• What you can see from this expression is that
  each digit is a placeholder for the power of the
  index of that placeholder of base 10, starting
  from the least significant digit with 10 raised
  to the power of zero (i.e. counting from the
  rightmost digit).

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• But why do we human beings use 10 based number
  system?

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• The most commonly accepted explanation is that
  our base-10 number system was adopted by our
  ancestors most likely because we have 10 fingers.
• Interestingly enough, maybe that is why digit in
  English also means a finger or toe.

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• We have reasons to ask a question in our minds
• If we happened to evolve to have eight fingers
  instead, would we probably have a base-8 number
  system?
• The answer is probably YES!

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Any other number systems?

• The good news about number systems is that it is
  not the only choice to have 10 different values
  in a digit.
• Actually, we can have base-anything number
  systems from a theoretical point of view.
• There are many good reasons to use different
  bases in different situations. For example, 7
  days/week, 12 months/year

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A generalized rule

• The following rules apply to base 10 and to any
  other base number system
• The system of base n requires n different symbols
  or values.
• The left most digit is the highest-order digit
  and represents the most significant digit, while
  the lowest-order digit is the least significant
  digit.
• A digit is represented as powers of the system's
  base.

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• Computers happen to operate using the base-2
  number system, also known as the binary number
  system, just like the base-10 number system is
  known as the decimal number system to human
  beings.

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The fundamental point

• Modern computers use binary number system, in
  which there are only zeros and ones. (Only two
  symbols)
• A bit to binary is similar a digit to a
  decimal information. (Again, the easiest way to
  understand bits is to compare them to something
  you know digits.)
• A bit has a single binary value, either 0 or 1.

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Binary vs. Decimal

• Binary is a base two system which works just like
  our decimal system.
• Considering the decimal number system, it has a
  set of values which range from 0 to 9.
• The binary number system is base 2 and therefore
  requires only two digits, 0 and 1.

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The fundamental point

• Binary representation of numbers and other
  information is the representation which can be
  understood by computer chips and can be saved in
  memory.
• It is important to computers because all computer
  data is ultimately represented by a series of
  zeros and ones, no matter you realize it or not.

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You might ask

• Why dont computers use the base-10 decimal
  system for numbers, counting and arithmetic?
• Why not 4 based, 7 based?
• Why 2 based?

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• We know that the computer doesn't have a real
  brain inside. In fact, it is made up mostly of
  semiconductor materials such as silicon. Yet, a
  computer acts in many ways as if it does have a
  real brain, because it can store (memorize) data
  and derive new information (operations) from the
  input data.

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Why binary?

• These questions can be answered by a series of
  relevant questions!
• How to store the values in hardware?
• How to automatically perform arithmetic
  operations on numbers?

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• The fundamental question is can we find out a
  physical material to stably maintain n different
  status?

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How to store?

• Advancement in material science guarantees that
  binary status can be represented with no
  ambiguity.
• Silicon and many other semiconductor materials
  can present one of two status at any given time,
  and can retain a status for a long time.
• Positive or negative, 5 volt or -5 volt.
• Think about 2 status in electronic world, if not
  One then Zero, very simple to implement in
  electronic world.

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• One the other hand, it is difficult, if not
  impossible, to find out a material to be able to
  maintain 10 different status stably.
• Generally speaking, the more status to maintain,
  the more difficult to find out such a material.

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How to calculate

• Another factor is how to implement proper digital
  circuits to perform arithmetic and logical
  operations based on a specific number system.
• It turns out that the binary system is the
  preferred way to implement CPUs to do various
  operations (arithmetic and logical operations).
  Not any other systems!

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• You could wire up and build computers that
  operate in base-10 (assume physically we can find
  out such kind of material.), but they would be
  fiendishly expensive right now. On the other
  hand, base-2 computers are relatively cheap.

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• Also because there must always be at least two
  symbols for an information processing system to
  be able to distinguish significances of different
  values and to manipulate on them, binary is the
  smallest numbering system that supports definite
  arithmetic and logic operations.

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The simplest answer is

• Basically speaking, binary system simplifies
  information representation and information
  processing in electronic world.
• Binary number system is the easiest one to
  implement from the hardware point of view.
• The binary number system suits a computer
  extremely well, because it allows simple CPU and
  memory designs.
• So computers use binary numbers.

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• Their CPU and memory are made up of millions of
  tiny switches that can be either ON or OFF. Two
  symbols, 0 and 1, can be used to stand for the
  two states of ON and OFF.

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• Since the computer is really made up of tiny
  switches that can be either OFF or ON, you can
  look at a binary number as a series of light
  switches. A 1 represents a switch that is ON, and
  a 0 means a switch that is OFF.

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• The computer's CPU needs only recognize two
  states, in the same way as a switch must always
  be open or closed, or an electrical flow on or
  off, a binary digit must always be one or zero.
• on or off,
• Yes or no,
• 1 or 0.
• But from this on-off, yes-no 1-0 state, all
  things may can be represented completely and
  calculated correctly.

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What is a feasible number system?

• Should be able to represent any information.
• Should be able to support arithmetic and logic
  operations.

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Bits

• The binary number system uses binary digits
  (bits) in place of decimal digits.
• A binary number is composed of only 0s and 1s,
  like this 1011.
• How do you figure out what the value of the
  binary number 1011 is in decimal world?

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How does it work?

• As we have shown that our decimal system is based
  on place or location. That is, the place of each
  digit decides the value of that digit.
• The binary system works in exactly the same way,
  except that its place value is based on the
  number two.

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• Therefore we have the one's place, the two's
  place, the four's place, the eight's place, the
  sixteen's place, and so on. Each place in the
  number represents two times (2X's) the place to
  its right.
• An example
• (1 23) (0 22) (1 21) (1 20)
• 8 0 2 1
• 11

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How you count or add in decimal?

• Considering the decimal number system, it has a
  set of values which range from 0 to 9.
• If you add 1 to 9, carry will happen.
• X Y is greater than 10 and carry 1 onto the
  next column on the left.
• If the sum is less than 10, put it down at the
  bottom and set the carry to zero. If it is not
  less than the base, subtract 10, put down the
  result, and set the carry to one.

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• With only two numerals, 1 (one) and 0 (zero),
  counting in binary is pretty simple. Just keep in
  mind the following
• 0 0 0
• 0 1 1
• 1 0 1
• 1 1 10
• When you look at this sequence,
• 0 and 1 are the same for decimal and binary
  number systems.
• As for the decimal number 2, you see for the
  first time carrying takes place in the binary
  system. If a bit is 1, and you add 1 to it, the
  bit becomes 0 and the next bit becomes 1.

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Bits

• 0 0
• 1 1
• 2 10
• 3 11
• 4 100
• 5 101
• 6 110
• 7 111
• 8 1000
• 9 1001
• 10 1010
• 11 1011
• 12 1100

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Bits

• Starting at zero and going through 12, counting
  in decimal and binary have been listed on the
  previous slide.
• Please also notice that, in the transition from
  decimal 11 to decimal 12, the carrying effect
  rolls over through 2 bits, turning 1011 into
  1100.

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decimal to binary

• Keep dividing by 2
• Ex 2 23710     
• 237 / 2 118    Remainder 1----------------------
  --------------------------------
• 118 / 2 59 Remainder 0---------------------
  ------------------------------
• 59 / 2 29 Remainder 1--------------------
  ----------------------------
• 29 / 2 14 Remainder 1--------------------
  ----------------------------
• 14 / 2 7 Remainder 0-------------------
  --------------------------
• 7 / 2 3 Remainder 1------------------
  ------------------------
• 3 / 2 1 Remainder 1------------------
  -----------------
• 1 / 2 0 Remainder 1------------------
  --------------
•                                          
  
• 
  v  v  v  v  v 
  v  v  v
• 
  1  1  1  0 
  1  1  0  1

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Binary arithmetic operation

• Look at adder in binary and decimal
• 3
• 3
• 6
• 11
• 11
• 110 (carry) which is 6 in decimal.

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How to add two numbers which are not necessarily
base 10

• Start with the rightmost column of digits (be
  sure the numbers are properly aligned with units
  digits under each other).
• Begin with carry zero.
• Add the digits in the current column plus the
  carry.
• If the sum is less than the base, put it down at
  the bottom and set the carry to zero. If it is
  not less than the base, subtract the base, put
  down the result, and set the carry to one.
• If you are not out of columns, move to the next
  one to the left, and go back to step c above.
• If the carry is not zero, write it down as the
  leftmost digit of the sum.
• Stop.

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More binary operations

• Likewise, other arithmetic operations such as
  subtraction, multiplication and division, as well
  as other logical operations can all be
  accomplished electronically in CPUs, but more
  complicated than binary adder.
• You just need to know that binary numbers can
  represent everything, support a complete set of
  arithmetic and logic operations. (Not the
  concentration of this class, take introduction to
  computer science or architecture course if want
  to explore more such as binary complementary code
  etc.!)

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• As you can see, numbers can become rather long
  and appear to be cumbersome in the binary system.
  For example, to show the number 10, we need four
  light switches, or four places.
• However, it is not a problem to computers at all!
  Because the real switches inside a computer are
  tiny and they are able to turn on and off very
  rapidly.

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The Hexadecimal System

• Although not a problem internally, long binary
  number seems a problem to display in some
  situations. A common practice to solve this
  problem is to use hexadecimal to represent Binary
  numbers more compactly externally.
• The hexadecimal system is base 16. Therefore, it
  requires 16 different symbols. The values 0
  through 9 are used, along with the letters A
  through F, which represent the decimal values 10
  through 15.
• 0..9, A, B, C, D, E, F
• 0..9, 10, 11, 12,13, 14, 15

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Hexadecimal lt-gtbinary
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Hexadecimal lt-gtbinary

• Group binary number 4 by 4 starting from the
  least significant position.

11,1101,1001
3D9
ED
1  1  1  0  ,1  1  0  1
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The Octal System

• The Octal system is base 8. Therefore it requires
  8 digits. The values 0 through 7 are used.
• Octal to hexadecimal conversion, or visa versa,
  is most easily performed by first converting to
  binary.

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• A binary number is converted to octal by grouping
  the bits in groups of three.

355
1  1  ,1  0  1 , 1  0  1
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• The binary, hexadecimal (hex) and octal system
  share one common feature they are all based on
  powers of 2.
• Each digit in the hex system is equivalent to a
  four-digit binary number and each digit in the
  octal system is equivalent to a 3-digit binary
  number.

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1 bit
1 byte
4 bytes 1 word System dependent.
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A bit

• A bit (from Binary digIT) is the smallest
  unit of memory, also the unit of measurement of
  data information.

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Bytes

• Since a single bit holds so little information,
  bits are rarely seen alone in computers. They are
  almost always bundled together into 8-bit
  collections, and these collections are called
  bytes.
• Bytes, larger units, then are treated as integral
  units of storage.

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Words

• On most machines, bytes are assembled into
  larger structures called words, where a word is
  usually defined to be the size required to hold
  an integer value.
• Some machines use two-byte words (16 bits), while
  some others use 4-byte words(32 bits) and some
  machines use less conventional sizes.

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Why are there 8 bits in a byte?

• A similar question is, "Why are there 12 eggs in
  a dozen?"
• Why your table has no larger or smaller working
  area?
• It targets at the most common situation.
• The 8-bit byte is something that people settled
  on through trial and error over the past 50
  years.
• To some extend, 8-bit is enough to represent all
  English characters and Arabic numbers. A byte
  used to be the basic unit to hold an individual
  character in a text document.

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One question?

• Can you use your one eye to show yes or no?
• Can you use your two eyes to represent four
  directions, north, east, south and west to your
  partner.
• Hint you can open and close your eyes to code
  different information.

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• With 8 bits in a byte, you can represent 256
  values ranging from 0 to 255, as shown here
• 0 00000000
• 1 00000001
• 2 00000010
• ...
• 254 11111110
• 255 11111111
• This is related to ASCII code!

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ASCII

• It is an acronym for the American Standard Code
  for Information Interchange.
• It is a standard seven-bit code that was first
  proposed by the American National Standards
  Institute or ANSI in 1963, and finalized in 1968
  as ANSI Standard X3.4.
• The purpose of ASCII was to provide a standard to
  code various symbols ( visible and invisible
  symbols)

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ASCII

• In the ASCII character set, each binary value
  between 0 and 127 represents a specific
  character.
• Most computers extend the ASCII character set to
  use the full range of 256 characters available in
  a byte. The upper 128 characters handle special
  things like accented characters from common
  foreign languages.

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• In general, ASCII works by assigning standard
  numeric values to letters, numbers, punctuation
  marks and other characters such as control codes.
  
• An uppercase "A," for example, is represented by
  the decimal number 65."

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Bytes ASCII

• By looking at the ASCII table, you can clearly
  see a one-to-one correspondence between each
  character and the ASCII code used.
• For example, 32 is the ASCII code for a space.
• We could expand these decimal numbers out to
  binary numbers (so 32 00100000), if we wanted
  to be technically correct -- that is how the
  computer really deals with things.

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Bytes ASCII

• Computers store text documents, both on disk and
  in memory, using these ASCII codes.
• For example, if you use Notepad in Windows
  XP/2000 to create a text file containing the
  words, "Four score and seven years ago," Notepad
  would use 1 byte of memory per character
  (including 1 byte for each space character
  between the words -- ASCII character 32).
• When Notepad stores the sentence in a file on
  disk, the file will also contain 1 byte per
  character and per space.
• Binary number is usually displayed as Hexadecimal
  to save display space.

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• Take a look at a file size now.
• Take a look at the space of your p drive

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Bytes ASCII

• If you were to look at the file as a computer
  looks at it, you would find that each byte
  contains not a letter but a number -- the number
  is the ASCII code corresponding to the character
  (see below). So on disk, the numbers for the file
  look like this
• F o u r a n d s e v e n
• 70 111 117 114 32 97 110 100 32 115 101 118 101
  110

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• Externally, it appears that human beings will use
  natural languages symbols to communicate with
  computer.
• But internally, computer will convert everything
  into binary data.
• Then process all information in binary world.
• Finally, computer will convert binary information
  back to symbols understandable to human beings .

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• When you type the letter A, the hardware logic
  built into the keyboard automatically translates
  that character into the ASCII code 65, which is
  then sent to the computer. Similarly, when the
  computer sends the ASCII code 65 to output
  devices, the output hardware instead draw letter
  A on your screen or your computer.

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revisit char data type

• In C, single characters are represented using
  the data type char, which is one of the most
  important scalar data types.
• char achar
• acharA
• achar65

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More on book

• Page 6 basic idea
• Page 31- collating sequence

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Character and integer

• A character and an integer (actually a small
  integer spanning only 8 bits) are actually
  indistinguishable on their own. If you want to
  use it as a char, it will be a char, if you want
  to use it as an integer, it will be an integer,
  as long as you know how to use proper C
  statements to express your intentions.

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• 1 bit
• 1 byte 8 bits
• 1 kb 210 bytes 1024 bytes !1000
• 1 Mb 1 k k bytes 210 210 bytes
• 1 G b 210 210 210 bytes
• 1 Terab 210 210 210 210 bytes

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Even larger capacity

• 1 petabyte 210 210 210 210 210 bytes (2
  to the 50th power )
• 1 exabyte 260
• 1 zettabyte 270
• 1 yottabyte 280

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Some interesting facts about what these
various-sized bytes can store

• 1 bit a binary decision
• 1 byte a character
• 5 Megabytes The complete works of Shakespeare
• 2 Gigabytes 20 meters of shelved books
• 10 Terabytes The printed collection of the US
  Library of Congress
• 200 Petabytes All printed material in the whole
  word.
• 5 Exabytes All words ever spoken by human beings

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CPU processes binary number

• The first microprocessor to make it into a home
  computer was the Intel 8080, a complete 8-bit
  computer on one chip, introduced in 1974.

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• PC market moved from the 8088 to the 80286, the
  80386, 80486, the Pentium, the Pentium II to the
  Pentium III to the Pentium 4.
• All of these microprocessors are made by Intel
  and all of them are improvements on the basic
  design of the 8088.
• The Pentium 4 can execute any piece of code that
  ran on the original 8088, but it does it about
  5,000 times faster!

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