Binary%20number,%20Bits%20and%20Bytes%20and%20memory

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Binary number, Bits and Bytesand memory

• Sen Zhang

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• Number systems
• Decimal
• Binary
• Bits
• bytes
• Numbers conversion among different systems

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Binary to Decimal

• To convert 1011 to its decimal value
• 1011 (It reads one zero one one in binary
  number)
• 10000101
• (1 23) (0 22) (1 21) (1 20)
• 8 0 2 1
• 11 (it reads eleven in decimal system)
• It is wrong to read 1011 one thousand and eleven,
  if you know it is a binary number.

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Decimal to Binary

• Keep dividing the decimal number by 2
• An Example to convert 23710 to binary value 1  1 
  1  0  1  1  0  1     
• 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

In the reversed order to get the result!
The result!
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The above two slides should be enough for you to
prepare exam.

• However, you should proceed reading the rest of
  the slides for better understanding to binary
  system if you are interested in computing
  technology.

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• Number systems
• Decimal
• Binary
• Bits
• bytes
• Numbers conversion among different systems

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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 to their
  meanings and 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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• But why do we human beings use 10 based number
  system?

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How to count?

• Fingers and toes? Cuts on trunk?
• 11111111111111111111111 (23)
• 11111 (five)
• 1111111111111111111111111111(28)
• 11111111111111111 (17 )
• Number system and calculating system
• 1, 2, 5 10, 11, 23
• 23528
• 23-517

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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, 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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• 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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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.
• 1011 has four bits
• 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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What is the value of the binary number 1011?

• 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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decimal to binary

• Keep dividing the decimal number 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

In the reversed order!
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1 bit

1 byte
8 bits
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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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• 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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Memory

• Where to save binary numbers in computer?
• In memory!
• What is memory?
• Memory is a space where you can save binary
  values, consisting of a sequence of units
  (counted in bytes).

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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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END