LSP 121

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LSP 121

• Number Systems
• andLogarithms

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Binary Numbers

• Why should anyone learn binary?
• Computers are based on the binary number system
  (on/off or 1/0)
• All music, video, data, and computer programs are
  stored in binary in computer memory/storage
• If your iPod / computer / flash drive has x
  storage capacity, what does that mean?

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Counting in binary
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The Pattern (think isomorphism)
Given a base B
Example for Base 2, binary
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Binary Numbers

• Before we discuss binary arithmetic, do you
  really understand decimal arithmetic?

1024 1 x 103 0 x 102 2 x 101 4 x 100

• Binary numbers are the same, except there are
  only 2 digits (0 and 1), and the base is 2

10010 1 x 24 0 x 23 0 x 22 1 x 21 0 x 20
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Binary Numbersconvert binary to decimal

• What is the decimal value of binary 10010101?
• This can be done using a grid with the successive
  powers of 2 written right to left.

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Convert decimal to binary

• What is the binary value of decimal 83?
• This can be done is two ways
• fill in the powers of 2 that add up to 83, or
• successively divide by 2, write remainder
  collect remainders beginning with last

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Binary Addition
Note that 1 1 1 0, write 0, carry 1Also
note that 1 1 1 1 1, write 1 carry 1.
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Binary Arithmetic

• Add the following two binary values
• 10011100
• 01011010
• So when a computer does arithmetic, it converts
  it to binary.
• When you type the letter n on the keyboard, it
  converts it to an 8-bit binary value.

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ASCII code

• Every character (keystroke) has a numeric
  equivalent, e.g., A, B, C, D is (in decimal) 65,
  66, 67, 68.
• The computer uses the binary value for each
  keystroke (including punctuation and the
  space).
• This is HELLO in ASCII code (binary)1001000
  1000101 1001100 1001100 1001111

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ASCII codes

• There are many sources (online) that indicate the
  ASCII codes for all keyboard characters as well
  as graphic, numeric and control characters.
• A good place to start ishttp//ascii-table.com

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Its true

• There are 10 kinds of people in the world
• Those that know binary and those that dont

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What is a Logarithm?

• A logarithm (or log) is a number that represents
  a power or exponent
• Why use logs?
• A simpler way to express large values
• Some things grow or shrink exponentially, so the
  log is a perfect numbering system

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Real numbers or logs?

• would you rather see 108,243,578 or8.03 (since
  108.03 108,243,578)
• would you rather say that an earthquake had the
  energy equivalent to423,427,834 or 8.6 Richter
  value
• The big number is the same as the small number if
  you use logs (logarithms)

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Constructing a logarithmic spiral
Start with symmetric spokes, draw perpendicular
to next spoke
As spokes increase, smooth curve starts to take
shape
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Logarithms and Geometry
A logarithmic spiral
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Logarithms in nature
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the Nautilus
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Example Sound

• Sound is a form of energy which obeys the Inverse
  Square Law
• This law states that sound decreases by 1/d2 as
  you get farther from the sound source,(d
  distance from sound source)
• The sound energy decreases inversely to the
  square of the distance from the source
• For example, if at 1 meter, the sound energy is 1
  unit, then, at 2 meters, it is ¼ units, at 3
  meters, it is 1/9 units

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Inverse-square law
This is true for all energy, such as, sound or
light intensities.
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Sounds

• The threshold of human hearing is considered 1
  (100)
• The rustle of leaves is 10 times louder than the
  threshold of hearing (101)
• A whisper is 100 times louder than the threshold
  of hearing (102)
• Busy street traffic is 108 times louder than the
  threshold of hearing

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Decibel

• These 10n values are clumsy to work with, so they
  created the term decibel (one tenth of a bel,
  remember Alexander Graham Bell?)
• A sound 108 is actually 80 decibels (dB)
• A sound 101 is 10 dB
• Or loudness in dB 10 log10 (intensity of sound
  intensity of softest audible sound)

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Remember this formula

• The most common approach to sound intensity
  measurement is to use the decibel scale

I/Io is the ratio of the intensities (no units
with this)its a pure number, such as, 10,000,000
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Dangerous Sounds

• A rock concert is typically around 100 - 120 dB
  (depends on where you sit/stand)
• The threshold of pain for the human ear is around
  120 dB
• Immediate perforation of the eardrum is 160 dB

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Log Base 10

• The log base 10 (written log10) is a very common
  log
• log10 x is the power to which 10 must be raised
  to obtain x
• Or better yet, 10 to what power equals x?
• log10 1000 ? (10 to what power 1000?) 3
• log10 10,000,000 7
• log10 0.1 -1
• log10 30 1.4777

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Log Base 2

• The log base 2 (written log2) is a very common
  log when dealing with computers (since computers
  use the binary number system which is base 2
  arithmetic)
• log2 x is read log base 2 of x it is the power
  to which 2 must be raised to obtain x
• Or simply, 2 to what power equals x?

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Log Base 2

• Log2 32 5, since 25 32
• Log2 1024 10
• Log2 40 5.32
• Dont have Log2 on your calculator? Take Log10
  of the same value and then divide by 0.301

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Example 1 Richter scale

• Richter value is given as R log(I) where I is
  the intensity of the earthquake
• E.g., if the intensity of an earthquake is given
  as 124,056 then the Richter value is
• R log(124056) 5.1 (use Excel or calculator)
• Conversely, if you know the Richter scale number
  you can find the intensity
• I 10R so if R 8.5 then
• I 108.5 316,227,766

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Example 2 sound levels

• Intensity ratio (I/Io) and sound level (L in
  decibels) are related by the following
• L 10 log(I/Io) where
• I intensity of the sound and
• Io intensity of threshold of sound (to humans)
• note also, dB decibel 1/10 of a Bel
• E.g., if the ratio is 100,000,000 find L (in dBs)
• L 10log(100000000) 80 dB
• dB and db are the same by the way

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Sound levels, part 2

• Also, if you are given the decibel value you can
  determine the ratio (I/Io)
• an average radio is rated at 70 dB, what is the
  ratio?
• use, I/Io 10(L/10) then
• I/Io 1070/10 107 10,000,000
• Jot it down I/Io 10(L/10)

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LOG expressions in Excel

• LOG(n)
• this will return the log base 10 of n
• 10y
• this will return n, if y LOG(n)
• LOG(128,2)
• this will return log base 2 of 128 (try it)
• LOG(a)/LOG(b)
• this also will return LOG a BASE b

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inverse of log functions

• Know the inverse of LOG functions
• if x log(n), how do you find n given x
• simply take 10x and you will get n
• Also,
• if x 10 log(y), how do you find y given x
• y 10(x/10)
• use this for Loudness logarithms which uses
• Loudness (in dB) 10 log(intensity ratio), and
  Intensity 10(L/10) where L loudness

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