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The Representation of Data
An Overview


Prof. David Bernstein
James Madison University

Computer Science Department
bernstdh@jmu.edu

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A Quiz
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What Is This?

binary-data
The Point of the Quiz
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  • It could be many things
    • Bits (i.e., binary values) can be and are used to represent a wide variety of things
  • We need contextual information
    • To interpret a bunch of bits we need to know the representation scheme being used
Why 0/1?
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  • Electronic/Magnetic Systems:
    • Positive/Negative
    • On/Off
    • Clockwise/Counterclockwise
  • Mechanical Systems:
    • Up/Down
    • Pits/Lands
    • Hole/Solid
    • Bump/Flat
An Easy First Example - The Counting Numbers
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counting-numbers
An Interesting Question and Answer
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  • The Question:
    • How many bits do we need to represent all of the counting numbers less than \(N\)?
  • Getting to the Answer:
    • With \(B\) bits, we can represent all of the counting numbers less than \(2^B\)
What About Negative Integers?
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  • An Obvious Place to Start:
    • Since there are two signs, use one bit (e.g., the left-most) to represent the sign
  • A Shotcoming of this Approach:
    • It results in both a +0 and a -0
Going Further: Negative Numbers (cont.)
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twos_complement
Going Further: Negative Numbers (cont.)
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/imgs
(Courtesy of xkcd)
What About Real Numbers?
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  • Think About Base 10:
    • The positions to the left of the decimal point are powers of 10 and the positions to the right of the decimal place are powers of 1/10
  • An Obvious Place to Start in Binary:
    • The positions to the left of the decimal point are powers of 2 and the positions to the right of the decimal place are powers of 1/2
Going Further: What About Real Numbers? (cont.)
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  • Terms:
    • Sign
    • Exponent
    • Mantissa
  • Normalization:
    • One digit left of the decimal
  • Example: +1.101101 x 23
    • Sign: +
    • Exponent: 3
    • Mantissa: 1.101101
Going Further: What About Real Numbers? (cont.)
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  • IEEE Short Real (Single Precision):
    • 1 bit for the sign
    • 8 bits for the exponent
    • 23 bits for the mantissa
  • IEEE Long Real (Double Precision):
    • 1 bit for the sign
    • 11 bits for the exponent
    • 52 bits for the mantissa
What About Characters?
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  • An Obvious Place to Start:
    • Count the number of characters
    • Determine the number of bits needed
    • Assign a binary number to each character
  • An Example:
    • There are 26 letters in the alphabet
    • 5 bits can represent \(2^5\) (i.e., 32) different things
    • Assign 00001 to A, 00010 to B, 00011 to C, ...., 11010 to Z
What About Characters? (cont.)
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  • The ASCII (American Standard Code for Information Interchange) Encoding:
    • ascii
  • Unicode:
    • A mapping for every character in every language (including many dead languages)
What About Other Things?
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  • Discrete Sets:
    • We can use the same approach as for characters
  • Continuous Sets:
    • We either have to sample the set (to create a discrete approximation) or describe the elements
What About Colors?
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  • A Sampling Scheme:
    • Use the fact that we have red, green and blue cones to think of colors as having a red, green and blue component
    • Think of each color as having a discrete number of levels (e.g., \(2^8 = 256\))
    • A palette of \(2^{24} = 16,777,216\) colors
  • A Description Scheme:
    • Use the wavelength
What About Pictures?
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  • A Sampling (Raster) Scheme:
    • Create a finite grid with equal sized cells (called picture elements or pixels)
    • raster
  • A Description (Vector) Scheme:
    • Use geometric shapes (e.g., points, lines, curves, rectangles, polygons, ellipses)
What About Audio?
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  • A Sampling Scheme:
    • Need to use both temporal sampling and amplitude sampling (called quantization)
    • soundwave-sampling soundwave-quantization
  • A Description Scheme:
    • Use something like standard musical notation
What About Programs?
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  • Getting Started:
    • Each processor is capable of executing a discrete set of operations
    • Each operation is given a code
  • The Next Step:
    • Each operation has a discrete number of operands, each of which is represented in binary
  • The Final Step:
    • A program is just a sequence operation codes and operand values
The Quiz Revisited
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  • What Is This?
    • binary-data
  • It Could Be Anything!
    • I could treat it as a number or bunch of numbers
    • I could treat it as a color or bunch of colors
    • I could treat it as a audio
    • I could treat it as a program
    • ...
There's Always More to Learn
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