Fundamentals of Data Representation

• Which set of numbers best describes the following?

  {…-2, -1, 0, 1, 2…}

  Integers
• What symbol is given to the set of integers?

  $Z$
• What set of numbers does the symbol R represent?

  Real numbers
• Is -6 an integer?

  Yes. Integers can be negative.
• What set of numbers should be used for counting?

  Natural numbers
• To which set of numbers does π belong?

  Irrational numbers
• Which set of numbers contains all real world quantities?

  Real numbers
• What type of number is used to describe the numerical positions of objects in relation to others?

  Ordinal numbers
• Which number base is used by humans for counting?

  Base 10 / Decimal / Denary
• Which number base can be denoted with a subscript 16?

  Hexadecimal
• What is the decimal equivalent of the hexadecimal digit E?

  14
• Which number base is useful as a shorthand representation for binary?

  Hexadecimal
• What is the decimal equivalent of the binary number 0101?

  5
• What is the binary equivalent of the hexadecimal number A3?

  10100011
• What is the binary equivalent of the decimal number 14?

  1110
• Describe what is meant by a bit

  The fundamental unit of information which can only take two values, 1 and 0.
• How many bits are there in a byte?

  8
• How many nibbles are there in a byte?

  2
• How many different values can be represented with 4 bits?

  16
• How many bits are there in a mebibit?

  2^20 = 1,048,576
• How many bits are there in a kilobyte?

  8000 (= 10^3 x 8)
• How many gigabytes are there in a terabyte?

  1000
• Which type of binary can represent negative numbers: signed or unsigned?

  Signed
• What is the range of decimal numbers that can be represented using unsigned binary with four bits?

  0 to 15 (= 0 to 2^4 - 1)
• How many unique bit patterns can be made with 8 bits?

  256
• Add the unsigned binary integers 1101 and 110

  10011
• Multiply the unsigned binary integers 11001001 and 110

  10010110110
• 011010110110 is a two’s complement signed binary number with an eight bit mantissa and four bit exponent. What is the number in decimal?

  53.5
• Which of the following could be a normalised two’s complement number? A - 11010110, B - 01101101, C - 00010101


  B - Normalised numbers must start with 01 or 10
• 01101101 is a fixed point number with four bits before and four bits after the binary point. What is the decimal equivalent of the number?

  6.8125 (0110.1101 = 4 + 2 + 1/2 + 1/4 + 1/16)
• Convert the number 38.75 to floating point binary with an eight bit mantissa and a four bit exponent

  0100110110110 (38.75 = 0100110.11, move binary point 6 positions to get 0.10011011, 6 = 0110, mantissa is 010011011 and the exponent is 0110)
• Convert the number -17.625 to floating point binary with a nine bit mantissa and a four bit exponent


  1011111010101 (-17.625 = 101111.101, move binary point 5 positions to get 1.01111101, 5 = 0101, mantissa is 101111101 and exponent is 0101)
• Why might fixed point and floating point representations of decimal numbers be inaccurate?

  Rounding errors can occur in values that cannot be represented exactly by binary.
• The number 12.53 is represented in fixed point binary as 1100.1. Calculate the absolute error in the representation.

  0.03 (1100.1 = 12.5, 12.53 - 12.5 = 0.03)
• The number 18.48 is represented in fixed point binary as 10010.1. Calculate the relative error in the representation.

  0.00108 (= 0.108%, 10010.1 = 18.5, 18.5 - 18.48 = 0.02, 0.02/18.48 = 0.00108…)
• Which allows for the greatest range of numbers with a given number of bits: fixed point or floating point?

  Floating point
• Why are floating point numbers normalised?

  To provide the maximum level of precision for a given number of bits.
• What error occurs when a number is too large to be represented with the available bits?

  Overflow
• What is meant by a character code?

  A decimal digit used to represent a character.
• For what does ASCII stand?


  American Standard Code for Information Interchange