Number systems

• Natural Numbers


  Denoted as N

  Postive whole numbers, including zero

  N = (0,1,2,3,4,5)

  used for counting
• Integer


  Denoted as Z

  Postive or negative whole numbers including zero

  N is a subset of Z
• Rational Numbers


  Denoted as Q

  Any positve or negative whole number that can be expressed as a fraction

  Z is a subset of Q
• Irrational Numbers


  A number that cannot be represented as a simple fraction (contains infinte numbers in decimals)

  e.g. Surds and Pi
• Real Numbers


  Denoted as R

  Any real world quantity expressed as a number

  positvee or negative with or without a fractional part

  super set of all other number systems
• Ordinal Numbers


  Number used to identify the numerical postion of an object in relation to other objects in a set

  E.g. 1st 2nd, 3rd
• What does the most significant bit (MSB) represent in signed binary numbers?

  The sign of the number
• What does an MSB of 0 indicate in signed binary representation?

  The number is positive
• What does an MSB of 1 indicate in signed binary representation?

  The number is negative
• What is the binary representation of +5?(using 4 bits)

  0101
• What is the binary representation of -5 in two's complement?

  1011
• What is the MSB allocated in two's complement representation?

  A negative place-value (or weight)
• What is the minimum value in two's complement for n bits?

  • 2^(n-1)
• What is the maximum value in two's complement for n bits?

  (2^n - 1)
• How do you perform binary subtraction?

  Convert the number to be subtracted to a negative number
• What is the range of values for unsigned fixed-point form?

  • Minimum non-zero: 1 in LSB position, 0's elsewhere
  • Maximum: all 1's for n bits
• What is the range of values for signed fixed-point form for positive numbers?

  • Minimum non-zero: 1 in LSB position, 0's elsewhere
  • Maximum: all 1's for n bits
• What is the range of values for signed fixed-point form for negative numbers?

  • Minimum: 1 in MSB position, 0's elsewhere
  • Maximum: all 1's for n bits
• What is the maximum negative number you can get with signed fixed-point form?


  • 1
• What is fixed-point form in binary representation?

  • The binary point is set in a fixed position
  • Does not need to be stored in memory
• What are the advantages and disadvantages of fixed-point form?

  Advantages:

  • Simplicity: same arithmetic as integers

  
  

  Disadvantages:

  • Limited range
  • Precision vs. range trade-off
• How do you convert unsigned fixed-point binary to denary using place values?

  1. Assign place values to each bit
  2. Sum the values of the bits that are 1
• What is the denary value of the binary number 011111011?

  125
• How do you convert 14 ⅛ to binary?

  0111010
• How do you convert a decimal number to an unsigned fixed-point binary number using specified bits?

  1. Determine the whole and fractional parts
  2. Use the specified number of bits for each part
• How do you convert 19 ⅛ to an unsigned fixed-point binary number using 5 bits for the whole part and 3 bits for the fractional part?

  10011.001
• What is the formula to find the greatest postive value that can be represented in two's complement using n bits ?


  (2^n-1)-1
• Provide an example where we use hexadecimal notation to represent data in computing and explain why we do not use binary?

  • We use hexidecimal notation for colour codes
  • Hexadecimals have a shorter representation of a number than binary
• What are the advantages of normalisation?


  • A unique representation of each number
  • makes sure they are as precise as possible in a given number of bits
• What are the advantages of floating-point ?

  • Unlike fixed-point systems, floating-point allows for greater precision when working with fractional numbers
  • Can represent a wide range of values that are either very large or very small
• What are the advantages of fixed-point system

  • Fixed-point is simpler system than floating point therefore arithemetic operations are quicker
  • Fixed-point systems require simpler hardware therefore they are easier and simpler to implement
• Explain the term mantissa in the context of the floating-point binary representation of numbers?

  • The mantissa is the part of a floating-point binary number that represents the significant digits of the value.
  • It is combined with the exponent to determine the overall number. In normalized form, the mantissa's most significant bit is typically 1.
• Explain the term exponent in the context of the floating-point binary representation of numbers?

  • The exponent in floating-point binary representation specifies the power of 2 by which the mantissa is multiplied.
  • It is stored in a biased form, allowing the representation of both positive and negative exponents.
• In the context of floating point, explain what overflow is and give an example of a situation which might cause overflow to occur in the context of floating point.

  • Overflow is when a number exceeds the maximum value that can be represented with the number of bits available
  • Example: if floating point system has a maximum exponent value of 127, trying to represent a value large than 2^127 will cause overflow.