outline

Internal Representation - 2

• real numbers
  • represented using scientific notation
  • 24378 = 2.378 x 10⁴
    • except it's base 2
  • components are sign, mantissa, and exponent

Why binary?

• Electronic bistable environment
  • on/off, high/low voltage
• Reliability
  • with only 2 values, can be widely separated
  • therefore clearly differentiated
  • "drift" causes less error

Limitations

• Fixed numbers of bits for a number
  • size depends on computer design
• Limits on size of integers
  • e.g., -32768 to +32767 (16 bit word)
• Limit on size of decimal numbers (exponent)
• Limit on accuracy of decimal numbers (mantissa)

Floating point number representation

• If x is any real number, its normal form representation is,
  

         x = f * 10E

• Ex:

             125.32   =    0.12532 * 103
            -125.32   =   -0.12532 * 103
               0.65   =    0.65 * 100

Floating point number representation 2

• The number f of the representation is called the mantissa and the E is the exponent.

diagram

• sign of mantissa					sign of exponent
  d1	d2	d3	d4	d5	d6	d7	d8
  	digits of mantissa					digits of exponent

Example 1

• Ex: Let x = 125.32. Then the representation is, x = 0.12532 * 10³

• 0

  1

  2

  5

  3

  0

  0

  3

Example 2

• Ex: Let x = -0.0325475. Then the representation is, x = -0.3255 * 10^-1

• 1

  3

  2

  5

  5

  1

  0

  1

Binary representation of decimal numbers

• Recall, the general binary representation of a positive integer is,
  d_n2ⁿ + d_{n - 1}2^{n - 1} + ... + d₁2¹ + d₀2⁰

Example 3.1

• Ex: Convert decimal 43 to binary.
  

  43/2 = 21 + 1   5/2 = 2 + 1
  21/2 = 10 + 1   2/2 = 1 + 0
  10/2 =  5 + 0   1/2 = 0 + 1
  

Example 3.2

• i.e. continue division by 2, noting all remainders, until a quotient of zero is reached. The resulting bit sequence, reads backwards,
  

  101011 = 43.    (32 + 8 + 2 + 1 = 43)

Binary representation of decimal numbers 2

• the general binary representation of a positive number less than 1 is,
  d_-12^-1 + d_-22^-2 + d_-32^-3...

Example 4.1

• Ex: Convert .625 into binary.
  

  .625 x 2 = 1.250 (extract the 1)
  .250 x 2 = 0.500 (extract the 0)
  .500 x 2 = 1.000 (extract the 1)

Example 4.2

• The digits extracted are taken in the order extracted. In this case, the result is .101 (1/2 + 1/8 = 5/8 = .625)

Example 5.1

• Ex: Convert 1/3 into binary.

Example 5.2

• Ans: .010101010...

Binary representation of decimal numbers 3

• One source of error in computations is due to back and forth conversion between decimal and binary.

Example 6.1

• Ex: Find the sum of .6 + .6?

Example 6.2

• 1. Convert .6 to binary:
  

  Ans:      .1001100110011...

Example 6.3

• 2. Find sum:
  

  Ans:     .10011001
           .10011001
          1.00110010

Example 6.4

• 3. Convert answer back to decimal:
  1 + 1/8 + 1/16 + 1/128 = 1.195 (actual sum = 1.2)

Example 6.5

• Error = 1.2 - 1.195 = .005 due to roundoff made during conversions.

Example 7.1

• Ex: Find a 16 bit representation of the approximation 1.414 to the square root of 2.

Example 7.2

• a) Convert left side of the decimal point.
• Ans: 1

Example 7.3

• b) Convert right side of the decimal point. Using multiplicative method we get,
  

  0.828 1.656 1.312 0.624 1.248 0.496
  0.992 1.984 1.968 1.936 1.872 1.744
  1.488 0.976 1.952

Example 7.4

• c) required 16 bit approximation is,
  

            1.0110 1001 1111 101

Example 7.5

• d) convert this back to decimal and find the roundoff error.

Example 8.1

• Ex: Express the 16 bit approximation
  11.0010 0100 0011 11 to pi in normalized form.

Example 8.2

• Ans: the normalized form is,
  

            .1100 1001 0000 1111 x 22

• In normalized form the leading 1 appears next to the decimal point.

Example 9.1

• Ex: Interpret the string 0110 1110 0000 0100 using the following convention:
  • the leading bit (most significant bit, MSB) is the sign of mantissa (sm).
  • next 11 bits corresponds to mantissa (m).
  • next bit tells the sign of the exponent (se).
  • next three bits corresponds to exponent (e).

Example 9.2

• Ans: the parts are:
  

            sm = 0; m = 11011100000; se = 0; e = 100

• Shifting the decimal point four places to the right (?) gives,
  1101.11, which is equal to 13.75