Number Representations & Computer Arithmetic (Fixed & Floating point) | Digital Logic - Computer Science Engineering (CSE) PDF Download

Floating Point Representation

1. To convert the floating point into decimal, we have 3 elements in a 32-bit floating point representation:
    i) Sign
    ii) Exponent
    iii) Mantissa

• Sign bit is the first bit of the binary representation. '1' implies negative number and '0' implies positive number.
  Example: 11000001110100000000000000000000 This is negative number.
• Exponent is decided by the next 8 bits of binary representation. 127 is the unique number for 32 bit floating point representation. It is known as bias. It is determined by 2^k-1 -1 where 'k' is the number of bits in exponent field.

  There are 2 exponent bits in 8-bit representation and 8 exponent bits in 32-bit representation.
  Thus
  bias = 3 for 8 bit conversion (2^2-1 -1 = 4-1 = 3)
  bias = 127 for 32 bit conversion. (2^8-1 -1 = 128-1 = 127)

  Example: 01000001110100000000000000000000
  10000011 = (131)₂
  131-127 = 4

  Hence the exponent of 2 will be 4 i.e. 2⁴ = 16.

• Mantissa is calculated from the remaining 24 bits of the binary representation. It consists of '1' and a fractional part which is determined by:

  Example:

  01000001110100000000000000000000

  The fractional part of mantissa is given by:

  1*(1/2) + 0*(1/4) + 1*(1/8) + 0*(1/16) +……… = 0.625

  Thus the mantissa will be 1 + 0.625 = 1.625

  The decimal number hence given as: Sign*Exponent*Mantissa = (-1)*(16)*(1.625) = -26

2. To convert the decimal into floating point, we have 3 elements in a 32-bit floating point representation:
    i) Sign (MSB)
    ii) Exponent (8 bits after MSB)
    iii) Mantissa (Remaining 23 bits)

Thus the floating point representation of -17 is 1 10000011 0001000000000000000000

• Sign bit is the first bit of the binary representation. '1' implies negative number and '0' implies positive number.
  Example: To convert -17 into 32-bit floating point representation Sign bit = 1
• Exponent is decided by the nearest smaller or equal to 2ⁿ number. For 17, 16 is the nearest 2ⁿ. Hence the exponent of 2 will be 4 since 2⁴ = 16. 127 is the unique number for 32 bit floating point representation. It is known as bias. It is determined by 2^k-1 -1 where 'k' is the number of bits in exponent field.

  Thus bias = 127 for 32 bit. (2^8-1 -1 = 128-1 = 127)

  Now, 127 + 4 = 131 i.e. 10000011 in binary representation.

• Mantissa: 17 in binary = 10001.

  Move the binary point so that there is only one bit from the left. Adjust the exponent of 2 so that the value does not change. This is normalizing the number. 1.0001 x 2⁴. Now, consider the fractional part and represented as 23 bits by adding zeros.

  00010000000000000000000