Question

10 p Assuming IEEE 754 single-precision floating point number representation, calculate the floating point number the following bit pattern represent. Show your work to get credit. 1100 0001 0110 0000 0000 0000 0000 0000 Τ Τ Τ. Arial (1200

Answer 1

In IEE 754 standard single precision floating pointing number representation has 32 bits which represent as follows

• Most significant bit (bit number 31) will denote the sign of the number. If it is '1', it means number is negative and if it is '0' then number is positive.
• The bits with bit number 30 to 23 will represent exponent part of the number
• The bits with bit number 22 to 0 will represent mantissa part of the number

The number in decimal format is given by <sign as +/-><number = 2^{(exponent - bias)} * (1 + mantissa)>

For 32bit representation, bias value is 127 in decimal representation.

Arrange the given 32bit representation as follows:

31 30 0 bit number bit value 1 1 0 23 22 0 0 0 0 1 0 1 Exponent 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 Mantissa Sign

• Here sign bit is '1'. So the number is negative.
• 
  27 Exponent (1000 00102 decimal (1x27) +Cox26)+(0125) +(0x24) + (0x2+) 7(0x2)+ equivalent (1x2)+(0x2) = 128+0+0+0+0+0+2+0 130 Matissa - 223 1100000000000000000000 2' + (1x2') + (1x22) +(0x2) + (0x2")+...+(ox223) = 0.5 +0.25 + otot- – – to 0.75
  
• Exponent is (1000 0010)₂ in binary. Its decimal equivalent is (130)₁₀. So exponent = 130.
• Mantissa is (11000000000000000000000)₂. Its decimal equivalent = (0.75)₁₀. So mantissa = 0.75.
• Hence our number = 2^{(exponent - bias)} * (1 + mantissa) = 2^{(130 - 127)} * (1 + 0.75) =2³ * 1.75 = 8 * 1.75 =14.

Therefore the decimal number represented by 32bit number is -14 (negative as sign bit is '1').

Answer = -14.