Answered: Show how each of the following…

Systems Architecture

7th Edition

ISBN:9781305080195

Author:Stephen D. Burd

Publisher:Stephen D. Burd

Chapter8: Data And Network Communication Technology

Section: Chapter Questions

Problem 31VE

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Question

Show how each of the following floating-point values would be stored using IEEE-754 single precision (be sure to indicate the sign bit, the exponent, and the significand fields):

a) 12.5 b) -1.5 c) 0.75 d) 26.625

Expert Solution

Step 1

Converting 12.5 to binary
   Convert decimal part first, then the fractional part
   > First convert 12 to binary
   Divide 12 successively by 2 until the quotient is 0
      > 12/2 = 6, remainder is 0
      > 6/2 = 3, remainder is 0
      > 3/2 = 1, remainder is 1
      > 1/2 = 0, remainder is 1
   Read remainders from the bottom to top as 1100
   So, 12 of decimal is 1100 in binary
   > Now, Convert 0.5 to binary
      > Multiply 0.5 with 2.     Since 1.0 is >= 1. then add 1 to result
      > This is equal to 1, so, stop calculating
   0.5 of decimal is .1 in binary
   so, 12.5 in binary is 1100.1
12.5 in simple binary => 1100.1
so, 12.5 in normal binary is 1100.1 => 1.1001 * 2^3

single precision:
--------------------
sign bit is 0(+ve)
exp bits are (127+3=130) => 10000010
   Divide 130 successively by 2 until the quotient is 0
      > 130/2 = 65, remainder is 0
      > 65/2 = 32, remainder is 1
      > 32/2 = 16, remainder is 0
      > 16/2 = 8, remainder is 0
      > 8/2 = 4, remainder is 0
      > 4/2 = 2, remainder is 0
      > 2/2 = 1, remainder is 0
      > 1/2 = 0, remainder is 1
   Read remainders from the bottom to top as 10000010
   So, 130 of decimal is 10000010 in binary
frac bits are 10010000000000000000000

so, 12.5 in single-precision format is 0 10000010 10010000000000000000000
in hexadecimal it is 0x41480000

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