Question

Determine the single precision and double precision machine representation of -200.75

single: ________________________(16) Double: ________________________ (16)

What is is the decimal number that corresponds to the following IEEE 32 bit floating point number?

1100 0001 0010 1000 0000 0000 0000 0000

would you mind showing me the steps how to solve this problem?

Answer 1

Answer:
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Determine the single precision and double precision machine representation of -200.75
single: 0xC348C000(16) Double: 0xC069180000000000 (16)

What is is the decimal number that corresponds to the following IEEE 32 bit floating point number?
1100 0001 0010 1000 0000 0000 0000 0000
Answer: -10.5

Explanation:
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1)
Converting 200.75 to binary
   Convert decimal part first, then the fractional part
   > First convert 200 to binary
   Divide 200 successively by 2 until the quotient is 0
      > 200/2 = 100, remainder is 0
      > 100/2 = 50, remainder is 0
      > 50/2 = 25, remainder is 0
      > 25/2 = 12, remainder is 1
      > 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 11001000
   So, 200 of decimal is 11001000 in binary
   > Now, Convert 0.75000000 to binary
      > Multiply 0.75000000 with 2.  Since 1.50000000 is >= 1. then add 1 to result
      > Multiply 0.50000000 with 2.  Since 1.00000000 is >= 1. then add 1 to result
      > This is equal to 1, so, stop calculating
   0.75 of decimal is .11 in binary
   so, 200.75 in binary is 11001000.11
-200.75 in simple binary => 11001000.11
so, -200.75 in normal binary is 11001000.11 => 1.100100011 * 2^7

single precision:
--------------------
sign bit is 1(-ve)
exp bits are (127+7=134) => 10000110
   Divide 134 successively by 2 until the quotient is 0
      > 134/2 = 67, remainder is 0
      > 67/2 = 33, remainder is 1
      > 33/2 = 16, remainder is 1
      > 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 10000110
   So, 134 of decimal is 10000110 in binary
frac bits are 10010001100000000000000

so, -200.75 in single-precision format is 1 10000110 10010001100000000000000
in hexadecimal it is 0xC348C000

2)
Converting 200.75 to binary
   Convert decimal part first, then the fractional part
   > First convert 200 to binary
   Divide 200 successively by 2 until the quotient is 0
      > 200/2 = 100, remainder is 0
      > 100/2 = 50, remainder is 0
      > 50/2 = 25, remainder is 0
      > 25/2 = 12, remainder is 1
      > 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 11001000
   So, 200 of decimal is 11001000 in binary
   > Now, Convert 0.75000000 to binary
      > Multiply 0.75000000 with 2.  Since 1.50000000 is >= 1. then add 1 to result
      > Multiply 0.50000000 with 2.  Since 1.00000000 is >= 1. then add 1 to result
      > This is equal to 1, so, stop calculating
   0.75 of decimal is .11 in binary
   so, 200.75 in binary is 11001000.11
-200.75 in simple binary => 11001000.11
so, -200.75 in normal binary is 11001000.11 => 1.100100011 * 2^7

64-bit format:
--------------------
sign bit is 1(-ve)
exp bits are (1023+7=1030) => 10000000110
   Divide 1030 successively by 2 until the quotient is 0
      > 1030/2 = 515, remainder is 0
      > 515/2 = 257, remainder is 1
      > 257/2 = 128, remainder is 1
      > 128/2 = 64, remainder is 0
      > 64/2 = 32, remainder is 0
      > 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 10000000110
   So, 1030 of decimal is 10000000110 in binary
frac bits are 1001000110000000000000000000000000000000000000000000

so, -200.75 in 64-bit format is 1 10000000110 1001000110000000000000000000000000000000000000000000
in hexadecimal it is 0xC069180000000000

3)
1 10000010 01010000000000000000000
sign bit is 1(-ve)
exp bits are 10000010
   => 10000010
   => 1x2^7+0x2^6+0x2^5+0x2^4+0x2^3+0x2^2+1x2^1+0x2^0
   => 1x128+0x64+0x32+0x16+0x8+0x4+1x2+0x1
   => 128+0+0+0+0+0+2+0
   => 130
in decimal it is 130
so, exponent/bias is 130-127 = 3
frac bits are 0101

IEEE-754 Decimal value is 1.frac * 2^exponent
IEEE-754 Decimal value is 1.0101 * 2^3
1.0101 in decimal is 1.3125
   => 1.0101
   => 1x2^0+0x2^-1+1x2^-2+0x2^-3+1x2^-4
   => 1x1+0x0.5+1x0.25+0x0.125+1x0.0625
   => 1+0.0+0.25+0.0+0.0625
   => 1.3125
so, 1.3125 * 2^3 in decimal is 10.5
so, 11000001001010000000000000000000 in IEEE-754 single precision format is -10.5
Answer: -10.5