Binary ↘ Float: The 32 Bit Single Precision IEEE 754 Binary Floating Point Standard Representation Number 0 - 1111 1110 - 011 0000 0000 0000 0000 1100 Converted and Written as a Base Ten Decimal System Number (as a Float)
0 - 1111 1110 - 011 0000 0000 0000 0000 1100: 32 bit single precision IEEE 754 binary floating point standard representation number converted to decimal system (base ten)
1. Identify the elements that make up the binary representation of the number:
The first bit (the leftmost) indicates the sign,
1 = negative, 0 = positive.
0
The next 8 bits contain the exponent:
1111 1110
The last 23 bits contain the mantissa:
011 0000 0000 0000 0000 1100
2. Convert the exponent from binary (from base 2) to decimal (in base 10).
The exponent is allways a positive integer.
1111 1110(2) =
1 × 27 + 1 × 26 + 1 × 25 + 1 × 24 + 1 × 23 + 1 × 22 + 1 × 21 + 0 × 20 =
128 + 64 + 32 + 16 + 8 + 4 + 2 + 0 =
128 + 64 + 32 + 16 + 8 + 4 + 2 =
254(10)
3. Adjust the exponent.
Subtract the excess bits: 2(8 - 1) - 1 = 127,
that is due to the 8 bit excess/bias notation.
The exponent, adjusted = 254 - 127 = 127
4. Convert the mantissa from binary (from base 2) to decimal (in base 10).
The mantissa represents the fractional part of the number (what comes after the whole part of the number, separated from it by a comma).
011 0000 0000 0000 0000 1100(2) =
0 × 2-1 + 1 × 2-2 + 1 × 2-3 + 0 × 2-4 + 0 × 2-5 + 0 × 2-6 + 0 × 2-7 + 0 × 2-8 + 0 × 2-9 + 0 × 2-10 + 0 × 2-11 + 0 × 2-12 + 0 × 2-13 + 0 × 2-14 + 0 × 2-15 + 0 × 2-16 + 0 × 2-17 + 0 × 2-18 + 0 × 2-19 + 1 × 2-20 + 1 × 2-21 + 0 × 2-22 + 0 × 2-23 =
0 + 0.25 + 0.125 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0.000 000 953 674 316 406 25 + 0.000 000 476 837 158 203 125 + 0 + 0 =
0.25 + 0.125 + 0.000 000 953 674 316 406 25 + 0.000 000 476 837 158 203 125 =
0.375 001 430 511 474 609 375(10)
5. Put all the numbers into expression to calculate the single precision floating point decimal value:
(-1)Sign × (1 + Mantissa) × 2(Adjusted exponent) =
(-1)0 × (1 + 0.375 001 430 511 474 609 375) × 2127 =
1.375 001 430 511 474 609 375 × 2127 =
233 944 370 647 060 437 451 115 129 976 356 077 568
0 - 1111 1110 - 011 0000 0000 0000 0000 1100 converted from a 32 bit single precision IEEE 754 binary floating point standard representation number to a decimal system number, written in base ten (float) = 233 944 370 647 060 437 451 115 129 976 356 077 568(10)
Spaces were used to group digits: for binary, by 4, for decimal, by 3.
More operations with 32 bit single precision IEEE 754 binary floating point standard representation numbers: