Binary ↘ Float: The 32 Bit Single Precision IEEE 754 Binary Floating Point Standard Representation Number 1 - 1111 1000 - 110 1101 1101 0100 0100 1101 Converted and Written as a Base Ten Decimal System Number (as a Float)
1 - 1111 1000 - 110 1101 1101 0100 0100 1101: 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.
1
The next 8 bits contain the exponent:
1111 1000
The last 23 bits contain the mantissa:
110 1101 1101 0100 0100 1101
2. Convert the exponent from binary (from base 2) to decimal (in base 10).
The exponent is allways a positive integer.
1111 1000(2) =
1 × 27 + 1 × 26 + 1 × 25 + 1 × 24 + 1 × 23 + 0 × 22 + 0 × 21 + 0 × 20 =
128 + 64 + 32 + 16 + 8 + 0 + 0 + 0 =
128 + 64 + 32 + 16 + 8 =
248(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 = 248 - 127 = 121
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).
110 1101 1101 0100 0100 1101(2) =
1 × 2-1 + 1 × 2-2 + 0 × 2-3 + 1 × 2-4 + 1 × 2-5 + 0 × 2-6 + 1 × 2-7 + 1 × 2-8 + 1 × 2-9 + 0 × 2-10 + 1 × 2-11 + 0 × 2-12 + 1 × 2-13 + 0 × 2-14 + 0 × 2-15 + 0 × 2-16 + 1 × 2-17 + 0 × 2-18 + 0 × 2-19 + 1 × 2-20 + 1 × 2-21 + 0 × 2-22 + 1 × 2-23 =
0.5 + 0.25 + 0 + 0.062 5 + 0.031 25 + 0 + 0.007 812 5 + 0.003 906 25 + 0.001 953 125 + 0 + 0.000 488 281 25 + 0 + 0.000 122 070 312 5 + 0 + 0 + 0 + 0.000 007 629 394 531 25 + 0 + 0 + 0.000 000 953 674 316 406 25 + 0.000 000 476 837 158 203 125 + 0 + 0.000 000 119 209 289 550 781 25 =
0.5 + 0.25 + 0.062 5 + 0.031 25 + 0.007 812 5 + 0.003 906 25 + 0.001 953 125 + 0.000 488 281 25 + 0.000 122 070 312 5 + 0.000 007 629 394 531 25 + 0.000 000 953 674 316 406 25 + 0.000 000 476 837 158 203 125 + 0.000 000 119 209 289 550 781 25 =
0.858 041 405 677 795 410 156 25(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)1 × (1 + 0.858 041 405 677 795 410 156 25) × 2121 =
-1.858 041 405 677 795 410 156 25 × 2121 =
-4 939 521 307 508 967 601 782 396 600 727 896 064
1 - 1111 1000 - 110 1101 1101 0100 0100 1101 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) = -4 939 521 307 508 967 601 782 396 600 727 896 064(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: