Binary ↘ Float: The 32 Bit Single Precision IEEE 754 Binary Floating Point Standard Representation Number 0 - 0101 0011 - 010 1001 1011 0000 0101 1111 Converted and Written as a Base Ten Decimal System Number (as a Float)
0 - 0101 0011 - 010 1001 1011 0000 0101 1111: 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:
0101 0011
The last 23 bits contain the mantissa:
010 1001 1011 0000 0101 1111
2. Convert the exponent from binary (from base 2) to decimal (in base 10).
The exponent is allways a positive integer.
0101 0011(2) =
0 × 27 + 1 × 26 + 0 × 25 + 1 × 24 + 0 × 23 + 0 × 22 + 1 × 21 + 1 × 20 =
0 + 64 + 0 + 16 + 0 + 0 + 2 + 1 =
64 + 16 + 2 + 1 =
83(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 = 83 - 127 = -44
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).
010 1001 1011 0000 0101 1111(2) =
0 × 2-1 + 1 × 2-2 + 0 × 2-3 + 1 × 2-4 + 0 × 2-5 + 0 × 2-6 + 1 × 2-7 + 1 × 2-8 + 0 × 2-9 + 1 × 2-10 + 1 × 2-11 + 0 × 2-12 + 0 × 2-13 + 0 × 2-14 + 0 × 2-15 + 0 × 2-16 + 1 × 2-17 + 0 × 2-18 + 1 × 2-19 + 1 × 2-20 + 1 × 2-21 + 1 × 2-22 + 1 × 2-23 =
0 + 0.25 + 0 + 0.062 5 + 0 + 0 + 0.007 812 5 + 0.003 906 25 + 0 + 0.000 976 562 5 + 0.000 488 281 25 + 0 + 0 + 0 + 0 + 0 + 0.000 007 629 394 531 25 + 0 + 0.000 001 907 348 632 812 5 + 0.000 000 953 674 316 406 25 + 0.000 000 476 837 158 203 125 + 0.000 000 238 418 579 101 562 5 + 0.000 000 119 209 289 550 781 25 =
0.25 + 0.062 5 + 0.007 812 5 + 0.003 906 25 + 0.000 976 562 5 + 0.000 488 281 25 + 0.000 007 629 394 531 25 + 0.000 001 907 348 632 812 5 + 0.000 000 953 674 316 406 25 + 0.000 000 476 837 158 203 125 + 0.000 000 238 418 579 101 562 5 + 0.000 000 119 209 289 550 781 25 =
0.325 694 918 632 507 324 218 75(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 + 0.325 694 918 632 507 324 218 75) × 2-44 =
1.325 694 918 632 507 324 218 75 × 2-44 =
0.000 000 000 000 075 357 031 541 472 413 449 511 68
0 - 0101 0011 - 010 1001 1011 0000 0101 1111 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) = 0.000 000 000 000 075 357 031 541 472 413 449 511 68(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: