Binary ↘ Float: The 32 Bit Single Precision IEEE 754 Binary Floating Point Standard Representation Number 0 - 0111 1101 - 000 0111 1000 0110 1011 1110 Converted and Written as a Base Ten Decimal System Number (as a Float)
0 - 0111 1101 - 000 0111 1000 0110 1011 1110: 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:
0111 1101
The last 23 bits contain the mantissa:
000 0111 1000 0110 1011 1110
2. Convert the exponent from binary (from base 2) to decimal (in base 10).
The exponent is allways a positive integer.
0111 1101(2) =
0 × 27 + 1 × 26 + 1 × 25 + 1 × 24 + 1 × 23 + 1 × 22 + 0 × 21 + 1 × 20 =
0 + 64 + 32 + 16 + 8 + 4 + 0 + 1 =
64 + 32 + 16 + 8 + 4 + 1 =
125(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 = 125 - 127 = -2
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).
000 0111 1000 0110 1011 1110(2) =
0 × 2-1 + 0 × 2-2 + 0 × 2-3 + 0 × 2-4 + 1 × 2-5 + 1 × 2-6 + 1 × 2-7 + 1 × 2-8 + 0 × 2-9 + 0 × 2-10 + 0 × 2-11 + 0 × 2-12 + 1 × 2-13 + 1 × 2-14 + 0 × 2-15 + 1 × 2-16 + 0 × 2-17 + 1 × 2-18 + 1 × 2-19 + 1 × 2-20 + 1 × 2-21 + 1 × 2-22 + 0 × 2-23 =
0 + 0 + 0 + 0 + 0.031 25 + 0.015 625 + 0.007 812 5 + 0.003 906 25 + 0 + 0 + 0 + 0 + 0.000 122 070 312 5 + 0.000 061 035 156 25 + 0 + 0.000 015 258 789 062 5 + 0 + 0.000 003 814 697 265 625 + 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 =
0.031 25 + 0.015 625 + 0.007 812 5 + 0.003 906 25 + 0.000 122 070 312 5 + 0.000 061 035 156 25 + 0.000 015 258 789 062 5 + 0.000 003 814 697 265 625 + 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 =
0.058 799 505 233 764 648 437 5(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 + 0.058 799 505 233 764 648 437 5) × 2-2 =
1.058 799 505 233 764 648 437 5 × 2-2 =
0.264 699 876 308 441 162 109 375
0 - 0111 1101 - 000 0111 1000 0110 1011 1110 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) = 0.264 699 876 308 441 162 109 375(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: