Binary ↘ Float: The 32 Bit Single Precision IEEE 754 Binary Floating Point Standard Representation Number 0 - 1100 0000 - 111 1010 1100 0000 0011 0101 Converted and Written as a Base Ten Decimal System Number (as a Float)
0 - 1100 0000 - 111 1010 1100 0000 0011 0101: 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:
1100 0000
The last 23 bits contain the mantissa:
111 1010 1100 0000 0011 0101
2. Convert the exponent from binary (from base 2) to decimal (in base 10).
The exponent is allways a positive integer.
1100 0000(2) =
1 × 27 + 1 × 26 + 0 × 25 + 0 × 24 + 0 × 23 + 0 × 22 + 0 × 21 + 0 × 20 =
128 + 64 + 0 + 0 + 0 + 0 + 0 + 0 =
128 + 64 =
192(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 = 192 - 127 = 65
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).
111 1010 1100 0000 0011 0101(2) =
1 × 2-1 + 1 × 2-2 + 1 × 2-3 + 1 × 2-4 + 0 × 2-5 + 1 × 2-6 + 0 × 2-7 + 1 × 2-8 + 1 × 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 + 1 × 2-18 + 1 × 2-19 + 0 × 2-20 + 1 × 2-21 + 0 × 2-22 + 1 × 2-23 =
0.5 + 0.25 + 0.125 + 0.062 5 + 0 + 0.015 625 + 0 + 0.003 906 25 + 0.001 953 125 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0.000 003 814 697 265 625 + 0.000 001 907 348 632 812 5 + 0 + 0.000 000 476 837 158 203 125 + 0 + 0.000 000 119 209 289 550 781 25 =
0.5 + 0.25 + 0.125 + 0.062 5 + 0.015 625 + 0.003 906 25 + 0.001 953 125 + 0.000 003 814 697 265 625 + 0.000 001 907 348 632 812 5 + 0.000 000 476 837 158 203 125 + 0.000 000 119 209 289 550 781 25 =
0.958 990 693 092 346 191 406 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)0 × (1 + 0.958 990 693 092 346 191 406 25) × 265 =
1.958 990 693 092 346 191 406 25 × 265 =
72 273 999 916 506 808 320
0 - 1100 0000 - 111 1010 1100 0000 0011 0101 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) = 72 273 999 916 506 808 320(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: