Binary ↘ Float: The 32 Bit Single Precision IEEE 754 Binary Floating Point Standard Representation Number 1 - 0100 0111 - 100 1011 1100 0000 0100 1100 Converted and Written as a Base Ten Decimal System Number (as a Float)
1 - 0100 0111 - 100 1011 1100 0000 0100 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.
1
The next 8 bits contain the exponent:
0100 0111
The last 23 bits contain the mantissa:
100 1011 1100 0000 0100 1100
2. Convert the exponent from binary (from base 2) to decimal (in base 10).
The exponent is allways a positive integer.
0100 0111(2) =
0 × 27 + 1 × 26 + 0 × 25 + 0 × 24 + 0 × 23 + 1 × 22 + 1 × 21 + 1 × 20 =
0 + 64 + 0 + 0 + 0 + 4 + 2 + 1 =
64 + 4 + 2 + 1 =
71(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 = 71 - 127 = -56
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).
100 1011 1100 0000 0100 1100(2) =
1 × 2-1 + 0 × 2-2 + 0 × 2-3 + 1 × 2-4 + 0 × 2-5 + 1 × 2-6 + 1 × 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 + 1 × 2-17 + 0 × 2-18 + 0 × 2-19 + 1 × 2-20 + 1 × 2-21 + 0 × 2-22 + 0 × 2-23 =
0.5 + 0 + 0 + 0.062 5 + 0 + 0.015 625 + 0.007 812 5 + 0.003 906 25 + 0.001 953 125 + 0 + 0 + 0 + 0 + 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 =
0.5 + 0.062 5 + 0.015 625 + 0.007 812 5 + 0.003 906 25 + 0.001 953 125 + 0.000 007 629 394 531 25 + 0.000 000 953 674 316 406 25 + 0.000 000 476 837 158 203 125 =
0.591 805 934 906 005 859 375(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)1 × (1 + 0.591 805 934 906 005 859 375) × 2-56 =
-1.591 805 934 906 005 859 375 × 2-56 =
-0.000 000 000 000 000 022 090 744 995 845 260 903 09
1 - 0100 0111 - 100 1011 1100 0000 0100 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) = -0.000 000 000 000 000 022 090 744 995 845 260 903 09(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: