Binary ↘ Float: The 32 Bit Single Precision IEEE 754 Binary Floating Point Standard Representation Number 1 - 1000 0110 - 100 0110 1100 0010 0110 0100 Converted and Written as a Base Ten Decimal System Number (as a Float)
1 - 1000 0110 - 100 0110 1100 0010 0110 0100: 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:
1000 0110
The last 23 bits contain the mantissa:
100 0110 1100 0010 0110 0100
2. Convert the exponent from binary (from base 2) to decimal (in base 10).
The exponent is allways a positive integer.
1000 0110(2) =
1 × 27 + 0 × 26 + 0 × 25 + 0 × 24 + 0 × 23 + 1 × 22 + 1 × 21 + 0 × 20 =
128 + 0 + 0 + 0 + 0 + 4 + 2 + 0 =
128 + 4 + 2 =
134(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 = 134 - 127 = 7
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 0110 1100 0010 0110 0100(2) =
1 × 2-1 + 0 × 2-2 + 0 × 2-3 + 0 × 2-4 + 1 × 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 + 1 × 2-14 + 0 × 2-15 + 0 × 2-16 + 1 × 2-17 + 1 × 2-18 + 0 × 2-19 + 0 × 2-20 + 1 × 2-21 + 0 × 2-22 + 0 × 2-23 =
0.5 + 0 + 0 + 0 + 0.031 25 + 0.015 625 + 0 + 0.003 906 25 + 0.001 953 125 + 0 + 0 + 0 + 0 + 0.000 061 035 156 25 + 0 + 0 + 0.000 007 629 394 531 25 + 0.000 003 814 697 265 625 + 0 + 0 + 0.000 000 476 837 158 203 125 + 0 + 0 =
0.5 + 0.031 25 + 0.015 625 + 0.003 906 25 + 0.001 953 125 + 0.000 061 035 156 25 + 0.000 007 629 394 531 25 + 0.000 003 814 697 265 625 + 0.000 000 476 837 158 203 125 =
0.552 807 331 085 205 078 125(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 + 0.552 807 331 085 205 078 125) × 27 =
-1.552 807 331 085 205 078 125 × 27 =
-198.759 338 378 906 25
1 - 1000 0110 - 100 0110 1100 0010 0110 0100 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) = -198.759 338 378 906 25(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: