Binary ↘ Float: The 32 Bit Single Precision IEEE 754 Binary Floating Point Standard Representation Number 1 - 1000 1000 - 100 1101 1000 0000 0000 0110 Converted and Written as a Base Ten Decimal System Number (as a Float)
1 - 1000 1000 - 100 1101 1000 0000 0000 0110: 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 1000
The last 23 bits contain the mantissa:
100 1101 1000 0000 0000 0110
2. Convert the exponent from binary (from base 2) to decimal (in base 10).
The exponent is allways a positive integer.
1000 1000(2) =
1 × 27 + 0 × 26 + 0 × 25 + 0 × 24 + 1 × 23 + 0 × 22 + 0 × 21 + 0 × 20 =
128 + 0 + 0 + 0 + 8 + 0 + 0 + 0 =
128 + 8 =
136(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 = 136 - 127 = 9
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 1101 1000 0000 0000 0110(2) =
1 × 2-1 + 0 × 2-2 + 0 × 2-3 + 1 × 2-4 + 1 × 2-5 + 0 × 2-6 + 1 × 2-7 + 1 × 2-8 + 0 × 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 + 0 × 2-18 + 0 × 2-19 + 0 × 2-20 + 1 × 2-21 + 1 × 2-22 + 0 × 2-23 =
0.5 + 0 + 0 + 0.062 5 + 0.031 25 + 0 + 0.007 812 5 + 0.003 906 25 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0.000 000 476 837 158 203 125 + 0.000 000 238 418 579 101 562 5 + 0 =
0.5 + 0.062 5 + 0.031 25 + 0.007 812 5 + 0.003 906 25 + 0.000 000 476 837 158 203 125 + 0.000 000 238 418 579 101 562 5 =
0.605 469 465 255 737 304 687 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)1 × (1 + 0.605 469 465 255 737 304 687 5) × 29 =
-1.605 469 465 255 737 304 687 5 × 29 =
-822.000 366 210 937 5
1 - 1000 1000 - 100 1101 1000 0000 0000 0110 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) = -822.000 366 210 937 5(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: