Binary ↘ Float: The 32 Bit Single Precision IEEE 754 Binary Floating Point Standard Representation Number 1 - 1010 1111 - 110 1000 0000 0000 0000 0001 Converted and Written as a Base Ten Decimal System Number (as a Float)
1 - 1010 1111 - 110 1000 0000 0000 0000 0001: 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:
1010 1111
The last 23 bits contain the mantissa:
110 1000 0000 0000 0000 0001
2. Convert the exponent from binary (from base 2) to decimal (in base 10).
The exponent is allways a positive integer.
1010 1111(2) =
1 × 27 + 0 × 26 + 1 × 25 + 0 × 24 + 1 × 23 + 1 × 22 + 1 × 21 + 1 × 20 =
128 + 0 + 32 + 0 + 8 + 4 + 2 + 1 =
128 + 32 + 8 + 4 + 2 + 1 =
175(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 = 175 - 127 = 48
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).
110 1000 0000 0000 0000 0001(2) =
1 × 2-1 + 1 × 2-2 + 0 × 2-3 + 1 × 2-4 + 0 × 2-5 + 0 × 2-6 + 0 × 2-7 + 0 × 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 + 0 × 2-21 + 0 × 2-22 + 1 × 2-23 =
0.5 + 0.25 + 0 + 0.062 5 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0.000 000 119 209 289 550 781 25 =
0.5 + 0.25 + 0.062 5 + 0.000 000 119 209 289 550 781 25 =
0.812 500 119 209 289 550 781 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)1 × (1 + 0.812 500 119 209 289 550 781 25) × 248 =
-1.812 500 119 209 289 550 781 25 × 248 =
-510 173 428 842 496
1 - 1010 1111 - 110 1000 0000 0000 0000 0001 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) = -510 173 428 842 496(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: