Binary ↘ Float: The 32 Bit Single Precision IEEE 754 Binary Floating Point Standard Representation Number 0 - 0111 1110 - 011 0101 0000 0110 0000 1011 Converted and Written as a Base Ten Decimal System Number (as a Float)
0 - 0111 1110 - 011 0101 0000 0110 0000 1011: 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:
0111 1110
The last 23 bits contain the mantissa:
011 0101 0000 0110 0000 1011
2. Convert the exponent from binary (from base 2) to decimal (in base 10).
The exponent is allways a positive integer.
0111 1110(2) =
0 × 27 + 1 × 26 + 1 × 25 + 1 × 24 + 1 × 23 + 1 × 22 + 1 × 21 + 0 × 20 =
0 + 64 + 32 + 16 + 8 + 4 + 2 + 0 =
64 + 32 + 16 + 8 + 4 + 2 =
126(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 = 126 - 127 = -1
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).
011 0101 0000 0110 0000 1011(2) =
0 × 2-1 + 1 × 2-2 + 1 × 2-3 + 0 × 2-4 + 1 × 2-5 + 0 × 2-6 + 1 × 2-7 + 0 × 2-8 + 0 × 2-9 + 0 × 2-10 + 0 × 2-11 + 0 × 2-12 + 1 × 2-13 + 1 × 2-14 + 0 × 2-15 + 0 × 2-16 + 0 × 2-17 + 0 × 2-18 + 0 × 2-19 + 1 × 2-20 + 0 × 2-21 + 1 × 2-22 + 1 × 2-23 =
0 + 0.25 + 0.125 + 0 + 0.031 25 + 0 + 0.007 812 5 + 0 + 0 + 0 + 0 + 0 + 0.000 122 070 312 5 + 0.000 061 035 156 25 + 0 + 0 + 0 + 0 + 0 + 0.000 000 953 674 316 406 25 + 0 + 0.000 000 238 418 579 101 562 5 + 0.000 000 119 209 289 550 781 25 =
0.25 + 0.125 + 0.031 25 + 0.007 812 5 + 0.000 122 070 312 5 + 0.000 061 035 156 25 + 0.000 000 953 674 316 406 25 + 0.000 000 238 418 579 101 562 5 + 0.000 000 119 209 289 550 781 25 =
0.414 246 916 770 935 058 593 75(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 + 0.414 246 916 770 935 058 593 75) × 2-1 =
1.414 246 916 770 935 058 593 75 × 2-1 =
0.707 123 458 385 467 529 296 875
0 - 0111 1110 - 011 0101 0000 0110 0000 1011 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) = 0.707 123 458 385 467 529 296 875(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: