Binary ↘ Float: The 32 Bit Single Precision IEEE 754 Binary Floating Point Standard Representation Number 1 - 0101 1101 - 110 1010 1111 1111 1010 1000 Converted and Written as a Base Ten Decimal System Number (as a Float)
1 - 0101 1101 - 110 1010 1111 1111 1010 1000: 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:
0101 1101
The last 23 bits contain the mantissa:
110 1010 1111 1111 1010 1000
2. Convert the exponent from binary (from base 2) to decimal (in base 10).
The exponent is allways a positive integer.
0101 1101(2) =
0 × 27 + 1 × 26 + 0 × 25 + 1 × 24 + 1 × 23 + 1 × 22 + 0 × 21 + 1 × 20 =
0 + 64 + 0 + 16 + 8 + 4 + 0 + 1 =
64 + 16 + 8 + 4 + 1 =
93(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 = 93 - 127 = -34
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 1010 1111 1111 1010 1000(2) =
1 × 2-1 + 1 × 2-2 + 0 × 2-3 + 1 × 2-4 + 0 × 2-5 + 1 × 2-6 + 0 × 2-7 + 1 × 2-8 + 1 × 2-9 + 1 × 2-10 + 1 × 2-11 + 1 × 2-12 + 1 × 2-13 + 1 × 2-14 + 1 × 2-15 + 1 × 2-16 + 0 × 2-17 + 1 × 2-18 + 0 × 2-19 + 1 × 2-20 + 0 × 2-21 + 0 × 2-22 + 0 × 2-23 =
0.5 + 0.25 + 0 + 0.062 5 + 0 + 0.015 625 + 0 + 0.003 906 25 + 0.001 953 125 + 0.000 976 562 5 + 0.000 488 281 25 + 0.000 244 140 625 + 0.000 122 070 312 5 + 0.000 061 035 156 25 + 0.000 030 517 578 125 + 0.000 015 258 789 062 5 + 0 + 0.000 003 814 697 265 625 + 0 + 0.000 000 953 674 316 406 25 + 0 + 0 + 0 =
0.5 + 0.25 + 0.062 5 + 0.015 625 + 0.003 906 25 + 0.001 953 125 + 0.000 976 562 5 + 0.000 488 281 25 + 0.000 244 140 625 + 0.000 122 070 312 5 + 0.000 061 035 156 25 + 0.000 030 517 578 125 + 0.000 015 258 789 062 5 + 0.000 003 814 697 265 625 + 0.000 000 953 674 316 406 25 =
0.835 927 009 582 519 531 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 + 0.835 927 009 582 519 531 25) × 2-34 =
-1.835 927 009 582 519 531 25 × 2-34 =
-0.000 000 000 106 865 016 835 655 524 118 919 856 84
1 - 0101 1101 - 110 1010 1111 1111 1010 1000 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) = -0.000 000 000 106 865 016 835 655 524 118 919 856 84(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: