Binary ↘ Float: The 32 Bit Single Precision IEEE 754 Binary Floating Point Standard Representation Number 0 - 0111 1110 - 010 0110 1110 1001 1001 0001 Converted and Written as a Base Ten Decimal System Number (as a Float)
0 - 0111 1110 - 010 0110 1110 1001 1001 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.
0
The next 8 bits contain the exponent:
0111 1110
The last 23 bits contain the mantissa:
010 0110 1110 1001 1001 0001
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).
010 0110 1110 1001 1001 0001(2) =
0 × 2-1 + 1 × 2-2 + 0 × 2-3 + 0 × 2-4 + 1 × 2-5 + 1 × 2-6 + 0 × 2-7 + 1 × 2-8 + 1 × 2-9 + 1 × 2-10 + 0 × 2-11 + 1 × 2-12 + 0 × 2-13 + 0 × 2-14 + 1 × 2-15 + 1 × 2-16 + 0 × 2-17 + 0 × 2-18 + 1 × 2-19 + 0 × 2-20 + 0 × 2-21 + 0 × 2-22 + 1 × 2-23 =
0 + 0.25 + 0 + 0 + 0.031 25 + 0.015 625 + 0 + 0.003 906 25 + 0.001 953 125 + 0.000 976 562 5 + 0 + 0.000 244 140 625 + 0 + 0 + 0.000 030 517 578 125 + 0.000 015 258 789 062 5 + 0 + 0 + 0.000 001 907 348 632 812 5 + 0 + 0 + 0 + 0.000 000 119 209 289 550 781 25 =
0.25 + 0.031 25 + 0.015 625 + 0.003 906 25 + 0.001 953 125 + 0.000 976 562 5 + 0.000 244 140 625 + 0.000 030 517 578 125 + 0.000 015 258 789 062 5 + 0.000 001 907 348 632 812 5 + 0.000 000 119 209 289 550 781 25 =
0.304 002 881 050 109 863 281 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)0 × (1 + 0.304 002 881 050 109 863 281 25) × 2-1 =
1.304 002 881 050 109 863 281 25 × 2-1 =
0.652 001 440 525 054 931 640 625
0 - 0111 1110 - 010 0110 1110 1001 1001 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) = 0.652 001 440 525 054 931 640 625(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: