Binary ↘ Float: The 32 Bit Single Precision IEEE 754 Binary Floating Point Standard Representation Number 1 - 1001 0110 - 111 1100 0111 0001 1101 0101 Converted and Written as a Base Ten Decimal System Number (as a Float)
1 - 1001 0110 - 111 1100 0111 0001 1101 0101: 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:
1001 0110
The last 23 bits contain the mantissa:
111 1100 0111 0001 1101 0101
2. Convert the exponent from binary (from base 2) to decimal (in base 10).
The exponent is allways a positive integer.
1001 0110(2) =
1 × 27 + 0 × 26 + 0 × 25 + 1 × 24 + 0 × 23 + 1 × 22 + 1 × 21 + 0 × 20 =
128 + 0 + 0 + 16 + 0 + 4 + 2 + 0 =
128 + 16 + 4 + 2 =
150(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 = 150 - 127 = 23
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).
111 1100 0111 0001 1101 0101(2) =
1 × 2-1 + 1 × 2-2 + 1 × 2-3 + 1 × 2-4 + 1 × 2-5 + 0 × 2-6 + 0 × 2-7 + 0 × 2-8 + 1 × 2-9 + 1 × 2-10 + 1 × 2-11 + 0 × 2-12 + 0 × 2-13 + 0 × 2-14 + 1 × 2-15 + 1 × 2-16 + 1 × 2-17 + 0 × 2-18 + 1 × 2-19 + 0 × 2-20 + 1 × 2-21 + 0 × 2-22 + 1 × 2-23 =
0.5 + 0.25 + 0.125 + 0.062 5 + 0.031 25 + 0 + 0 + 0 + 0.001 953 125 + 0.000 976 562 5 + 0.000 488 281 25 + 0 + 0 + 0 + 0.000 030 517 578 125 + 0.000 015 258 789 062 5 + 0.000 007 629 394 531 25 + 0 + 0.000 001 907 348 632 812 5 + 0 + 0.000 000 476 837 158 203 125 + 0 + 0.000 000 119 209 289 550 781 25 =
0.5 + 0.25 + 0.125 + 0.062 5 + 0.031 25 + 0.001 953 125 + 0.000 976 562 5 + 0.000 488 281 25 + 0.000 030 517 578 125 + 0.000 015 258 789 062 5 + 0.000 007 629 394 531 25 + 0.000 001 907 348 632 812 5 + 0.000 000 476 837 158 203 125 + 0.000 000 119 209 289 550 781 25 =
0.972 223 877 906 799 316 406 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 + 0.972 223 877 906 799 316 406 25) × 223 =
-1.972 223 877 906 799 316 406 25 × 223 =
-16 544 213
1 - 1001 0110 - 111 1100 0111 0001 1101 0101 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) = -16 544 213(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: