Binary ↘ Float: The 32 Bit Single Precision IEEE 754 Binary Floating Point Standard Representation Number 0 - 1000 0010 - 111 0001 1000 0101 0001 1011 Converted and Written as a Base Ten Decimal System Number (as a Float)
0 - 1000 0010 - 111 0001 1000 0101 0001 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:
1000 0010
The last 23 bits contain the mantissa:
111 0001 1000 0101 0001 1011
2. Convert the exponent from binary (from base 2) to decimal (in base 10).
The exponent is allways a positive integer.
1000 0010(2) =
1 × 27 + 0 × 26 + 0 × 25 + 0 × 24 + 0 × 23 + 0 × 22 + 1 × 21 + 0 × 20 =
128 + 0 + 0 + 0 + 0 + 0 + 2 + 0 =
128 + 2 =
130(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 = 130 - 127 = 3
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 0001 1000 0101 0001 1011(2) =
1 × 2-1 + 1 × 2-2 + 1 × 2-3 + 0 × 2-4 + 0 × 2-5 + 0 × 2-6 + 1 × 2-7 + 1 × 2-8 + 0 × 2-9 + 0 × 2-10 + 0 × 2-11 + 0 × 2-12 + 1 × 2-13 + 0 × 2-14 + 1 × 2-15 + 0 × 2-16 + 0 × 2-17 + 0 × 2-18 + 1 × 2-19 + 1 × 2-20 + 0 × 2-21 + 1 × 2-22 + 1 × 2-23 =
0.5 + 0.25 + 0.125 + 0 + 0 + 0 + 0.007 812 5 + 0.003 906 25 + 0 + 0 + 0 + 0 + 0.000 122 070 312 5 + 0 + 0.000 030 517 578 125 + 0 + 0 + 0 + 0.000 001 907 348 632 812 5 + 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 =
0.5 + 0.25 + 0.125 + 0.007 812 5 + 0.003 906 25 + 0.000 122 070 312 5 + 0.000 030 517 578 125 + 0.000 001 907 348 632 812 5 + 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 =
0.886 874 556 541 442 871 093 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 + 0.886 874 556 541 442 871 093 75) × 23 =
1.886 874 556 541 442 871 093 75 × 23 =
15.094 996 452 331 542 968 75
0 - 1000 0010 - 111 0001 1000 0101 0001 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) = 15.094 996 452 331 542 968 75(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: