What are the steps to convert
0 - 1000 0010 - 100 0101 1000 0101 0010 1011, a 32 bit single precision IEEE 754 binary floating point representation standard to decimal?
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:
100 0101 1000 0101 0010 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).
100 0101 1000 0101 0010 1011(2) =
1 × 2-1 + 0 × 2-2 + 0 × 2-3 + 0 × 2-4 + 1 × 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 + 1 × 2-18 + 0 × 2-19 + 1 × 2-20 + 0 × 2-21 + 1 × 2-22 + 1 × 2-23 =
0.5 + 0 + 0 + 0 + 0.031 25 + 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 + 0.000 003 814 697 265 625 + 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 =
0.5 + 0.031 25 + 0.007 812 5 + 0.003 906 25 + 0.000 122 070 312 5 + 0.000 030 517 578 125 + 0.000 003 814 697 265 625 + 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 =
0.543 126 463 890 075 683 593 75(10)
= 12.345 011 711 120 605 468 75
0 - 1000 0010 - 100 0101 1000 0101 0010 1011, a 32 bit single precision IEEE 754 binary floating point representation standard to a decimal number, written in base ten (float) = 12.345 011 711 120 605 468 75(10)
Spaces were used to group digits: for binary, by 4, for decimal, by 3.