Binary ↘ Float: The 32 Bit Single Precision IEEE 754 Binary Floating Point Standard Representation Number 1 - 1000 1111 - 011 0111 0100 0000 0010 1001 Converted and Written as a Base Ten Decimal System Number (as a Float)
1 - 1000 1111 - 011 0111 0100 0000 0010 1001: 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:
1000 1111
The last 23 bits contain the mantissa:
011 0111 0100 0000 0010 1001
2. Convert the exponent from binary (from base 2) to decimal (in base 10).
The exponent is allways a positive integer.
1000 1111(2) =
1 × 27 + 0 × 26 + 0 × 25 + 0 × 24 + 1 × 23 + 1 × 22 + 1 × 21 + 1 × 20 =
128 + 0 + 0 + 0 + 8 + 4 + 2 + 1 =
128 + 8 + 4 + 2 + 1 =
143(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 = 143 - 127 = 16
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).
011 0111 0100 0000 0010 1001(2) =
0 × 2-1 + 1 × 2-2 + 1 × 2-3 + 0 × 2-4 + 1 × 2-5 + 1 × 2-6 + 1 × 2-7 + 0 × 2-8 + 1 × 2-9 + 0 × 2-10 + 0 × 2-11 + 0 × 2-12 + 0 × 2-13 + 0 × 2-14 + 0 × 2-15 + 0 × 2-16 + 0 × 2-17 + 1 × 2-18 + 0 × 2-19 + 1 × 2-20 + 0 × 2-21 + 0 × 2-22 + 1 × 2-23 =
0 + 0.25 + 0.125 + 0 + 0.031 25 + 0.015 625 + 0.007 812 5 + 0 + 0.001 953 125 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0.000 003 814 697 265 625 + 0 + 0.000 000 953 674 316 406 25 + 0 + 0 + 0.000 000 119 209 289 550 781 25 =
0.25 + 0.125 + 0.031 25 + 0.015 625 + 0.007 812 5 + 0.001 953 125 + 0.000 003 814 697 265 625 + 0.000 000 953 674 316 406 25 + 0.000 000 119 209 289 550 781 25 =
0.431 645 512 580 871 582 031 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 + 0.431 645 512 580 871 582 031 25) × 216 =
-1.431 645 512 580 871 582 031 25 × 216 =
-93 824.320 312 5
1 - 1000 1111 - 011 0111 0100 0000 0010 1001 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) = -93 824.320 312 5(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: