Binary ↘ Float: The 32 Bit Single Precision IEEE 754 Binary Floating Point Standard Representation Number 0 - 0111 1010 - 111 0101 1100 0010 1100 1010 Converted and Written as a Base Ten Decimal System Number (as a Float)
0 - 0111 1010 - 111 0101 1100 0010 1100 1010: 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 1010
The last 23 bits contain the mantissa:
111 0101 1100 0010 1100 1010
2. Convert the exponent from binary (from base 2) to decimal (in base 10).
The exponent is allways a positive integer.
0111 1010(2) =
0 × 27 + 1 × 26 + 1 × 25 + 1 × 24 + 1 × 23 + 0 × 22 + 1 × 21 + 0 × 20 =
0 + 64 + 32 + 16 + 8 + 0 + 2 + 0 =
64 + 32 + 16 + 8 + 2 =
122(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 = 122 - 127 = -5
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 0101 1100 0010 1100 1010(2) =
1 × 2-1 + 1 × 2-2 + 1 × 2-3 + 0 × 2-4 + 1 × 2-5 + 0 × 2-6 + 1 × 2-7 + 1 × 2-8 + 1 × 2-9 + 0 × 2-10 + 0 × 2-11 + 0 × 2-12 + 0 × 2-13 + 1 × 2-14 + 0 × 2-15 + 1 × 2-16 + 1 × 2-17 + 0 × 2-18 + 0 × 2-19 + 1 × 2-20 + 0 × 2-21 + 1 × 2-22 + 0 × 2-23 =
0.5 + 0.25 + 0.125 + 0 + 0.031 25 + 0 + 0.007 812 5 + 0.003 906 25 + 0.001 953 125 + 0 + 0 + 0 + 0 + 0.000 061 035 156 25 + 0 + 0.000 015 258 789 062 5 + 0.000 007 629 394 531 25 + 0 + 0 + 0.000 000 953 674 316 406 25 + 0 + 0.000 000 238 418 579 101 562 5 + 0 =
0.5 + 0.25 + 0.125 + 0.031 25 + 0.007 812 5 + 0.003 906 25 + 0.001 953 125 + 0.000 061 035 156 25 + 0.000 015 258 789 062 5 + 0.000 007 629 394 531 25 + 0.000 000 953 674 316 406 25 + 0.000 000 238 418 579 101 562 5 =
0.920 006 990 432 739 257 812 5(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 + 0.920 006 990 432 739 257 812 5) × 2-5 =
1.920 006 990 432 739 257 812 5 × 2-5 =
0.060 000 218 451 023 101 806 640 625
0 - 0111 1010 - 111 0101 1100 0010 1100 1010 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) = 0.060 000 218 451 023 101 806 640 625(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: