Binary ↘ Float: The 32 Bit Single Precision IEEE 754 Binary Floating Point Standard Representation Number 1 - 0111 0110 - 010 0110 0111 0110 0101 0001 Converted and Written as a Base Ten Decimal System Number (as a Float)
1 - 0111 0110 - 010 0110 0111 0110 0101 0001: 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:
0111 0110
The last 23 bits contain the mantissa:
010 0110 0111 0110 0101 0001
2. Convert the exponent from binary (from base 2) to decimal (in base 10).
The exponent is allways a positive integer.
0111 0110(2) =
0 × 27 + 1 × 26 + 1 × 25 + 1 × 24 + 0 × 23 + 1 × 22 + 1 × 21 + 0 × 20 =
0 + 64 + 32 + 16 + 0 + 4 + 2 + 0 =
64 + 32 + 16 + 4 + 2 =
118(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 = 118 - 127 = -9
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).
010 0110 0111 0110 0101 0001(2) =
0 × 2-1 + 1 × 2-2 + 0 × 2-3 + 0 × 2-4 + 1 × 2-5 + 1 × 2-6 + 0 × 2-7 + 0 × 2-8 + 1 × 2-9 + 1 × 2-10 + 1 × 2-11 + 0 × 2-12 + 1 × 2-13 + 1 × 2-14 + 0 × 2-15 + 0 × 2-16 + 1 × 2-17 + 0 × 2-18 + 1 × 2-19 + 0 × 2-20 + 0 × 2-21 + 0 × 2-22 + 1 × 2-23 =
0 + 0.25 + 0 + 0 + 0.031 25 + 0.015 625 + 0 + 0 + 0.001 953 125 + 0.000 976 562 5 + 0.000 488 281 25 + 0 + 0.000 122 070 312 5 + 0.000 061 035 156 25 + 0 + 0 + 0.000 007 629 394 531 25 + 0 + 0.000 001 907 348 632 812 5 + 0 + 0 + 0 + 0.000 000 119 209 289 550 781 25 =
0.25 + 0.031 25 + 0.015 625 + 0.001 953 125 + 0.000 976 562 5 + 0.000 488 281 25 + 0.000 122 070 312 5 + 0.000 061 035 156 25 + 0.000 007 629 394 531 25 + 0.000 001 907 348 632 812 5 + 0.000 000 119 209 289 550 781 25 =
0.300 485 730 171 203 613 281 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 + 0.300 485 730 171 203 613 281 25) × 2-9 =
-1.300 485 730 171 203 613 281 25 × 2-9 =
-0.002 540 011 191 740 632 057 189 941 406 25
1 - 0111 0110 - 010 0110 0111 0110 0101 0001 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) = -0.002 540 011 191 740 632 057 189 941 406 25(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: