Binary ↘ Float: The 32 Bit Single Precision IEEE 754 Binary Floating Point Standard Representation Number 0 - 0111 1100 - 010 1000 0000 0000 0100 1100 Converted and Written as a Base Ten Decimal System Number (as a Float)
0 - 0111 1100 - 010 1000 0000 0000 0100 1100: 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 1100
The last 23 bits contain the mantissa:
010 1000 0000 0000 0100 1100
2. Convert the exponent from binary (from base 2) to decimal (in base 10).
The exponent is allways a positive integer.
0111 1100(2) =
0 × 27 + 1 × 26 + 1 × 25 + 1 × 24 + 1 × 23 + 1 × 22 + 0 × 21 + 0 × 20 =
0 + 64 + 32 + 16 + 8 + 4 + 0 + 0 =
64 + 32 + 16 + 8 + 4 =
124(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 = 124 - 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).
010 1000 0000 0000 0100 1100(2) =
0 × 2-1 + 1 × 2-2 + 0 × 2-3 + 1 × 2-4 + 0 × 2-5 + 0 × 2-6 + 0 × 2-7 + 0 × 2-8 + 0 × 2-9 + 0 × 2-10 + 0 × 2-11 + 0 × 2-12 + 0 × 2-13 + 0 × 2-14 + 0 × 2-15 + 0 × 2-16 + 1 × 2-17 + 0 × 2-18 + 0 × 2-19 + 1 × 2-20 + 1 × 2-21 + 0 × 2-22 + 0 × 2-23 =
0 + 0.25 + 0 + 0.062 5 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0.000 007 629 394 531 25 + 0 + 0 + 0.000 000 953 674 316 406 25 + 0.000 000 476 837 158 203 125 + 0 + 0 =
0.25 + 0.062 5 + 0.000 007 629 394 531 25 + 0.000 000 953 674 316 406 25 + 0.000 000 476 837 158 203 125 =
0.312 509 059 906 005 859 375(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 + 0.312 509 059 906 005 859 375) × 2-3 =
1.312 509 059 906 005 859 375 × 2-3 =
0.164 063 632 488 250 732 421 875
0 - 0111 1100 - 010 1000 0000 0000 0100 1100 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) = 0.164 063 632 488 250 732 421 875(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: