Binary ↘ Float: The 32 Bit Single Precision IEEE 754 Binary Floating Point Standard Representation Number 0 - 0101 1001 - 110 1001 0000 1000 0000 0000 Converted and Written as a Base Ten Decimal System Number (as a Float)
0 - 0101 1001 - 110 1001 0000 1000 0000 0000: 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:
0101 1001
The last 23 bits contain the mantissa:
110 1001 0000 1000 0000 0000
2. Convert the exponent from binary (from base 2) to decimal (in base 10).
The exponent is allways a positive integer.
0101 1001(2) =
0 × 27 + 1 × 26 + 0 × 25 + 1 × 24 + 1 × 23 + 0 × 22 + 0 × 21 + 1 × 20 =
0 + 64 + 0 + 16 + 8 + 0 + 0 + 1 =
64 + 16 + 8 + 1 =
89(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 = 89 - 127 = -38
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).
110 1001 0000 1000 0000 0000(2) =
1 × 2-1 + 1 × 2-2 + 0 × 2-3 + 1 × 2-4 + 0 × 2-5 + 0 × 2-6 + 1 × 2-7 + 0 × 2-8 + 0 × 2-9 + 0 × 2-10 + 0 × 2-11 + 1 × 2-12 + 0 × 2-13 + 0 × 2-14 + 0 × 2-15 + 0 × 2-16 + 0 × 2-17 + 0 × 2-18 + 0 × 2-19 + 0 × 2-20 + 0 × 2-21 + 0 × 2-22 + 0 × 2-23 =
0.5 + 0.25 + 0 + 0.062 5 + 0 + 0 + 0.007 812 5 + 0 + 0 + 0 + 0 + 0.000 244 140 625 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 =
0.5 + 0.25 + 0.062 5 + 0.007 812 5 + 0.000 244 140 625 =
0.820 556 640 625(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 + 0.820 556 640 625) × 2-38 =
1.820 556 640 625 × 2-38 =
0.000 000 000 006 623 146 475 703 833 857 551 217 07
0 - 0101 1001 - 110 1001 0000 1000 0000 0000 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) = 0.000 000 000 006 623 146 475 703 833 857 551 217 07(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: