Binary ↘ Float: The 32 Bit Single Precision IEEE 754 Binary Floating Point Standard Representation Number 1 - 1000 0101 - 010 0010 1111 0000 0010 0000 Converted and Written as a Base Ten Decimal System Number (as a Float)
1 - 1000 0101 - 010 0010 1111 0000 0010 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.
1
The next 8 bits contain the exponent:
1000 0101
The last 23 bits contain the mantissa:
010 0010 1111 0000 0010 0000
2. Convert the exponent from binary (from base 2) to decimal (in base 10).
The exponent is allways a positive integer.
1000 0101(2) =
1 × 27 + 0 × 26 + 0 × 25 + 0 × 24 + 0 × 23 + 1 × 22 + 0 × 21 + 1 × 20 =
128 + 0 + 0 + 0 + 0 + 4 + 0 + 1 =
128 + 4 + 1 =
133(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 = 133 - 127 = 6
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 0010 1111 0000 0010 0000(2) =
0 × 2-1 + 1 × 2-2 + 0 × 2-3 + 0 × 2-4 + 0 × 2-5 + 1 × 2-6 + 0 × 2-7 + 1 × 2-8 + 1 × 2-9 + 1 × 2-10 + 1 × 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 + 0 × 2-20 + 0 × 2-21 + 0 × 2-22 + 0 × 2-23 =
0 + 0.25 + 0 + 0 + 0 + 0.015 625 + 0 + 0.003 906 25 + 0.001 953 125 + 0.000 976 562 5 + 0.000 488 281 25 + 0 + 0 + 0 + 0 + 0 + 0 + 0.000 003 814 697 265 625 + 0 + 0 + 0 + 0 + 0 =
0.25 + 0.015 625 + 0.003 906 25 + 0.001 953 125 + 0.000 976 562 5 + 0.000 488 281 25 + 0.000 003 814 697 265 625 =
0.272 953 033 447 265 625(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)1 × (1 + 0.272 953 033 447 265 625) × 26 =
-1.272 953 033 447 265 625 × 26 =
-81.468 994 140 625
1 - 1000 0101 - 010 0010 1111 0000 0010 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) = -81.468 994 140 625(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: