Binary ↘ Double: The 64 Bit Double Precision IEEE 754 Binary Floating Point Standard Representation Number 0 - 100 0000 0011 - 1011 1001 0001 0000 0000 0000 0000 0000 0000 0000 0000 0000 0100 Converted and Written as a Base Ten Decimal System Number (as a Double)
0 - 100 0000 0011 - 1011 1001 0001 0000 0000 0000 0000 0000 0000 0000 0000 0000 0100: 64 bit double 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 11 bits contain the exponent:
100 0000 0011
The last 52 bits contain the mantissa:
1011 1001 0001 0000 0000 0000 0000 0000 0000 0000 0000 0000 0100
2. Convert the exponent from binary (from base 2) to decimal (in base 10).
The exponent is allways a positive integer.
100 0000 0011(2) =
1 × 210 + 0 × 29 + 0 × 28 + 0 × 27 + 0 × 26 + 0 × 25 + 0 × 24 + 0 × 23 + 0 × 22 + 1 × 21 + 1 × 20 =
1,024 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 2 + 1 =
1,024 + 2 + 1 =
1,027(10)
3. Adjust the exponent.
Subtract the excess bits: 2(11 - 1) - 1 = 1023,
that is due to the 11 bit excess/bias notation.
The exponent, adjusted = 1,027 - 1023 = 4
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).
1011 1001 0001 0000 0000 0000 0000 0000 0000 0000 0000 0000 0100(2) =
1 × 2-1 + 0 × 2-2 + 1 × 2-3 + 1 × 2-4 + 1 × 2-5 + 0 × 2-6 + 0 × 2-7 + 1 × 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 × 2-24 + 0 × 2-25 + 0 × 2-26 + 0 × 2-27 + 0 × 2-28 + 0 × 2-29 + 0 × 2-30 + 0 × 2-31 + 0 × 2-32 + 0 × 2-33 + 0 × 2-34 + 0 × 2-35 + 0 × 2-36 + 0 × 2-37 + 0 × 2-38 + 0 × 2-39 + 0 × 2-40 + 0 × 2-41 + 0 × 2-42 + 0 × 2-43 + 0 × 2-44 + 0 × 2-45 + 0 × 2-46 + 0 × 2-47 + 0 × 2-48 + 0 × 2-49 + 1 × 2-50 + 0 × 2-51 + 0 × 2-52 =
0.5 + 0 + 0.125 + 0.062 5 + 0.031 25 + 0 + 0 + 0.003 906 25 + 0 + 0 + 0 + 0.000 244 140 625 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0.000 000 000 000 000 888 178 419 700 125 232 338 905 334 472 656 25 + 0 + 0 =
0.5 + 0.125 + 0.062 5 + 0.031 25 + 0.003 906 25 + 0.000 244 140 625 + 0.000 000 000 000 000 888 178 419 700 125 232 338 905 334 472 656 25 =
0.722 900 390 625 000 888 178 419 700 125 232 338 905 334 472 656 25(10)
5. Put all the numbers into expression to calculate the double precision floating point decimal value:
(-1)Sign × (1 + Mantissa) × 2(Adjusted exponent) =
(-1)0 × (1 + 0.722 900 390 625 000 888 178 419 700 125 232 338 905 334 472 656 25) × 24 =
1.722 900 390 625 000 888 178 419 700 125 232 338 905 334 472 656 25 × 24 =
27.566 406 250 000 014 210 854 715 202 003 717 422 485 351 562 5
0 - 100 0000 0011 - 1011 1001 0001 0000 0000 0000 0000 0000 0000 0000 0000 0000 0100 converted from a 64 bit double precision IEEE 754 binary floating point standard representation number to a decimal system number, written in base ten (double) = 27.566 406 250 000 014 210 854 715 202 003 717 422 485 351 562 5(10)
Spaces were used to group digits: for binary, by 4, for decimal, by 3.
More operations with 64 bit double precision IEEE 754 binary floating point standard representation numbers: