Binary ↘ Double: The 64 Bit Double Precision IEEE 754 Binary Floating Point Standard Representation Number 1 - 011 1111 1110 - 0011 1000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0100 Converted and Written as a Base Ten Decimal System Number (as a Double)
1 - 011 1111 1110 - 0011 1000 0000 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.
1
The next 11 bits contain the exponent:
011 1111 1110
The last 52 bits contain the mantissa:
0011 1000 0000 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.
011 1111 1110(2) =
0 × 210 + 1 × 29 + 1 × 28 + 1 × 27 + 1 × 26 + 1 × 25 + 1 × 24 + 1 × 23 + 1 × 22 + 1 × 21 + 0 × 20 =
0 + 512 + 256 + 128 + 64 + 32 + 16 + 8 + 4 + 2 + 0 =
512 + 256 + 128 + 64 + 32 + 16 + 8 + 4 + 2 =
1,022(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 = 1,022 - 1023 = -1
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).
0011 1000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0100(2) =
0 × 2-1 + 0 × 2-2 + 1 × 2-3 + 1 × 2-4 + 1 × 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 + 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 + 0 + 0.125 + 0.062 5 + 0.031 25 + 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 + 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 =
0.125 + 0.062 5 + 0.031 25 + 0.000 000 000 000 000 888 178 419 700 125 232 338 905 334 472 656 25 =
0.218 750 000 000 000 888 178 419 700 125 232 338 905 334 472 656 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)1 × (1 + 0.218 750 000 000 000 888 178 419 700 125 232 338 905 334 472 656 25) × 2-1 =
-1.218 750 000 000 000 888 178 419 700 125 232 338 905 334 472 656 25 × 2-1 =
-0.609 375 000 000 000 444 089 209 850 062 616 169 452 667 236 328 125
1 - 011 1111 1110 - 0011 1000 0000 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) = -0.609 375 000 000 000 444 089 209 850 062 616 169 452 667 236 328 125(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: