Binary ↘ Double: The 64 Bit Double Precision IEEE 754 Binary Floating Point Standard Representation Number 1 - 111 1011 0000 - 1111 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0001 Converted and Written as a Base Ten Decimal System Number (as a Double)
1 - 111 1011 0000 - 1111 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0001: 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:
111 1011 0000
The last 52 bits contain the mantissa:
1111 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0001
2. Convert the exponent from binary (from base 2) to decimal (in base 10).
The exponent is allways a positive integer.
111 1011 0000(2) =
1 × 210 + 1 × 29 + 1 × 28 + 1 × 27 + 0 × 26 + 1 × 25 + 1 × 24 + 0 × 23 + 0 × 22 + 0 × 21 + 0 × 20 =
1,024 + 512 + 256 + 128 + 0 + 32 + 16 + 0 + 0 + 0 + 0 =
1,024 + 512 + 256 + 128 + 32 + 16 =
1,968(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 = 1,968 - 1023 = 945
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).
1111 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0001(2) =
1 × 2-1 + 1 × 2-2 + 1 × 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 + 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 + 0 × 2-50 + 0 × 2-51 + 1 × 2-52 =
0.5 + 0.25 + 0.125 + 0.062 5 + 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 + 0 + 0 + 0.000 000 000 000 000 222 044 604 925 031 308 084 726 333 618 164 062 5 =
0.5 + 0.25 + 0.125 + 0.062 5 + 0.000 000 000 000 000 222 044 604 925 031 308 084 726 333 618 164 062 5 =
0.937 500 000 000 000 222 044 604 925 031 308 084 726 333 618 164 062 5(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 + 0.937 500 000 000 000 222 044 604 925 031 308 084 726 333 618 164 062 5) × 2945 =
-1.937 500 000 000 000 222 044 604 925 031 308 084 726 333 618 164 062 5 × 2945 =
-576 219 052 035 141 002 871 065 609 991 242 261 093 336 442 707 591 167 740 027 350 000 117 270 644 745 323 082 990 541 457 868 020 155 703 991 425 848 263 891 350 525 939 789 735 822 062 391 788 012 870 073 808 273 508 540 837 814 701 126 483 686 425 894 128 657 397 918 966 313 545 380 366 150 313 729 052 664 412 433 330 322 361 966 779 824 243 040 327 411 531 684 882 939 904
1 - 111 1011 0000 - 1111 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0001 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) = -576 219 052 035 141 002 871 065 609 991 242 261 093 336 442 707 591 167 740 027 350 000 117 270 644 745 323 082 990 541 457 868 020 155 703 991 425 848 263 891 350 525 939 789 735 822 062 391 788 012 870 073 808 273 508 540 837 814 701 126 483 686 425 894 128 657 397 918 966 313 545 380 366 150 313 729 052 664 412 433 330 322 361 966 779 824 243 040 327 411 531 684 882 939 904(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: