Binary ↘ Double: The 64 Bit Double Precision IEEE 754 Binary Floating Point Standard Representation Number 0 - 101 1111 0000 - 0011 0010 1000 0000 0000 0000 0000 0000 0000 0000 0000 0101 0100 Converted and Written as a Base Ten Decimal System Number (as a Double)
0 - 101 1111 0000 - 0011 0010 1000 0000 0000 0000 0000 0000 0000 0000 0000 0101 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:
101 1111 0000
The last 52 bits contain the mantissa:
0011 0010 1000 0000 0000 0000 0000 0000 0000 0000 0000 0101 0100
2. Convert the exponent from binary (from base 2) to decimal (in base 10).
The exponent is allways a positive integer.
101 1111 0000(2) =
1 × 210 + 0 × 29 + 1 × 28 + 1 × 27 + 1 × 26 + 1 × 25 + 1 × 24 + 0 × 23 + 0 × 22 + 0 × 21 + 0 × 20 =
1,024 + 0 + 256 + 128 + 64 + 32 + 16 + 0 + 0 + 0 + 0 =
1,024 + 256 + 128 + 64 + 32 + 16 =
1,520(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 = 1,520 - 1023 = 497
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 0010 1000 0000 0000 0000 0000 0000 0000 0000 0000 0101 0100(2) =
0 × 2-1 + 0 × 2-2 + 1 × 2-3 + 1 × 2-4 + 0 × 2-5 + 0 × 2-6 + 1 × 2-7 + 0 × 2-8 + 1 × 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 + 1 × 2-46 + 0 × 2-47 + 1 × 2-48 + 0 × 2-49 + 1 × 2-50 + 0 × 2-51 + 0 × 2-52 =
0 + 0 + 0.125 + 0.062 5 + 0 + 0 + 0.007 812 5 + 0 + 0.001 953 125 + 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 + 0.000 000 000 000 014 210 854 715 202 003 717 422 485 351 562 5 + 0 + 0.000 000 000 000 003 552 713 678 800 500 929 355 621 337 890 625 + 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 + 0.007 812 5 + 0.001 953 125 + 0.000 000 000 000 014 210 854 715 202 003 717 422 485 351 562 5 + 0.000 000 000 000 003 552 713 678 800 500 929 355 621 337 890 625 + 0.000 000 000 000 000 888 178 419 700 125 232 338 905 334 472 656 25 =
0.197 265 625 000 018 651 746 813 702 629 879 117 012 023 925 781 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 + 0.197 265 625 000 018 651 746 813 702 629 879 117 012 023 925 781 25) × 2497 =
1.197 265 625 000 018 651 746 813 702 629 879 117 012 023 925 781 25 × 2497 =
489 889 756 503 995 660 692 856 393 949 332 769 928 498 833 379 739 305 996 313 000 199 723 291 876 599 575 946 764 586 819 742 593 056 201 815 164 458 486 524 213 725 722 267 654 149 862 605 193 216
0 - 101 1111 0000 - 0011 0010 1000 0000 0000 0000 0000 0000 0000 0000 0000 0101 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) = 489 889 756 503 995 660 692 856 393 949 332 769 928 498 833 379 739 305 996 313 000 199 723 291 876 599 575 946 764 586 819 742 593 056 201 815 164 458 486 524 213 725 722 267 654 149 862 605 193 216(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: