Binary ↘ Double: The 64 Bit Double Precision IEEE 754 Binary Floating Point Standard Representation Number 0 - 011 1010 1111 - 0100 0000 0001 0000 0000 1000 0000 0010 0000 1001 0000 0011 1011 Converted and Written as a Base Ten Decimal System Number (as a Double)
0 - 011 1010 1111 - 0100 0000 0001 0000 0000 1000 0000 0010 0000 1001 0000 0011 1011: 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:
011 1010 1111
The last 52 bits contain the mantissa:
0100 0000 0001 0000 0000 1000 0000 0010 0000 1001 0000 0011 1011
2. Convert the exponent from binary (from base 2) to decimal (in base 10).
The exponent is allways a positive integer.
011 1010 1111(2) =
0 × 210 + 1 × 29 + 1 × 28 + 1 × 27 + 0 × 26 + 1 × 25 + 0 × 24 + 1 × 23 + 1 × 22 + 1 × 21 + 1 × 20 =
0 + 512 + 256 + 128 + 0 + 32 + 0 + 8 + 4 + 2 + 1 =
512 + 256 + 128 + 32 + 8 + 4 + 2 + 1 =
943(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 = 943 - 1023 = -80
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).
0100 0000 0001 0000 0000 1000 0000 0010 0000 1001 0000 0011 1011(2) =
0 × 2-1 + 1 × 2-2 + 0 × 2-3 + 0 × 2-4 + 0 × 2-5 + 0 × 2-6 + 0 × 2-7 + 0 × 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 + 1 × 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 + 1 × 2-31 + 0 × 2-32 + 0 × 2-33 + 0 × 2-34 + 0 × 2-35 + 0 × 2-36 + 1 × 2-37 + 0 × 2-38 + 0 × 2-39 + 1 × 2-40 + 0 × 2-41 + 0 × 2-42 + 0 × 2-43 + 0 × 2-44 + 0 × 2-45 + 0 × 2-46 + 1 × 2-47 + 1 × 2-48 + 1 × 2-49 + 0 × 2-50 + 1 × 2-51 + 1 × 2-52 =
0 + 0.25 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0.000 244 140 625 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0.000 000 476 837 158 203 125 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0.000 000 000 465 661 287 307 739 257 812 5 + 0 + 0 + 0 + 0 + 0 + 0.000 000 000 007 275 957 614 183 425 903 320 312 5 + 0 + 0 + 0.000 000 000 000 909 494 701 772 928 237 915 039 062 5 + 0 + 0 + 0 + 0 + 0 + 0 + 0.000 000 000 000 007 105 427 357 601 001 858 711 242 675 781 25 + 0.000 000 000 000 003 552 713 678 800 500 929 355 621 337 890 625 + 0.000 000 000 000 001 776 356 839 400 250 464 677 810 668 945 312 5 + 0 + 0.000 000 000 000 000 444 089 209 850 062 616 169 452 667 236 328 125 + 0.000 000 000 000 000 222 044 604 925 031 308 084 726 333 618 164 062 5 =
0.25 + 0.000 244 140 625 + 0.000 000 476 837 158 203 125 + 0.000 000 000 465 661 287 307 739 257 812 5 + 0.000 000 000 007 275 957 614 183 425 903 320 312 5 + 0.000 000 000 000 909 494 701 772 928 237 915 039 062 5 + 0.000 000 000 000 007 105 427 357 601 001 858 711 242 675 781 25 + 0.000 000 000 000 003 552 713 678 800 500 929 355 621 337 890 625 + 0.000 000 000 000 001 776 356 839 400 250 464 677 810 668 945 312 5 + 0.000 000 000 000 000 444 089 209 850 062 616 169 452 667 236 328 125 + 0.000 000 000 000 000 222 044 604 925 031 308 084 726 333 618 164 062 5 =
0.250 244 617 936 018 043 380 386 188 800 912 350 416 183 471 679 687 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)0 × (1 + 0.250 244 617 936 018 043 380 386 188 800 912 350 416 183 471 679 687 5) × 2-80 =
1.250 244 617 936 018 043 380 386 188 800 912 350 416 183 471 679 687 5 × 2-80 =
0.000 000 000 000 000 000 000 001 034 178 108 905 441 456 044 079 163 084 508 199 386 514 432 465 902 331 286 409 932 399 432 409 777 517 705 151 694 826 781 749 725 341 796 875
0 - 011 1010 1111 - 0100 0000 0001 0000 0000 1000 0000 0010 0000 1001 0000 0011 1011 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) = 0.000 000 000 000 000 000 000 001 034 178 108 905 441 456 044 079 163 084 508 199 386 514 432 465 902 331 286 409 932 399 432 409 777 517 705 151 694 826 781 749 725 341 796 875(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: