Binary ↘ Double: The 64 Bit Double Precision IEEE 754 Binary Floating Point Standard Representation Number 1 - 111 0001 0011 - 1011 0100 0110 0100 0101 0111 0001 0111 1000 1000 0000 1110 1111 Converted and Written as a Base Ten Decimal System Number (as a Double)
1 - 111 0001 0011 - 1011 0100 0110 0100 0101 0111 0001 0111 1000 1000 0000 1110 1111: 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 0001 0011
The last 52 bits contain the mantissa:
1011 0100 0110 0100 0101 0111 0001 0111 1000 1000 0000 1110 1111
2. Convert the exponent from binary (from base 2) to decimal (in base 10).
The exponent is allways a positive integer.
111 0001 0011(2) =
1 × 210 + 1 × 29 + 1 × 28 + 0 × 27 + 0 × 26 + 0 × 25 + 1 × 24 + 0 × 23 + 0 × 22 + 1 × 21 + 1 × 20 =
1,024 + 512 + 256 + 0 + 0 + 0 + 16 + 0 + 0 + 2 + 1 =
1,024 + 512 + 256 + 16 + 2 + 1 =
1,811(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 = 1,811 - 1023 = 788
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 0100 0110 0100 0101 0111 0001 0111 1000 1000 0000 1110 1111(2) =
1 × 2-1 + 0 × 2-2 + 1 × 2-3 + 1 × 2-4 + 0 × 2-5 + 1 × 2-6 + 0 × 2-7 + 0 × 2-8 + 0 × 2-9 + 1 × 2-10 + 1 × 2-11 + 0 × 2-12 + 0 × 2-13 + 1 × 2-14 + 0 × 2-15 + 0 × 2-16 + 0 × 2-17 + 1 × 2-18 + 0 × 2-19 + 1 × 2-20 + 0 × 2-21 + 1 × 2-22 + 1 × 2-23 + 1 × 2-24 + 0 × 2-25 + 0 × 2-26 + 0 × 2-27 + 1 × 2-28 + 0 × 2-29 + 1 × 2-30 + 1 × 2-31 + 1 × 2-32 + 1 × 2-33 + 0 × 2-34 + 0 × 2-35 + 0 × 2-36 + 1 × 2-37 + 0 × 2-38 + 0 × 2-39 + 0 × 2-40 + 0 × 2-41 + 0 × 2-42 + 0 × 2-43 + 0 × 2-44 + 1 × 2-45 + 1 × 2-46 + 1 × 2-47 + 0 × 2-48 + 1 × 2-49 + 1 × 2-50 + 1 × 2-51 + 1 × 2-52 =
0.5 + 0 + 0.125 + 0.062 5 + 0 + 0.015 625 + 0 + 0 + 0 + 0.000 976 562 5 + 0.000 488 281 25 + 0 + 0 + 0.000 061 035 156 25 + 0 + 0 + 0 + 0.000 003 814 697 265 625 + 0 + 0.000 000 953 674 316 406 25 + 0 + 0.000 000 238 418 579 101 562 5 + 0.000 000 119 209 289 550 781 25 + 0.000 000 059 604 644 775 390 625 + 0 + 0 + 0 + 0.000 000 003 725 290 298 461 914 062 5 + 0 + 0.000 000 000 931 322 574 615 478 515 625 + 0.000 000 000 465 661 287 307 739 257 812 5 + 0.000 000 000 232 830 643 653 869 628 906 25 + 0.000 000 000 116 415 321 826 934 814 453 125 + 0 + 0 + 0 + 0.000 000 000 007 275 957 614 183 425 903 320 312 5 + 0 + 0 + 0 + 0 + 0 + 0 + 0 + 0.000 000 000 000 028 421 709 430 404 007 434 844 970 703 125 + 0.000 000 000 000 014 210 854 715 202 003 717 422 485 351 562 5 + 0.000 000 000 000 007 105 427 357 601 001 858 711 242 675 781 25 + 0 + 0.000 000 000 000 001 776 356 839 400 250 464 677 810 668 945 312 5 + 0.000 000 000 000 000 888 178 419 700 125 232 338 905 334 472 656 25 + 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 =
0.5 + 0.125 + 0.062 5 + 0.015 625 + 0.000 976 562 5 + 0.000 488 281 25 + 0.000 061 035 156 25 + 0.000 003 814 697 265 625 + 0.000 000 953 674 316 406 25 + 0.000 000 238 418 579 101 562 5 + 0.000 000 119 209 289 550 781 25 + 0.000 000 059 604 644 775 390 625 + 0.000 000 003 725 290 298 461 914 062 5 + 0.000 000 000 931 322 574 615 478 515 625 + 0.000 000 000 465 661 287 307 739 257 812 5 + 0.000 000 000 232 830 643 653 869 628 906 25 + 0.000 000 000 116 415 321 826 934 814 453 125 + 0.000 000 000 007 275 957 614 183 425 903 320 312 5 + 0.000 000 000 000 028 421 709 430 404 007 434 844 970 703 125 + 0.000 000 000 000 014 210 854 715 202 003 717 422 485 351 562 5 + 0.000 000 000 000 007 105 427 357 601 001 858 711 242 675 781 25 + 0.000 000 000 000 001 776 356 839 400 250 464 677 810 668 945 312 5 + 0.000 000 000 000 000 888 178 419 700 125 232 338 905 334 472 656 25 + 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 =
0.704 656 069 989 194 611 125 071 787 682 827 562 093 734 741 210 937 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 + 0.704 656 069 989 194 611 125 071 787 682 827 562 093 734 741 210 937 5) × 2788 =
-1.704 656 069 989 194 611 125 071 787 682 827 562 093 734 741 210 937 5 × 2788 =
-2 775 066 229 927 783 369 153 994 699 578 563 314 415 159 703 626 481 724 191 583 252 358 522 476 494 420 165 582 274 554 717 826 822 727 142 738 396 987 793 987 261 173 508 980 566 467 993 536 980 434 900 422 607 989 147 993 841 001 637 459 638 215 817 172 685 371 325 631 519 295 468 345 664 248 238 275 167 256 576
1 - 111 0001 0011 - 1011 0100 0110 0100 0101 0111 0001 0111 1000 1000 0000 1110 1111 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) = -2 775 066 229 927 783 369 153 994 699 578 563 314 415 159 703 626 481 724 191 583 252 358 522 476 494 420 165 582 274 554 717 826 822 727 142 738 396 987 793 987 261 173 508 980 566 467 993 536 980 434 900 422 607 989 147 993 841 001 637 459 638 215 817 172 685 371 325 631 519 295 468 345 664 248 238 275 167 256 576(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: