Binary ↘ Double: The 64 Bit Double Precision IEEE 754 Binary Floating Point Standard Representation Number 0 - 111 1111 1101 - 0101 0101 0101 0101 0101 0101 0101 0101 0101 0101 0101 0101 1010 Converted and Written as a Base Ten Decimal System Number (as a Double)
0 - 111 1111 1101 - 0101 0101 0101 0101 0101 0101 0101 0101 0101 0101 0101 0101 1010: 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:
111 1111 1101
The last 52 bits contain the mantissa:
0101 0101 0101 0101 0101 0101 0101 0101 0101 0101 0101 0101 1010
2. Convert the exponent from binary (from base 2) to decimal (in base 10).
The exponent is allways a positive integer.
111 1111 1101(2) =
1 × 210 + 1 × 29 + 1 × 28 + 1 × 27 + 1 × 26 + 1 × 25 + 1 × 24 + 1 × 23 + 1 × 22 + 0 × 21 + 1 × 20 =
1,024 + 512 + 256 + 128 + 64 + 32 + 16 + 8 + 4 + 0 + 1 =
1,024 + 512 + 256 + 128 + 64 + 32 + 16 + 8 + 4 + 1 =
2,045(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 = 2,045 - 1023 = 1022
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).
0101 0101 0101 0101 0101 0101 0101 0101 0101 0101 0101 0101 1010(2) =
0 × 2-1 + 1 × 2-2 + 0 × 2-3 + 1 × 2-4 + 0 × 2-5 + 1 × 2-6 + 0 × 2-7 + 1 × 2-8 + 0 × 2-9 + 1 × 2-10 + 0 × 2-11 + 1 × 2-12 + 0 × 2-13 + 1 × 2-14 + 0 × 2-15 + 1 × 2-16 + 0 × 2-17 + 1 × 2-18 + 0 × 2-19 + 1 × 2-20 + 0 × 2-21 + 1 × 2-22 + 0 × 2-23 + 1 × 2-24 + 0 × 2-25 + 1 × 2-26 + 0 × 2-27 + 1 × 2-28 + 0 × 2-29 + 1 × 2-30 + 0 × 2-31 + 1 × 2-32 + 0 × 2-33 + 1 × 2-34 + 0 × 2-35 + 1 × 2-36 + 0 × 2-37 + 1 × 2-38 + 0 × 2-39 + 1 × 2-40 + 0 × 2-41 + 1 × 2-42 + 0 × 2-43 + 1 × 2-44 + 0 × 2-45 + 1 × 2-46 + 0 × 2-47 + 1 × 2-48 + 1 × 2-49 + 0 × 2-50 + 1 × 2-51 + 0 × 2-52 =
0 + 0.25 + 0 + 0.062 5 + 0 + 0.015 625 + 0 + 0.003 906 25 + 0 + 0.000 976 562 5 + 0 + 0.000 244 140 625 + 0 + 0.000 061 035 156 25 + 0 + 0.000 015 258 789 062 5 + 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 + 0.000 000 059 604 644 775 390 625 + 0 + 0.000 000 014 901 161 193 847 656 25 + 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 + 0.000 000 000 232 830 643 653 869 628 906 25 + 0 + 0.000 000 000 058 207 660 913 467 407 226 562 5 + 0 + 0.000 000 000 014 551 915 228 366 851 806 640 625 + 0 + 0.000 000 000 003 637 978 807 091 712 951 660 156 25 + 0 + 0.000 000 000 000 909 494 701 772 928 237 915 039 062 5 + 0 + 0.000 000 000 000 227 373 675 443 232 059 478 759 765 625 + 0 + 0.000 000 000 000 056 843 418 860 808 014 869 689 941 406 25 + 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 + 0.000 000 000 000 001 776 356 839 400 250 464 677 810 668 945 312 5 + 0 + 0.000 000 000 000 000 444 089 209 850 062 616 169 452 667 236 328 125 + 0 =
0.25 + 0.062 5 + 0.015 625 + 0.003 906 25 + 0.000 976 562 5 + 0.000 244 140 625 + 0.000 061 035 156 25 + 0.000 015 258 789 062 5 + 0.000 003 814 697 265 625 + 0.000 000 953 674 316 406 25 + 0.000 000 238 418 579 101 562 5 + 0.000 000 059 604 644 775 390 625 + 0.000 000 014 901 161 193 847 656 25 + 0.000 000 003 725 290 298 461 914 062 5 + 0.000 000 000 931 322 574 615 478 515 625 + 0.000 000 000 232 830 643 653 869 628 906 25 + 0.000 000 000 058 207 660 913 467 407 226 562 5 + 0.000 000 000 014 551 915 228 366 851 806 640 625 + 0.000 000 000 003 637 978 807 091 712 951 660 156 25 + 0.000 000 000 000 909 494 701 772 928 237 915 039 062 5 + 0.000 000 000 000 227 373 675 443 232 059 478 759 765 625 + 0.000 000 000 000 056 843 418 860 808 014 869 689 941 406 25 + 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 + 0.000 000 000 000 001 776 356 839 400 250 464 677 810 668 945 312 5 + 0.000 000 000 000 000 444 089 209 850 062 616 169 452 667 236 328 125 =
0.333 333 333 333 334 369 541 489 650 146 104 395 389 556 884 765 625(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 + 0.333 333 333 333 334 369 541 489 650 146 104 395 389 556 884 765 625) × 21022 =
1.333 333 333 333 334 369 541 489 650 146 104 395 389 556 884 765 625 × 21022 =
59 923 104 495 410 576 827 250 728 836 429 763 102 629 203 490 976 760 665 006 103 879 920 193 661 269 311 245 846 969 383 297 380 322 725 899 454 790 156 537 210 022 954 096 361 233 521 168 872 730 343 458 706 597 924 454 354 969 225 566 442 059 081 150 115 697 861 352 498 036 184 857 707 600 139 047 471 543 548 323 000 584 819 794 406 434 015 892 441 246 136 909 080 467 543 756 643 340 817 901 639 696 384
0 - 111 1111 1101 - 0101 0101 0101 0101 0101 0101 0101 0101 0101 0101 0101 0101 1010 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) = 59 923 104 495 410 576 827 250 728 836 429 763 102 629 203 490 976 760 665 006 103 879 920 193 661 269 311 245 846 969 383 297 380 322 725 899 454 790 156 537 210 022 954 096 361 233 521 168 872 730 343 458 706 597 924 454 354 969 225 566 442 059 081 150 115 697 861 352 498 036 184 857 707 600 139 047 471 543 548 323 000 584 819 794 406 434 015 892 441 246 136 909 080 467 543 756 643 340 817 901 639 696 384(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: