0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 151 98 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 151 98(10) to 64 bit double precision IEEE 754 binary floating point representation standard (1 bit for sign, 11 bits for exponent, 52 bits for mantissa)

What are the steps to convert decimal number
0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 151 98(10) to 64 bit double precision IEEE 754 binary floating point representation (1 bit for sign, 11 bits for exponent, 52 bits for mantissa)

1. First, convert to binary (in base 2) the integer part: 0.
Divide the number repeatedly by 2.

Keep track of each remainder.

We stop when we get a quotient that is equal to zero.


  • division = quotient + remainder;
  • 0 ÷ 2 = 0 + 0;

2. Construct the base 2 representation of the integer part of the number.

Take all the remainders starting from the bottom of the list constructed above.

0(10) =

0(2)


3. Convert to binary (base 2) the fractional part: 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 151 98.

Multiply it repeatedly by 2.

Keep track of each integer part of the results.

Stop when we get a fractional part that is equal to zero.


  • #) multiplying = integer + fractional part;
  • 1) 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 151 98 × 2 = 0 + 0.000 000 000 007 269 999 999 999 999 516 522 114 064 601 356 670 303 96;
  • 2) 0.000 000 000 007 269 999 999 999 999 516 522 114 064 601 356 670 303 96 × 2 = 0 + 0.000 000 000 014 539 999 999 999 999 033 044 228 129 202 713 340 607 92;
  • 3) 0.000 000 000 014 539 999 999 999 999 033 044 228 129 202 713 340 607 92 × 2 = 0 + 0.000 000 000 029 079 999 999 999 998 066 088 456 258 405 426 681 215 84;
  • 4) 0.000 000 000 029 079 999 999 999 998 066 088 456 258 405 426 681 215 84 × 2 = 0 + 0.000 000 000 058 159 999 999 999 996 132 176 912 516 810 853 362 431 68;
  • 5) 0.000 000 000 058 159 999 999 999 996 132 176 912 516 810 853 362 431 68 × 2 = 0 + 0.000 000 000 116 319 999 999 999 992 264 353 825 033 621 706 724 863 36;
  • 6) 0.000 000 000 116 319 999 999 999 992 264 353 825 033 621 706 724 863 36 × 2 = 0 + 0.000 000 000 232 639 999 999 999 984 528 707 650 067 243 413 449 726 72;
  • 7) 0.000 000 000 232 639 999 999 999 984 528 707 650 067 243 413 449 726 72 × 2 = 0 + 0.000 000 000 465 279 999 999 999 969 057 415 300 134 486 826 899 453 44;
  • 8) 0.000 000 000 465 279 999 999 999 969 057 415 300 134 486 826 899 453 44 × 2 = 0 + 0.000 000 000 930 559 999 999 999 938 114 830 600 268 973 653 798 906 88;
  • 9) 0.000 000 000 930 559 999 999 999 938 114 830 600 268 973 653 798 906 88 × 2 = 0 + 0.000 000 001 861 119 999 999 999 876 229 661 200 537 947 307 597 813 76;
  • 10) 0.000 000 001 861 119 999 999 999 876 229 661 200 537 947 307 597 813 76 × 2 = 0 + 0.000 000 003 722 239 999 999 999 752 459 322 401 075 894 615 195 627 52;
  • 11) 0.000 000 003 722 239 999 999 999 752 459 322 401 075 894 615 195 627 52 × 2 = 0 + 0.000 000 007 444 479 999 999 999 504 918 644 802 151 789 230 391 255 04;
  • 12) 0.000 000 007 444 479 999 999 999 504 918 644 802 151 789 230 391 255 04 × 2 = 0 + 0.000 000 014 888 959 999 999 999 009 837 289 604 303 578 460 782 510 08;
  • 13) 0.000 000 014 888 959 999 999 999 009 837 289 604 303 578 460 782 510 08 × 2 = 0 + 0.000 000 029 777 919 999 999 998 019 674 579 208 607 156 921 565 020 16;
  • 14) 0.000 000 029 777 919 999 999 998 019 674 579 208 607 156 921 565 020 16 × 2 = 0 + 0.000 000 059 555 839 999 999 996 039 349 158 417 214 313 843 130 040 32;
  • 15) 0.000 000 059 555 839 999 999 996 039 349 158 417 214 313 843 130 040 32 × 2 = 0 + 0.000 000 119 111 679 999 999 992 078 698 316 834 428 627 686 260 080 64;
  • 16) 0.000 000 119 111 679 999 999 992 078 698 316 834 428 627 686 260 080 64 × 2 = 0 + 0.000 000 238 223 359 999 999 984 157 396 633 668 857 255 372 520 161 28;
  • 17) 0.000 000 238 223 359 999 999 984 157 396 633 668 857 255 372 520 161 28 × 2 = 0 + 0.000 000 476 446 719 999 999 968 314 793 267 337 714 510 745 040 322 56;
  • 18) 0.000 000 476 446 719 999 999 968 314 793 267 337 714 510 745 040 322 56 × 2 = 0 + 0.000 000 952 893 439 999 999 936 629 586 534 675 429 021 490 080 645 12;
  • 19) 0.000 000 952 893 439 999 999 936 629 586 534 675 429 021 490 080 645 12 × 2 = 0 + 0.000 001 905 786 879 999 999 873 259 173 069 350 858 042 980 161 290 24;
  • 20) 0.000 001 905 786 879 999 999 873 259 173 069 350 858 042 980 161 290 24 × 2 = 0 + 0.000 003 811 573 759 999 999 746 518 346 138 701 716 085 960 322 580 48;
  • 21) 0.000 003 811 573 759 999 999 746 518 346 138 701 716 085 960 322 580 48 × 2 = 0 + 0.000 007 623 147 519 999 999 493 036 692 277 403 432 171 920 645 160 96;
  • 22) 0.000 007 623 147 519 999 999 493 036 692 277 403 432 171 920 645 160 96 × 2 = 0 + 0.000 015 246 295 039 999 998 986 073 384 554 806 864 343 841 290 321 92;
  • 23) 0.000 015 246 295 039 999 998 986 073 384 554 806 864 343 841 290 321 92 × 2 = 0 + 0.000 030 492 590 079 999 997 972 146 769 109 613 728 687 682 580 643 84;
  • 24) 0.000 030 492 590 079 999 997 972 146 769 109 613 728 687 682 580 643 84 × 2 = 0 + 0.000 060 985 180 159 999 995 944 293 538 219 227 457 375 365 161 287 68;
  • 25) 0.000 060 985 180 159 999 995 944 293 538 219 227 457 375 365 161 287 68 × 2 = 0 + 0.000 121 970 360 319 999 991 888 587 076 438 454 914 750 730 322 575 36;
  • 26) 0.000 121 970 360 319 999 991 888 587 076 438 454 914 750 730 322 575 36 × 2 = 0 + 0.000 243 940 720 639 999 983 777 174 152 876 909 829 501 460 645 150 72;
  • 27) 0.000 243 940 720 639 999 983 777 174 152 876 909 829 501 460 645 150 72 × 2 = 0 + 0.000 487 881 441 279 999 967 554 348 305 753 819 659 002 921 290 301 44;
  • 28) 0.000 487 881 441 279 999 967 554 348 305 753 819 659 002 921 290 301 44 × 2 = 0 + 0.000 975 762 882 559 999 935 108 696 611 507 639 318 005 842 580 602 88;
  • 29) 0.000 975 762 882 559 999 935 108 696 611 507 639 318 005 842 580 602 88 × 2 = 0 + 0.001 951 525 765 119 999 870 217 393 223 015 278 636 011 685 161 205 76;
  • 30) 0.001 951 525 765 119 999 870 217 393 223 015 278 636 011 685 161 205 76 × 2 = 0 + 0.003 903 051 530 239 999 740 434 786 446 030 557 272 023 370 322 411 52;
  • 31) 0.003 903 051 530 239 999 740 434 786 446 030 557 272 023 370 322 411 52 × 2 = 0 + 0.007 806 103 060 479 999 480 869 572 892 061 114 544 046 740 644 823 04;
  • 32) 0.007 806 103 060 479 999 480 869 572 892 061 114 544 046 740 644 823 04 × 2 = 0 + 0.015 612 206 120 959 998 961 739 145 784 122 229 088 093 481 289 646 08;
  • 33) 0.015 612 206 120 959 998 961 739 145 784 122 229 088 093 481 289 646 08 × 2 = 0 + 0.031 224 412 241 919 997 923 478 291 568 244 458 176 186 962 579 292 16;
  • 34) 0.031 224 412 241 919 997 923 478 291 568 244 458 176 186 962 579 292 16 × 2 = 0 + 0.062 448 824 483 839 995 846 956 583 136 488 916 352 373 925 158 584 32;
  • 35) 0.062 448 824 483 839 995 846 956 583 136 488 916 352 373 925 158 584 32 × 2 = 0 + 0.124 897 648 967 679 991 693 913 166 272 977 832 704 747 850 317 168 64;
  • 36) 0.124 897 648 967 679 991 693 913 166 272 977 832 704 747 850 317 168 64 × 2 = 0 + 0.249 795 297 935 359 983 387 826 332 545 955 665 409 495 700 634 337 28;
  • 37) 0.249 795 297 935 359 983 387 826 332 545 955 665 409 495 700 634 337 28 × 2 = 0 + 0.499 590 595 870 719 966 775 652 665 091 911 330 818 991 401 268 674 56;
  • 38) 0.499 590 595 870 719 966 775 652 665 091 911 330 818 991 401 268 674 56 × 2 = 0 + 0.999 181 191 741 439 933 551 305 330 183 822 661 637 982 802 537 349 12;
  • 39) 0.999 181 191 741 439 933 551 305 330 183 822 661 637 982 802 537 349 12 × 2 = 1 + 0.998 362 383 482 879 867 102 610 660 367 645 323 275 965 605 074 698 24;
  • 40) 0.998 362 383 482 879 867 102 610 660 367 645 323 275 965 605 074 698 24 × 2 = 1 + 0.996 724 766 965 759 734 205 221 320 735 290 646 551 931 210 149 396 48;
  • 41) 0.996 724 766 965 759 734 205 221 320 735 290 646 551 931 210 149 396 48 × 2 = 1 + 0.993 449 533 931 519 468 410 442 641 470 581 293 103 862 420 298 792 96;
  • 42) 0.993 449 533 931 519 468 410 442 641 470 581 293 103 862 420 298 792 96 × 2 = 1 + 0.986 899 067 863 038 936 820 885 282 941 162 586 207 724 840 597 585 92;
  • 43) 0.986 899 067 863 038 936 820 885 282 941 162 586 207 724 840 597 585 92 × 2 = 1 + 0.973 798 135 726 077 873 641 770 565 882 325 172 415 449 681 195 171 84;
  • 44) 0.973 798 135 726 077 873 641 770 565 882 325 172 415 449 681 195 171 84 × 2 = 1 + 0.947 596 271 452 155 747 283 541 131 764 650 344 830 899 362 390 343 68;
  • 45) 0.947 596 271 452 155 747 283 541 131 764 650 344 830 899 362 390 343 68 × 2 = 1 + 0.895 192 542 904 311 494 567 082 263 529 300 689 661 798 724 780 687 36;
  • 46) 0.895 192 542 904 311 494 567 082 263 529 300 689 661 798 724 780 687 36 × 2 = 1 + 0.790 385 085 808 622 989 134 164 527 058 601 379 323 597 449 561 374 72;
  • 47) 0.790 385 085 808 622 989 134 164 527 058 601 379 323 597 449 561 374 72 × 2 = 1 + 0.580 770 171 617 245 978 268 329 054 117 202 758 647 194 899 122 749 44;
  • 48) 0.580 770 171 617 245 978 268 329 054 117 202 758 647 194 899 122 749 44 × 2 = 1 + 0.161 540 343 234 491 956 536 658 108 234 405 517 294 389 798 245 498 88;
  • 49) 0.161 540 343 234 491 956 536 658 108 234 405 517 294 389 798 245 498 88 × 2 = 0 + 0.323 080 686 468 983 913 073 316 216 468 811 034 588 779 596 490 997 76;
  • 50) 0.323 080 686 468 983 913 073 316 216 468 811 034 588 779 596 490 997 76 × 2 = 0 + 0.646 161 372 937 967 826 146 632 432 937 622 069 177 559 192 981 995 52;
  • 51) 0.646 161 372 937 967 826 146 632 432 937 622 069 177 559 192 981 995 52 × 2 = 1 + 0.292 322 745 875 935 652 293 264 865 875 244 138 355 118 385 963 991 04;
  • 52) 0.292 322 745 875 935 652 293 264 865 875 244 138 355 118 385 963 991 04 × 2 = 0 + 0.584 645 491 751 871 304 586 529 731 750 488 276 710 236 771 927 982 08;
  • 53) 0.584 645 491 751 871 304 586 529 731 750 488 276 710 236 771 927 982 08 × 2 = 1 + 0.169 290 983 503 742 609 173 059 463 500 976 553 420 473 543 855 964 16;
  • 54) 0.169 290 983 503 742 609 173 059 463 500 976 553 420 473 543 855 964 16 × 2 = 0 + 0.338 581 967 007 485 218 346 118 927 001 953 106 840 947 087 711 928 32;
  • 55) 0.338 581 967 007 485 218 346 118 927 001 953 106 840 947 087 711 928 32 × 2 = 0 + 0.677 163 934 014 970 436 692 237 854 003 906 213 681 894 175 423 856 64;
  • 56) 0.677 163 934 014 970 436 692 237 854 003 906 213 681 894 175 423 856 64 × 2 = 1 + 0.354 327 868 029 940 873 384 475 708 007 812 427 363 788 350 847 713 28;
  • 57) 0.354 327 868 029 940 873 384 475 708 007 812 427 363 788 350 847 713 28 × 2 = 0 + 0.708 655 736 059 881 746 768 951 416 015 624 854 727 576 701 695 426 56;
  • 58) 0.708 655 736 059 881 746 768 951 416 015 624 854 727 576 701 695 426 56 × 2 = 1 + 0.417 311 472 119 763 493 537 902 832 031 249 709 455 153 403 390 853 12;
  • 59) 0.417 311 472 119 763 493 537 902 832 031 249 709 455 153 403 390 853 12 × 2 = 0 + 0.834 622 944 239 526 987 075 805 664 062 499 418 910 306 806 781 706 24;
  • 60) 0.834 622 944 239 526 987 075 805 664 062 499 418 910 306 806 781 706 24 × 2 = 1 + 0.669 245 888 479 053 974 151 611 328 124 998 837 820 613 613 563 412 48;
  • 61) 0.669 245 888 479 053 974 151 611 328 124 998 837 820 613 613 563 412 48 × 2 = 1 + 0.338 491 776 958 107 948 303 222 656 249 997 675 641 227 227 126 824 96;
  • 62) 0.338 491 776 958 107 948 303 222 656 249 997 675 641 227 227 126 824 96 × 2 = 0 + 0.676 983 553 916 215 896 606 445 312 499 995 351 282 454 454 253 649 92;
  • 63) 0.676 983 553 916 215 896 606 445 312 499 995 351 282 454 454 253 649 92 × 2 = 1 + 0.353 967 107 832 431 793 212 890 624 999 990 702 564 908 908 507 299 84;
  • 64) 0.353 967 107 832 431 793 212 890 624 999 990 702 564 908 908 507 299 84 × 2 = 0 + 0.707 934 215 664 863 586 425 781 249 999 981 405 129 817 817 014 599 68;
  • 65) 0.707 934 215 664 863 586 425 781 249 999 981 405 129 817 817 014 599 68 × 2 = 1 + 0.415 868 431 329 727 172 851 562 499 999 962 810 259 635 634 029 199 36;
  • 66) 0.415 868 431 329 727 172 851 562 499 999 962 810 259 635 634 029 199 36 × 2 = 0 + 0.831 736 862 659 454 345 703 124 999 999 925 620 519 271 268 058 398 72;
  • 67) 0.831 736 862 659 454 345 703 124 999 999 925 620 519 271 268 058 398 72 × 2 = 1 + 0.663 473 725 318 908 691 406 249 999 999 851 241 038 542 536 116 797 44;
  • 68) 0.663 473 725 318 908 691 406 249 999 999 851 241 038 542 536 116 797 44 × 2 = 1 + 0.326 947 450 637 817 382 812 499 999 999 702 482 077 085 072 233 594 88;
  • 69) 0.326 947 450 637 817 382 812 499 999 999 702 482 077 085 072 233 594 88 × 2 = 0 + 0.653 894 901 275 634 765 624 999 999 999 404 964 154 170 144 467 189 76;
  • 70) 0.653 894 901 275 634 765 624 999 999 999 404 964 154 170 144 467 189 76 × 2 = 1 + 0.307 789 802 551 269 531 249 999 999 998 809 928 308 340 288 934 379 52;
  • 71) 0.307 789 802 551 269 531 249 999 999 998 809 928 308 340 288 934 379 52 × 2 = 0 + 0.615 579 605 102 539 062 499 999 999 997 619 856 616 680 577 868 759 04;
  • 72) 0.615 579 605 102 539 062 499 999 999 997 619 856 616 680 577 868 759 04 × 2 = 1 + 0.231 159 210 205 078 124 999 999 999 995 239 713 233 361 155 737 518 08;
  • 73) 0.231 159 210 205 078 124 999 999 999 995 239 713 233 361 155 737 518 08 × 2 = 0 + 0.462 318 420 410 156 249 999 999 999 990 479 426 466 722 311 475 036 16;
  • 74) 0.462 318 420 410 156 249 999 999 999 990 479 426 466 722 311 475 036 16 × 2 = 0 + 0.924 636 840 820 312 499 999 999 999 980 958 852 933 444 622 950 072 32;
  • 75) 0.924 636 840 820 312 499 999 999 999 980 958 852 933 444 622 950 072 32 × 2 = 1 + 0.849 273 681 640 624 999 999 999 999 961 917 705 866 889 245 900 144 64;
  • 76) 0.849 273 681 640 624 999 999 999 999 961 917 705 866 889 245 900 144 64 × 2 = 1 + 0.698 547 363 281 249 999 999 999 999 923 835 411 733 778 491 800 289 28;
  • 77) 0.698 547 363 281 249 999 999 999 999 923 835 411 733 778 491 800 289 28 × 2 = 1 + 0.397 094 726 562 499 999 999 999 999 847 670 823 467 556 983 600 578 56;
  • 78) 0.397 094 726 562 499 999 999 999 999 847 670 823 467 556 983 600 578 56 × 2 = 0 + 0.794 189 453 124 999 999 999 999 999 695 341 646 935 113 967 201 157 12;
  • 79) 0.794 189 453 124 999 999 999 999 999 695 341 646 935 113 967 201 157 12 × 2 = 1 + 0.588 378 906 249 999 999 999 999 999 390 683 293 870 227 934 402 314 24;
  • 80) 0.588 378 906 249 999 999 999 999 999 390 683 293 870 227 934 402 314 24 × 2 = 1 + 0.176 757 812 499 999 999 999 999 998 781 366 587 740 455 868 804 628 48;
  • 81) 0.176 757 812 499 999 999 999 999 998 781 366 587 740 455 868 804 628 48 × 2 = 0 + 0.353 515 624 999 999 999 999 999 997 562 733 175 480 911 737 609 256 96;
  • 82) 0.353 515 624 999 999 999 999 999 997 562 733 175 480 911 737 609 256 96 × 2 = 0 + 0.707 031 249 999 999 999 999 999 995 125 466 350 961 823 475 218 513 92;
  • 83) 0.707 031 249 999 999 999 999 999 995 125 466 350 961 823 475 218 513 92 × 2 = 1 + 0.414 062 499 999 999 999 999 999 990 250 932 701 923 646 950 437 027 84;
  • 84) 0.414 062 499 999 999 999 999 999 990 250 932 701 923 646 950 437 027 84 × 2 = 0 + 0.828 124 999 999 999 999 999 999 980 501 865 403 847 293 900 874 055 68;
  • 85) 0.828 124 999 999 999 999 999 999 980 501 865 403 847 293 900 874 055 68 × 2 = 1 + 0.656 249 999 999 999 999 999 999 961 003 730 807 694 587 801 748 111 36;
  • 86) 0.656 249 999 999 999 999 999 999 961 003 730 807 694 587 801 748 111 36 × 2 = 1 + 0.312 499 999 999 999 999 999 999 922 007 461 615 389 175 603 496 222 72;
  • 87) 0.312 499 999 999 999 999 999 999 922 007 461 615 389 175 603 496 222 72 × 2 = 0 + 0.624 999 999 999 999 999 999 999 844 014 923 230 778 351 206 992 445 44;
  • 88) 0.624 999 999 999 999 999 999 999 844 014 923 230 778 351 206 992 445 44 × 2 = 1 + 0.249 999 999 999 999 999 999 999 688 029 846 461 556 702 413 984 890 88;
  • 89) 0.249 999 999 999 999 999 999 999 688 029 846 461 556 702 413 984 890 88 × 2 = 0 + 0.499 999 999 999 999 999 999 999 376 059 692 923 113 404 827 969 781 76;
  • 90) 0.499 999 999 999 999 999 999 999 376 059 692 923 113 404 827 969 781 76 × 2 = 0 + 0.999 999 999 999 999 999 999 998 752 119 385 846 226 809 655 939 563 52;
  • 91) 0.999 999 999 999 999 999 999 998 752 119 385 846 226 809 655 939 563 52 × 2 = 1 + 0.999 999 999 999 999 999 999 997 504 238 771 692 453 619 311 879 127 04;

We didn't get any fractional part that was equal to zero. But we had enough iterations (over Mantissa limit) and at least one integer that was different from zero => FULL STOP (Losing precision - the converted number we get in the end will be just a very good approximation of the initial one).


4. Construct the base 2 representation of the fractional part of the number.

Take all the integer parts of the multiplying operations, starting from the top of the constructed list above:


0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 151 98(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0011 1111 1111 0010 1001 0101 1010 1011 0101 0011 1011 0010 1101 001(2)

5. Positive number before normalization:

0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 151 98(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0011 1111 1111 0010 1001 0101 1010 1011 0101 0011 1011 0010 1101 001(2)

6. Normalize the binary representation of the number.

Shift the decimal mark 39 positions to the right, so that only one non zero digit remains to the left of it:


0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 151 98(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0011 1111 1111 0010 1001 0101 1010 1011 0101 0011 1011 0010 1101 001(2) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0011 1111 1111 0010 1001 0101 1010 1011 0101 0011 1011 0010 1101 001(2) × 20 =

1.1111 1111 1001 0100 1010 1101 0101 1010 1001 1101 1001 0110 1001(2) × 2-39


7. Up to this moment, there are the following elements that would feed into the 64 bit double precision IEEE 754 binary floating point representation:

Sign 0 (a positive number)

Exponent (unadjusted): -39

Mantissa (not normalized):
1.1111 1111 1001 0100 1010 1101 0101 1010 1001 1101 1001 0110 1001


8. Adjust the exponent.

Use the 11 bit excess/bias notation:


Exponent (adjusted) =

Exponent (unadjusted) + 2(11-1) - 1 =

-39 + 2(11-1) - 1 =

(-39 + 1 023)(10) =

984(10)


9. Convert the adjusted exponent from the decimal (base 10) to 11 bit binary.

Use the same technique of repeatedly dividing by 2:


  • division = quotient + remainder;
  • 984 ÷ 2 = 492 + 0;
  • 492 ÷ 2 = 246 + 0;
  • 246 ÷ 2 = 123 + 0;
  • 123 ÷ 2 = 61 + 1;
  • 61 ÷ 2 = 30 + 1;
  • 30 ÷ 2 = 15 + 0;
  • 15 ÷ 2 = 7 + 1;
  • 7 ÷ 2 = 3 + 1;
  • 3 ÷ 2 = 1 + 1;
  • 1 ÷ 2 = 0 + 1;

10. Construct the base 2 representation of the adjusted exponent.

Take all the remainders starting from the bottom of the list constructed above.


Exponent (adjusted) =

984(10) =

011 1101 1000(2)


11. Normalize the mantissa.

a) Remove the leading (the leftmost) bit, since it's allways 1, and the decimal point, if the case.

b) Adjust its length to 52 bits, only if necessary (not the case here).


Mantissa (normalized) =

1. 1111 1111 1001 0100 1010 1101 0101 1010 1001 1101 1001 0110 1001 =

1111 1111 1001 0100 1010 1101 0101 1010 1001 1101 1001 0110 1001


12. The three elements that make up the number's 64 bit double precision IEEE 754 binary floating point representation:

Sign (1 bit) =
0 (a positive number)

Exponent (11 bits) =
011 1101 1000

Mantissa (52 bits) =
1111 1111 1001 0100 1010 1101 0101 1010 1001 1101 1001 0110 1001


Decimal number 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 151 98 converted to 64 bit double precision IEEE 754 binary floating point representation:

0 - 011 1101 1000 - 1111 1111 1001 0100 1010 1101 0101 1010 1001 1101 1001 0110 1001

Share

» Convert decimal 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 03 to 64 bit double precision IEEE 754 binary floating point representation standard

» Calculations Performed by Our Visitors: Decimal Numbers Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard. Data organized on a Monthly Basis

» Month 07, 2026 [July]: Decimal Numbers Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard. Calculations performed during the month of: July - by our visitors

» Improve your social skills: The Art of Strategic Ambiguity. Gemini Notebook YouTube Video of 6.59 minutes

How to convert numbers from the decimal system (base ten) to 64 bit double precision IEEE 754 binary floating point standard

Follow the steps below to convert a base 10 decimal number to 64 bit double precision IEEE 754 binary floating point:

  • 1. If the number to be converted is negative, start with its the positive version.
  • 2. First convert the integer part. Divide repeatedly by 2 the positive representation of the integer number that is to be converted to binary, until we get a quotient that is equal to zero, keeping track of each remainder.
  • 3. Construct the base 2 representation of the positive integer part of the number, by taking all the remainders from the previous operations, starting from the bottom of the list constructed above. Thus, the last remainder of the divisions becomes the first symbol (the leftmost) of the base two number, while the first remainder becomes the last symbol (the rightmost).
  • 4. Then convert the fractional part. Multiply the number repeatedly by 2, until we get a fractional part that is equal to zero, keeping track of each integer part of the results.
  • 5. Construct the base 2 representation of the fractional part of the number, by taking all the integer parts of the multiplying operations, starting from the top of the list constructed above (they should appear in the binary representation, from left to right, in the order they have been calculated).
  • 6. Normalize the binary representation of the number, shifting the decimal mark (the decimal point) "n" positions either to the left, or to the right, so that only one non zero digit remains to the left of the decimal mark.
  • 7. Adjust the exponent in 11 bit excess/bias notation and then convert it from decimal (base 10) to 11 bit binary, by using the same technique of repeatedly dividing by 2, as shown above:
    Exponent (adjusted) = Exponent (unadjusted) + 2(11-1) - 1
  • 8. Normalize mantissa, remove the leading (leftmost) bit, since it's allways '1' (and the decimal mark, if the case) and adjust its length to 52 bits, either by removing the excess bits from the right (losing precision...) or by adding extra bits set on '0' to the right.
  • 9. Sign (it takes 1 bit) is either 1 for a negative or 0 for a positive number.

Example: convert the negative number -31.640 215 from the decimal system (base ten) to 64 bit double precision IEEE 754 binary floating point:

  • 1. Start with the positive version of the number:

    |-31.640 215| = 31.640 215

  • 2. First convert the integer part, 31. Divide it repeatedly by 2, keeping track of each remainder, until we get a quotient that is equal to zero:
    • division = quotient + remainder;
    • 31 ÷ 2 = 15 + 1;
    • 15 ÷ 2 = 7 + 1;
    • 7 ÷ 2 = 3 + 1;
    • 3 ÷ 2 = 1 + 1;
    • 1 ÷ 2 = 0 + 1;
    • We have encountered a quotient that is ZERO => FULL STOP
  • 3. Construct the base 2 representation of the integer part of the number by taking all the remainders of the previous dividing operations, starting from the bottom of the list constructed above:

    31(10) = 1 1111(2)

  • 4. Then, convert the fractional part, 0.640 215. Multiply repeatedly by 2, keeping track of each integer part of the results, until we get a fractional part that is equal to zero:
    • #) multiplying = integer + fractional part;
    • 1) 0.640 215 × 2 = 1 + 0.280 43;
    • 2) 0.280 43 × 2 = 0 + 0.560 86;
    • 3) 0.560 86 × 2 = 1 + 0.121 72;
    • 4) 0.121 72 × 2 = 0 + 0.243 44;
    • 5) 0.243 44 × 2 = 0 + 0.486 88;
    • 6) 0.486 88 × 2 = 0 + 0.973 76;
    • 7) 0.973 76 × 2 = 1 + 0.947 52;
    • 8) 0.947 52 × 2 = 1 + 0.895 04;
    • 9) 0.895 04 × 2 = 1 + 0.790 08;
    • 10) 0.790 08 × 2 = 1 + 0.580 16;
    • 11) 0.580 16 × 2 = 1 + 0.160 32;
    • 12) 0.160 32 × 2 = 0 + 0.320 64;
    • 13) 0.320 64 × 2 = 0 + 0.641 28;
    • 14) 0.641 28 × 2 = 1 + 0.282 56;
    • 15) 0.282 56 × 2 = 0 + 0.565 12;
    • 16) 0.565 12 × 2 = 1 + 0.130 24;
    • 17) 0.130 24 × 2 = 0 + 0.260 48;
    • 18) 0.260 48 × 2 = 0 + 0.520 96;
    • 19) 0.520 96 × 2 = 1 + 0.041 92;
    • 20) 0.041 92 × 2 = 0 + 0.083 84;
    • 21) 0.083 84 × 2 = 0 + 0.167 68;
    • 22) 0.167 68 × 2 = 0 + 0.335 36;
    • 23) 0.335 36 × 2 = 0 + 0.670 72;
    • 24) 0.670 72 × 2 = 1 + 0.341 44;
    • 25) 0.341 44 × 2 = 0 + 0.682 88;
    • 26) 0.682 88 × 2 = 1 + 0.365 76;
    • 27) 0.365 76 × 2 = 0 + 0.731 52;
    • 28) 0.731 52 × 2 = 1 + 0.463 04;
    • 29) 0.463 04 × 2 = 0 + 0.926 08;
    • 30) 0.926 08 × 2 = 1 + 0.852 16;
    • 31) 0.852 16 × 2 = 1 + 0.704 32;
    • 32) 0.704 32 × 2 = 1 + 0.408 64;
    • 33) 0.408 64 × 2 = 0 + 0.817 28;
    • 34) 0.817 28 × 2 = 1 + 0.634 56;
    • 35) 0.634 56 × 2 = 1 + 0.269 12;
    • 36) 0.269 12 × 2 = 0 + 0.538 24;
    • 37) 0.538 24 × 2 = 1 + 0.076 48;
    • 38) 0.076 48 × 2 = 0 + 0.152 96;
    • 39) 0.152 96 × 2 = 0 + 0.305 92;
    • 40) 0.305 92 × 2 = 0 + 0.611 84;
    • 41) 0.611 84 × 2 = 1 + 0.223 68;
    • 42) 0.223 68 × 2 = 0 + 0.447 36;
    • 43) 0.447 36 × 2 = 0 + 0.894 72;
    • 44) 0.894 72 × 2 = 1 + 0.789 44;
    • 45) 0.789 44 × 2 = 1 + 0.578 88;
    • 46) 0.578 88 × 2 = 1 + 0.157 76;
    • 47) 0.157 76 × 2 = 0 + 0.315 52;
    • 48) 0.315 52 × 2 = 0 + 0.631 04;
    • 49) 0.631 04 × 2 = 1 + 0.262 08;
    • 50) 0.262 08 × 2 = 0 + 0.524 16;
    • 51) 0.524 16 × 2 = 1 + 0.048 32;
    • 52) 0.048 32 × 2 = 0 + 0.096 64;
    • 53) 0.096 64 × 2 = 0 + 0.193 28;
    • We didn't get any fractional part that was equal to zero. But we had enough iterations (over Mantissa limit = 52) and at least one integer part that was different from zero => FULL STOP (losing precision...).
  • 5. Construct the base 2 representation of the fractional part of the number, by taking all the integer parts of the previous multiplying operations, starting from the top of the constructed list above:

    0.640 215(10) = 0.1010 0011 1110 0101 0010 0001 0101 0111 0110 1000 1001 1100 1010 0(2)

  • 6. Summarizing - the positive number before normalization:

    31.640 215(10) = 1 1111.1010 0011 1110 0101 0010 0001 0101 0111 0110 1000 1001 1100 1010 0(2)

  • 7. Normalize the binary representation of the number, shifting the decimal mark 4 positions to the left so that only one non-zero digit stays to the left of the decimal mark:

    31.640 215(10) =
    1 1111.1010 0011 1110 0101 0010 0001 0101 0111 0110 1000 1001 1100 1010 0(2) =
    1 1111.1010 0011 1110 0101 0010 0001 0101 0111 0110 1000 1001 1100 1010 0(2) × 20 =
    1.1111 1010 0011 1110 0101 0010 0001 0101 0111 0110 1000 1001 1100 1010 0(2) × 24

  • 8. Up to this moment, there are the following elements that would feed into the 64 bit double precision IEEE 754 binary floating point representation:

    Sign: 1 (a negative number)

    Exponent (unadjusted): 4

    Mantissa (not-normalized): 1.1111 1010 0011 1110 0101 0010 0001 0101 0111 0110 1000 1001 1100 1010 0

  • 9. Adjust the exponent in 11 bit excess/bias notation and then convert it from decimal (base 10) to 11 bit binary (base 2), by using the same technique of repeatedly dividing it by 2, as shown above:

    Exponent (adjusted) = Exponent (unadjusted) + 2(11-1) - 1 = (4 + 1023)(10) = 1027(10) =
    100 0000 0011(2)

  • 10. Normalize mantissa, remove the leading (leftmost) bit, since it's allways '1' (and the decimal sign) and adjust its length to 52 bits, by removing the excess bits, from the right (losing precision...):

    Mantissa (not-normalized): 1.1111 1010 0011 1110 0101 0010 0001 0101 0111 0110 1000 1001 1100 1010 0

    Mantissa (normalized): 1111 1010 0011 1110 0101 0010 0001 0101 0111 0110 1000 1001 1100

  • Conclusion:

    Sign (1 bit) = 1 (a negative number)

    Exponent (8 bits) = 100 0000 0011

    Mantissa (52 bits) = 1111 1010 0011 1110 0101 0010 0001 0101 0111 0110 1000 1001 1100

  • Number -31.640 215, converted from decimal system (base 10) to 64 bit double precision IEEE 754 binary floating point =
    1 - 100 0000 0011 - 1111 1010 0011 1110 0101 0010 0001 0101 0111 0110 1000 1001 1100