0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 336 55 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 336 55(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
0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 336 55(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: 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 336 55.

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) 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 336 55 × 2 = 0 + 0.000 000 000 007 269 999 999 999 999 516 522 114 064 601 356 673 1;
  • 2) 0.000 000 000 007 269 999 999 999 999 516 522 114 064 601 356 673 1 × 2 = 0 + 0.000 000 000 014 539 999 999 999 999 033 044 228 129 202 713 346 2;
  • 3) 0.000 000 000 014 539 999 999 999 999 033 044 228 129 202 713 346 2 × 2 = 0 + 0.000 000 000 029 079 999 999 999 998 066 088 456 258 405 426 692 4;
  • 4) 0.000 000 000 029 079 999 999 999 998 066 088 456 258 405 426 692 4 × 2 = 0 + 0.000 000 000 058 159 999 999 999 996 132 176 912 516 810 853 384 8;
  • 5) 0.000 000 000 058 159 999 999 999 996 132 176 912 516 810 853 384 8 × 2 = 0 + 0.000 000 000 116 319 999 999 999 992 264 353 825 033 621 706 769 6;
  • 6) 0.000 000 000 116 319 999 999 999 992 264 353 825 033 621 706 769 6 × 2 = 0 + 0.000 000 000 232 639 999 999 999 984 528 707 650 067 243 413 539 2;
  • 7) 0.000 000 000 232 639 999 999 999 984 528 707 650 067 243 413 539 2 × 2 = 0 + 0.000 000 000 465 279 999 999 999 969 057 415 300 134 486 827 078 4;
  • 8) 0.000 000 000 465 279 999 999 999 969 057 415 300 134 486 827 078 4 × 2 = 0 + 0.000 000 000 930 559 999 999 999 938 114 830 600 268 973 654 156 8;
  • 9) 0.000 000 000 930 559 999 999 999 938 114 830 600 268 973 654 156 8 × 2 = 0 + 0.000 000 001 861 119 999 999 999 876 229 661 200 537 947 308 313 6;
  • 10) 0.000 000 001 861 119 999 999 999 876 229 661 200 537 947 308 313 6 × 2 = 0 + 0.000 000 003 722 239 999 999 999 752 459 322 401 075 894 616 627 2;
  • 11) 0.000 000 003 722 239 999 999 999 752 459 322 401 075 894 616 627 2 × 2 = 0 + 0.000 000 007 444 479 999 999 999 504 918 644 802 151 789 233 254 4;
  • 12) 0.000 000 007 444 479 999 999 999 504 918 644 802 151 789 233 254 4 × 2 = 0 + 0.000 000 014 888 959 999 999 999 009 837 289 604 303 578 466 508 8;
  • 13) 0.000 000 014 888 959 999 999 999 009 837 289 604 303 578 466 508 8 × 2 = 0 + 0.000 000 029 777 919 999 999 998 019 674 579 208 607 156 933 017 6;
  • 14) 0.000 000 029 777 919 999 999 998 019 674 579 208 607 156 933 017 6 × 2 = 0 + 0.000 000 059 555 839 999 999 996 039 349 158 417 214 313 866 035 2;
  • 15) 0.000 000 059 555 839 999 999 996 039 349 158 417 214 313 866 035 2 × 2 = 0 + 0.000 000 119 111 679 999 999 992 078 698 316 834 428 627 732 070 4;
  • 16) 0.000 000 119 111 679 999 999 992 078 698 316 834 428 627 732 070 4 × 2 = 0 + 0.000 000 238 223 359 999 999 984 157 396 633 668 857 255 464 140 8;
  • 17) 0.000 000 238 223 359 999 999 984 157 396 633 668 857 255 464 140 8 × 2 = 0 + 0.000 000 476 446 719 999 999 968 314 793 267 337 714 510 928 281 6;
  • 18) 0.000 000 476 446 719 999 999 968 314 793 267 337 714 510 928 281 6 × 2 = 0 + 0.000 000 952 893 439 999 999 936 629 586 534 675 429 021 856 563 2;
  • 19) 0.000 000 952 893 439 999 999 936 629 586 534 675 429 021 856 563 2 × 2 = 0 + 0.000 001 905 786 879 999 999 873 259 173 069 350 858 043 713 126 4;
  • 20) 0.000 001 905 786 879 999 999 873 259 173 069 350 858 043 713 126 4 × 2 = 0 + 0.000 003 811 573 759 999 999 746 518 346 138 701 716 087 426 252 8;
  • 21) 0.000 003 811 573 759 999 999 746 518 346 138 701 716 087 426 252 8 × 2 = 0 + 0.000 007 623 147 519 999 999 493 036 692 277 403 432 174 852 505 6;
  • 22) 0.000 007 623 147 519 999 999 493 036 692 277 403 432 174 852 505 6 × 2 = 0 + 0.000 015 246 295 039 999 998 986 073 384 554 806 864 349 705 011 2;
  • 23) 0.000 015 246 295 039 999 998 986 073 384 554 806 864 349 705 011 2 × 2 = 0 + 0.000 030 492 590 079 999 997 972 146 769 109 613 728 699 410 022 4;
  • 24) 0.000 030 492 590 079 999 997 972 146 769 109 613 728 699 410 022 4 × 2 = 0 + 0.000 060 985 180 159 999 995 944 293 538 219 227 457 398 820 044 8;
  • 25) 0.000 060 985 180 159 999 995 944 293 538 219 227 457 398 820 044 8 × 2 = 0 + 0.000 121 970 360 319 999 991 888 587 076 438 454 914 797 640 089 6;
  • 26) 0.000 121 970 360 319 999 991 888 587 076 438 454 914 797 640 089 6 × 2 = 0 + 0.000 243 940 720 639 999 983 777 174 152 876 909 829 595 280 179 2;
  • 27) 0.000 243 940 720 639 999 983 777 174 152 876 909 829 595 280 179 2 × 2 = 0 + 0.000 487 881 441 279 999 967 554 348 305 753 819 659 190 560 358 4;
  • 28) 0.000 487 881 441 279 999 967 554 348 305 753 819 659 190 560 358 4 × 2 = 0 + 0.000 975 762 882 559 999 935 108 696 611 507 639 318 381 120 716 8;
  • 29) 0.000 975 762 882 559 999 935 108 696 611 507 639 318 381 120 716 8 × 2 = 0 + 0.001 951 525 765 119 999 870 217 393 223 015 278 636 762 241 433 6;
  • 30) 0.001 951 525 765 119 999 870 217 393 223 015 278 636 762 241 433 6 × 2 = 0 + 0.003 903 051 530 239 999 740 434 786 446 030 557 273 524 482 867 2;
  • 31) 0.003 903 051 530 239 999 740 434 786 446 030 557 273 524 482 867 2 × 2 = 0 + 0.007 806 103 060 479 999 480 869 572 892 061 114 547 048 965 734 4;
  • 32) 0.007 806 103 060 479 999 480 869 572 892 061 114 547 048 965 734 4 × 2 = 0 + 0.015 612 206 120 959 998 961 739 145 784 122 229 094 097 931 468 8;
  • 33) 0.015 612 206 120 959 998 961 739 145 784 122 229 094 097 931 468 8 × 2 = 0 + 0.031 224 412 241 919 997 923 478 291 568 244 458 188 195 862 937 6;
  • 34) 0.031 224 412 241 919 997 923 478 291 568 244 458 188 195 862 937 6 × 2 = 0 + 0.062 448 824 483 839 995 846 956 583 136 488 916 376 391 725 875 2;
  • 35) 0.062 448 824 483 839 995 846 956 583 136 488 916 376 391 725 875 2 × 2 = 0 + 0.124 897 648 967 679 991 693 913 166 272 977 832 752 783 451 750 4;
  • 36) 0.124 897 648 967 679 991 693 913 166 272 977 832 752 783 451 750 4 × 2 = 0 + 0.249 795 297 935 359 983 387 826 332 545 955 665 505 566 903 500 8;
  • 37) 0.249 795 297 935 359 983 387 826 332 545 955 665 505 566 903 500 8 × 2 = 0 + 0.499 590 595 870 719 966 775 652 665 091 911 331 011 133 807 001 6;
  • 38) 0.499 590 595 870 719 966 775 652 665 091 911 331 011 133 807 001 6 × 2 = 0 + 0.999 181 191 741 439 933 551 305 330 183 822 662 022 267 614 003 2;
  • 39) 0.999 181 191 741 439 933 551 305 330 183 822 662 022 267 614 003 2 × 2 = 1 + 0.998 362 383 482 879 867 102 610 660 367 645 324 044 535 228 006 4;
  • 40) 0.998 362 383 482 879 867 102 610 660 367 645 324 044 535 228 006 4 × 2 = 1 + 0.996 724 766 965 759 734 205 221 320 735 290 648 089 070 456 012 8;
  • 41) 0.996 724 766 965 759 734 205 221 320 735 290 648 089 070 456 012 8 × 2 = 1 + 0.993 449 533 931 519 468 410 442 641 470 581 296 178 140 912 025 6;
  • 42) 0.993 449 533 931 519 468 410 442 641 470 581 296 178 140 912 025 6 × 2 = 1 + 0.986 899 067 863 038 936 820 885 282 941 162 592 356 281 824 051 2;
  • 43) 0.986 899 067 863 038 936 820 885 282 941 162 592 356 281 824 051 2 × 2 = 1 + 0.973 798 135 726 077 873 641 770 565 882 325 184 712 563 648 102 4;
  • 44) 0.973 798 135 726 077 873 641 770 565 882 325 184 712 563 648 102 4 × 2 = 1 + 0.947 596 271 452 155 747 283 541 131 764 650 369 425 127 296 204 8;
  • 45) 0.947 596 271 452 155 747 283 541 131 764 650 369 425 127 296 204 8 × 2 = 1 + 0.895 192 542 904 311 494 567 082 263 529 300 738 850 254 592 409 6;
  • 46) 0.895 192 542 904 311 494 567 082 263 529 300 738 850 254 592 409 6 × 2 = 1 + 0.790 385 085 808 622 989 134 164 527 058 601 477 700 509 184 819 2;
  • 47) 0.790 385 085 808 622 989 134 164 527 058 601 477 700 509 184 819 2 × 2 = 1 + 0.580 770 171 617 245 978 268 329 054 117 202 955 401 018 369 638 4;
  • 48) 0.580 770 171 617 245 978 268 329 054 117 202 955 401 018 369 638 4 × 2 = 1 + 0.161 540 343 234 491 956 536 658 108 234 405 910 802 036 739 276 8;
  • 49) 0.161 540 343 234 491 956 536 658 108 234 405 910 802 036 739 276 8 × 2 = 0 + 0.323 080 686 468 983 913 073 316 216 468 811 821 604 073 478 553 6;
  • 50) 0.323 080 686 468 983 913 073 316 216 468 811 821 604 073 478 553 6 × 2 = 0 + 0.646 161 372 937 967 826 146 632 432 937 623 643 208 146 957 107 2;
  • 51) 0.646 161 372 937 967 826 146 632 432 937 623 643 208 146 957 107 2 × 2 = 1 + 0.292 322 745 875 935 652 293 264 865 875 247 286 416 293 914 214 4;
  • 52) 0.292 322 745 875 935 652 293 264 865 875 247 286 416 293 914 214 4 × 2 = 0 + 0.584 645 491 751 871 304 586 529 731 750 494 572 832 587 828 428 8;
  • 53) 0.584 645 491 751 871 304 586 529 731 750 494 572 832 587 828 428 8 × 2 = 1 + 0.169 290 983 503 742 609 173 059 463 500 989 145 665 175 656 857 6;
  • 54) 0.169 290 983 503 742 609 173 059 463 500 989 145 665 175 656 857 6 × 2 = 0 + 0.338 581 967 007 485 218 346 118 927 001 978 291 330 351 313 715 2;
  • 55) 0.338 581 967 007 485 218 346 118 927 001 978 291 330 351 313 715 2 × 2 = 0 + 0.677 163 934 014 970 436 692 237 854 003 956 582 660 702 627 430 4;
  • 56) 0.677 163 934 014 970 436 692 237 854 003 956 582 660 702 627 430 4 × 2 = 1 + 0.354 327 868 029 940 873 384 475 708 007 913 165 321 405 254 860 8;
  • 57) 0.354 327 868 029 940 873 384 475 708 007 913 165 321 405 254 860 8 × 2 = 0 + 0.708 655 736 059 881 746 768 951 416 015 826 330 642 810 509 721 6;
  • 58) 0.708 655 736 059 881 746 768 951 416 015 826 330 642 810 509 721 6 × 2 = 1 + 0.417 311 472 119 763 493 537 902 832 031 652 661 285 621 019 443 2;
  • 59) 0.417 311 472 119 763 493 537 902 832 031 652 661 285 621 019 443 2 × 2 = 0 + 0.834 622 944 239 526 987 075 805 664 063 305 322 571 242 038 886 4;
  • 60) 0.834 622 944 239 526 987 075 805 664 063 305 322 571 242 038 886 4 × 2 = 1 + 0.669 245 888 479 053 974 151 611 328 126 610 645 142 484 077 772 8;
  • 61) 0.669 245 888 479 053 974 151 611 328 126 610 645 142 484 077 772 8 × 2 = 1 + 0.338 491 776 958 107 948 303 222 656 253 221 290 284 968 155 545 6;
  • 62) 0.338 491 776 958 107 948 303 222 656 253 221 290 284 968 155 545 6 × 2 = 0 + 0.676 983 553 916 215 896 606 445 312 506 442 580 569 936 311 091 2;
  • 63) 0.676 983 553 916 215 896 606 445 312 506 442 580 569 936 311 091 2 × 2 = 1 + 0.353 967 107 832 431 793 212 890 625 012 885 161 139 872 622 182 4;
  • 64) 0.353 967 107 832 431 793 212 890 625 012 885 161 139 872 622 182 4 × 2 = 0 + 0.707 934 215 664 863 586 425 781 250 025 770 322 279 745 244 364 8;
  • 65) 0.707 934 215 664 863 586 425 781 250 025 770 322 279 745 244 364 8 × 2 = 1 + 0.415 868 431 329 727 172 851 562 500 051 540 644 559 490 488 729 6;
  • 66) 0.415 868 431 329 727 172 851 562 500 051 540 644 559 490 488 729 6 × 2 = 0 + 0.831 736 862 659 454 345 703 125 000 103 081 289 118 980 977 459 2;
  • 67) 0.831 736 862 659 454 345 703 125 000 103 081 289 118 980 977 459 2 × 2 = 1 + 0.663 473 725 318 908 691 406 250 000 206 162 578 237 961 954 918 4;
  • 68) 0.663 473 725 318 908 691 406 250 000 206 162 578 237 961 954 918 4 × 2 = 1 + 0.326 947 450 637 817 382 812 500 000 412 325 156 475 923 909 836 8;
  • 69) 0.326 947 450 637 817 382 812 500 000 412 325 156 475 923 909 836 8 × 2 = 0 + 0.653 894 901 275 634 765 625 000 000 824 650 312 951 847 819 673 6;
  • 70) 0.653 894 901 275 634 765 625 000 000 824 650 312 951 847 819 673 6 × 2 = 1 + 0.307 789 802 551 269 531 250 000 001 649 300 625 903 695 639 347 2;
  • 71) 0.307 789 802 551 269 531 250 000 001 649 300 625 903 695 639 347 2 × 2 = 0 + 0.615 579 605 102 539 062 500 000 003 298 601 251 807 391 278 694 4;
  • 72) 0.615 579 605 102 539 062 500 000 003 298 601 251 807 391 278 694 4 × 2 = 1 + 0.231 159 210 205 078 125 000 000 006 597 202 503 614 782 557 388 8;
  • 73) 0.231 159 210 205 078 125 000 000 006 597 202 503 614 782 557 388 8 × 2 = 0 + 0.462 318 420 410 156 250 000 000 013 194 405 007 229 565 114 777 6;
  • 74) 0.462 318 420 410 156 250 000 000 013 194 405 007 229 565 114 777 6 × 2 = 0 + 0.924 636 840 820 312 500 000 000 026 388 810 014 459 130 229 555 2;
  • 75) 0.924 636 840 820 312 500 000 000 026 388 810 014 459 130 229 555 2 × 2 = 1 + 0.849 273 681 640 625 000 000 000 052 777 620 028 918 260 459 110 4;
  • 76) 0.849 273 681 640 625 000 000 000 052 777 620 028 918 260 459 110 4 × 2 = 1 + 0.698 547 363 281 250 000 000 000 105 555 240 057 836 520 918 220 8;
  • 77) 0.698 547 363 281 250 000 000 000 105 555 240 057 836 520 918 220 8 × 2 = 1 + 0.397 094 726 562 500 000 000 000 211 110 480 115 673 041 836 441 6;
  • 78) 0.397 094 726 562 500 000 000 000 211 110 480 115 673 041 836 441 6 × 2 = 0 + 0.794 189 453 125 000 000 000 000 422 220 960 231 346 083 672 883 2;
  • 79) 0.794 189 453 125 000 000 000 000 422 220 960 231 346 083 672 883 2 × 2 = 1 + 0.588 378 906 250 000 000 000 000 844 441 920 462 692 167 345 766 4;
  • 80) 0.588 378 906 250 000 000 000 000 844 441 920 462 692 167 345 766 4 × 2 = 1 + 0.176 757 812 500 000 000 000 001 688 883 840 925 384 334 691 532 8;
  • 81) 0.176 757 812 500 000 000 000 001 688 883 840 925 384 334 691 532 8 × 2 = 0 + 0.353 515 625 000 000 000 000 003 377 767 681 850 768 669 383 065 6;
  • 82) 0.353 515 625 000 000 000 000 003 377 767 681 850 768 669 383 065 6 × 2 = 0 + 0.707 031 250 000 000 000 000 006 755 535 363 701 537 338 766 131 2;
  • 83) 0.707 031 250 000 000 000 000 006 755 535 363 701 537 338 766 131 2 × 2 = 1 + 0.414 062 500 000 000 000 000 013 511 070 727 403 074 677 532 262 4;
  • 84) 0.414 062 500 000 000 000 000 013 511 070 727 403 074 677 532 262 4 × 2 = 0 + 0.828 125 000 000 000 000 000 027 022 141 454 806 149 355 064 524 8;
  • 85) 0.828 125 000 000 000 000 000 027 022 141 454 806 149 355 064 524 8 × 2 = 1 + 0.656 250 000 000 000 000 000 054 044 282 909 612 298 710 129 049 6;
  • 86) 0.656 250 000 000 000 000 000 054 044 282 909 612 298 710 129 049 6 × 2 = 1 + 0.312 500 000 000 000 000 000 108 088 565 819 224 597 420 258 099 2;
  • 87) 0.312 500 000 000 000 000 000 108 088 565 819 224 597 420 258 099 2 × 2 = 0 + 0.625 000 000 000 000 000 000 216 177 131 638 449 194 840 516 198 4;
  • 88) 0.625 000 000 000 000 000 000 216 177 131 638 449 194 840 516 198 4 × 2 = 1 + 0.250 000 000 000 000 000 000 432 354 263 276 898 389 681 032 396 8;
  • 89) 0.250 000 000 000 000 000 000 432 354 263 276 898 389 681 032 396 8 × 2 = 0 + 0.500 000 000 000 000 000 000 864 708 526 553 796 779 362 064 793 6;
  • 90) 0.500 000 000 000 000 000 000 864 708 526 553 796 779 362 064 793 6 × 2 = 1 + 0.000 000 000 000 000 000 001 729 417 053 107 593 558 724 129 587 2;
  • 91) 0.000 000 000 000 000 000 001 729 417 053 107 593 558 724 129 587 2 × 2 = 0 + 0.000 000 000 000 000 000 003 458 834 106 215 187 117 448 259 174 4;

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:


0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 336 55(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0011 1111 1111 0010 1001 0101 1010 1011 0101 0011 1011 0010 1101 010(2)

5. Positive number before normalization:

0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 336 55(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0011 1111 1111 0010 1001 0101 1010 1011 0101 0011 1011 0010 1101 010(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:


0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 336 55(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0011 1111 1111 0010 1001 0101 1010 1011 0101 0011 1011 0010 1101 010(2) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0011 1111 1111 0010 1001 0101 1010 1011 0101 0011 1011 0010 1101 010(2) × 20 =

1.1111 1111 1001 0100 1010 1101 0101 1010 1001 1101 1001 0110 1010(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 1010


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 1010 =

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


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 1010


Decimal number 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 336 55 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 1010

Share

» Convert decimal 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 336 86 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: My Manager Only Says “We” When Referring to “My Work”. Gemini Notebook YouTube Video of 11.13 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