0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 336 09 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 09(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 09(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 09.

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 09 × 2 = 0 + 0.000 000 000 007 269 999 999 999 999 516 522 114 064 601 356 672 18;
  • 2) 0.000 000 000 007 269 999 999 999 999 516 522 114 064 601 356 672 18 × 2 = 0 + 0.000 000 000 014 539 999 999 999 999 033 044 228 129 202 713 344 36;
  • 3) 0.000 000 000 014 539 999 999 999 999 033 044 228 129 202 713 344 36 × 2 = 0 + 0.000 000 000 029 079 999 999 999 998 066 088 456 258 405 426 688 72;
  • 4) 0.000 000 000 029 079 999 999 999 998 066 088 456 258 405 426 688 72 × 2 = 0 + 0.000 000 000 058 159 999 999 999 996 132 176 912 516 810 853 377 44;
  • 5) 0.000 000 000 058 159 999 999 999 996 132 176 912 516 810 853 377 44 × 2 = 0 + 0.000 000 000 116 319 999 999 999 992 264 353 825 033 621 706 754 88;
  • 6) 0.000 000 000 116 319 999 999 999 992 264 353 825 033 621 706 754 88 × 2 = 0 + 0.000 000 000 232 639 999 999 999 984 528 707 650 067 243 413 509 76;
  • 7) 0.000 000 000 232 639 999 999 999 984 528 707 650 067 243 413 509 76 × 2 = 0 + 0.000 000 000 465 279 999 999 999 969 057 415 300 134 486 827 019 52;
  • 8) 0.000 000 000 465 279 999 999 999 969 057 415 300 134 486 827 019 52 × 2 = 0 + 0.000 000 000 930 559 999 999 999 938 114 830 600 268 973 654 039 04;
  • 9) 0.000 000 000 930 559 999 999 999 938 114 830 600 268 973 654 039 04 × 2 = 0 + 0.000 000 001 861 119 999 999 999 876 229 661 200 537 947 308 078 08;
  • 10) 0.000 000 001 861 119 999 999 999 876 229 661 200 537 947 308 078 08 × 2 = 0 + 0.000 000 003 722 239 999 999 999 752 459 322 401 075 894 616 156 16;
  • 11) 0.000 000 003 722 239 999 999 999 752 459 322 401 075 894 616 156 16 × 2 = 0 + 0.000 000 007 444 479 999 999 999 504 918 644 802 151 789 232 312 32;
  • 12) 0.000 000 007 444 479 999 999 999 504 918 644 802 151 789 232 312 32 × 2 = 0 + 0.000 000 014 888 959 999 999 999 009 837 289 604 303 578 464 624 64;
  • 13) 0.000 000 014 888 959 999 999 999 009 837 289 604 303 578 464 624 64 × 2 = 0 + 0.000 000 029 777 919 999 999 998 019 674 579 208 607 156 929 249 28;
  • 14) 0.000 000 029 777 919 999 999 998 019 674 579 208 607 156 929 249 28 × 2 = 0 + 0.000 000 059 555 839 999 999 996 039 349 158 417 214 313 858 498 56;
  • 15) 0.000 000 059 555 839 999 999 996 039 349 158 417 214 313 858 498 56 × 2 = 0 + 0.000 000 119 111 679 999 999 992 078 698 316 834 428 627 716 997 12;
  • 16) 0.000 000 119 111 679 999 999 992 078 698 316 834 428 627 716 997 12 × 2 = 0 + 0.000 000 238 223 359 999 999 984 157 396 633 668 857 255 433 994 24;
  • 17) 0.000 000 238 223 359 999 999 984 157 396 633 668 857 255 433 994 24 × 2 = 0 + 0.000 000 476 446 719 999 999 968 314 793 267 337 714 510 867 988 48;
  • 18) 0.000 000 476 446 719 999 999 968 314 793 267 337 714 510 867 988 48 × 2 = 0 + 0.000 000 952 893 439 999 999 936 629 586 534 675 429 021 735 976 96;
  • 19) 0.000 000 952 893 439 999 999 936 629 586 534 675 429 021 735 976 96 × 2 = 0 + 0.000 001 905 786 879 999 999 873 259 173 069 350 858 043 471 953 92;
  • 20) 0.000 001 905 786 879 999 999 873 259 173 069 350 858 043 471 953 92 × 2 = 0 + 0.000 003 811 573 759 999 999 746 518 346 138 701 716 086 943 907 84;
  • 21) 0.000 003 811 573 759 999 999 746 518 346 138 701 716 086 943 907 84 × 2 = 0 + 0.000 007 623 147 519 999 999 493 036 692 277 403 432 173 887 815 68;
  • 22) 0.000 007 623 147 519 999 999 493 036 692 277 403 432 173 887 815 68 × 2 = 0 + 0.000 015 246 295 039 999 998 986 073 384 554 806 864 347 775 631 36;
  • 23) 0.000 015 246 295 039 999 998 986 073 384 554 806 864 347 775 631 36 × 2 = 0 + 0.000 030 492 590 079 999 997 972 146 769 109 613 728 695 551 262 72;
  • 24) 0.000 030 492 590 079 999 997 972 146 769 109 613 728 695 551 262 72 × 2 = 0 + 0.000 060 985 180 159 999 995 944 293 538 219 227 457 391 102 525 44;
  • 25) 0.000 060 985 180 159 999 995 944 293 538 219 227 457 391 102 525 44 × 2 = 0 + 0.000 121 970 360 319 999 991 888 587 076 438 454 914 782 205 050 88;
  • 26) 0.000 121 970 360 319 999 991 888 587 076 438 454 914 782 205 050 88 × 2 = 0 + 0.000 243 940 720 639 999 983 777 174 152 876 909 829 564 410 101 76;
  • 27) 0.000 243 940 720 639 999 983 777 174 152 876 909 829 564 410 101 76 × 2 = 0 + 0.000 487 881 441 279 999 967 554 348 305 753 819 659 128 820 203 52;
  • 28) 0.000 487 881 441 279 999 967 554 348 305 753 819 659 128 820 203 52 × 2 = 0 + 0.000 975 762 882 559 999 935 108 696 611 507 639 318 257 640 407 04;
  • 29) 0.000 975 762 882 559 999 935 108 696 611 507 639 318 257 640 407 04 × 2 = 0 + 0.001 951 525 765 119 999 870 217 393 223 015 278 636 515 280 814 08;
  • 30) 0.001 951 525 765 119 999 870 217 393 223 015 278 636 515 280 814 08 × 2 = 0 + 0.003 903 051 530 239 999 740 434 786 446 030 557 273 030 561 628 16;
  • 31) 0.003 903 051 530 239 999 740 434 786 446 030 557 273 030 561 628 16 × 2 = 0 + 0.007 806 103 060 479 999 480 869 572 892 061 114 546 061 123 256 32;
  • 32) 0.007 806 103 060 479 999 480 869 572 892 061 114 546 061 123 256 32 × 2 = 0 + 0.015 612 206 120 959 998 961 739 145 784 122 229 092 122 246 512 64;
  • 33) 0.015 612 206 120 959 998 961 739 145 784 122 229 092 122 246 512 64 × 2 = 0 + 0.031 224 412 241 919 997 923 478 291 568 244 458 184 244 493 025 28;
  • 34) 0.031 224 412 241 919 997 923 478 291 568 244 458 184 244 493 025 28 × 2 = 0 + 0.062 448 824 483 839 995 846 956 583 136 488 916 368 488 986 050 56;
  • 35) 0.062 448 824 483 839 995 846 956 583 136 488 916 368 488 986 050 56 × 2 = 0 + 0.124 897 648 967 679 991 693 913 166 272 977 832 736 977 972 101 12;
  • 36) 0.124 897 648 967 679 991 693 913 166 272 977 832 736 977 972 101 12 × 2 = 0 + 0.249 795 297 935 359 983 387 826 332 545 955 665 473 955 944 202 24;
  • 37) 0.249 795 297 935 359 983 387 826 332 545 955 665 473 955 944 202 24 × 2 = 0 + 0.499 590 595 870 719 966 775 652 665 091 911 330 947 911 888 404 48;
  • 38) 0.499 590 595 870 719 966 775 652 665 091 911 330 947 911 888 404 48 × 2 = 0 + 0.999 181 191 741 439 933 551 305 330 183 822 661 895 823 776 808 96;
  • 39) 0.999 181 191 741 439 933 551 305 330 183 822 661 895 823 776 808 96 × 2 = 1 + 0.998 362 383 482 879 867 102 610 660 367 645 323 791 647 553 617 92;
  • 40) 0.998 362 383 482 879 867 102 610 660 367 645 323 791 647 553 617 92 × 2 = 1 + 0.996 724 766 965 759 734 205 221 320 735 290 647 583 295 107 235 84;
  • 41) 0.996 724 766 965 759 734 205 221 320 735 290 647 583 295 107 235 84 × 2 = 1 + 0.993 449 533 931 519 468 410 442 641 470 581 295 166 590 214 471 68;
  • 42) 0.993 449 533 931 519 468 410 442 641 470 581 295 166 590 214 471 68 × 2 = 1 + 0.986 899 067 863 038 936 820 885 282 941 162 590 333 180 428 943 36;
  • 43) 0.986 899 067 863 038 936 820 885 282 941 162 590 333 180 428 943 36 × 2 = 1 + 0.973 798 135 726 077 873 641 770 565 882 325 180 666 360 857 886 72;
  • 44) 0.973 798 135 726 077 873 641 770 565 882 325 180 666 360 857 886 72 × 2 = 1 + 0.947 596 271 452 155 747 283 541 131 764 650 361 332 721 715 773 44;
  • 45) 0.947 596 271 452 155 747 283 541 131 764 650 361 332 721 715 773 44 × 2 = 1 + 0.895 192 542 904 311 494 567 082 263 529 300 722 665 443 431 546 88;
  • 46) 0.895 192 542 904 311 494 567 082 263 529 300 722 665 443 431 546 88 × 2 = 1 + 0.790 385 085 808 622 989 134 164 527 058 601 445 330 886 863 093 76;
  • 47) 0.790 385 085 808 622 989 134 164 527 058 601 445 330 886 863 093 76 × 2 = 1 + 0.580 770 171 617 245 978 268 329 054 117 202 890 661 773 726 187 52;
  • 48) 0.580 770 171 617 245 978 268 329 054 117 202 890 661 773 726 187 52 × 2 = 1 + 0.161 540 343 234 491 956 536 658 108 234 405 781 323 547 452 375 04;
  • 49) 0.161 540 343 234 491 956 536 658 108 234 405 781 323 547 452 375 04 × 2 = 0 + 0.323 080 686 468 983 913 073 316 216 468 811 562 647 094 904 750 08;
  • 50) 0.323 080 686 468 983 913 073 316 216 468 811 562 647 094 904 750 08 × 2 = 0 + 0.646 161 372 937 967 826 146 632 432 937 623 125 294 189 809 500 16;
  • 51) 0.646 161 372 937 967 826 146 632 432 937 623 125 294 189 809 500 16 × 2 = 1 + 0.292 322 745 875 935 652 293 264 865 875 246 250 588 379 619 000 32;
  • 52) 0.292 322 745 875 935 652 293 264 865 875 246 250 588 379 619 000 32 × 2 = 0 + 0.584 645 491 751 871 304 586 529 731 750 492 501 176 759 238 000 64;
  • 53) 0.584 645 491 751 871 304 586 529 731 750 492 501 176 759 238 000 64 × 2 = 1 + 0.169 290 983 503 742 609 173 059 463 500 985 002 353 518 476 001 28;
  • 54) 0.169 290 983 503 742 609 173 059 463 500 985 002 353 518 476 001 28 × 2 = 0 + 0.338 581 967 007 485 218 346 118 927 001 970 004 707 036 952 002 56;
  • 55) 0.338 581 967 007 485 218 346 118 927 001 970 004 707 036 952 002 56 × 2 = 0 + 0.677 163 934 014 970 436 692 237 854 003 940 009 414 073 904 005 12;
  • 56) 0.677 163 934 014 970 436 692 237 854 003 940 009 414 073 904 005 12 × 2 = 1 + 0.354 327 868 029 940 873 384 475 708 007 880 018 828 147 808 010 24;
  • 57) 0.354 327 868 029 940 873 384 475 708 007 880 018 828 147 808 010 24 × 2 = 0 + 0.708 655 736 059 881 746 768 951 416 015 760 037 656 295 616 020 48;
  • 58) 0.708 655 736 059 881 746 768 951 416 015 760 037 656 295 616 020 48 × 2 = 1 + 0.417 311 472 119 763 493 537 902 832 031 520 075 312 591 232 040 96;
  • 59) 0.417 311 472 119 763 493 537 902 832 031 520 075 312 591 232 040 96 × 2 = 0 + 0.834 622 944 239 526 987 075 805 664 063 040 150 625 182 464 081 92;
  • 60) 0.834 622 944 239 526 987 075 805 664 063 040 150 625 182 464 081 92 × 2 = 1 + 0.669 245 888 479 053 974 151 611 328 126 080 301 250 364 928 163 84;
  • 61) 0.669 245 888 479 053 974 151 611 328 126 080 301 250 364 928 163 84 × 2 = 1 + 0.338 491 776 958 107 948 303 222 656 252 160 602 500 729 856 327 68;
  • 62) 0.338 491 776 958 107 948 303 222 656 252 160 602 500 729 856 327 68 × 2 = 0 + 0.676 983 553 916 215 896 606 445 312 504 321 205 001 459 712 655 36;
  • 63) 0.676 983 553 916 215 896 606 445 312 504 321 205 001 459 712 655 36 × 2 = 1 + 0.353 967 107 832 431 793 212 890 625 008 642 410 002 919 425 310 72;
  • 64) 0.353 967 107 832 431 793 212 890 625 008 642 410 002 919 425 310 72 × 2 = 0 + 0.707 934 215 664 863 586 425 781 250 017 284 820 005 838 850 621 44;
  • 65) 0.707 934 215 664 863 586 425 781 250 017 284 820 005 838 850 621 44 × 2 = 1 + 0.415 868 431 329 727 172 851 562 500 034 569 640 011 677 701 242 88;
  • 66) 0.415 868 431 329 727 172 851 562 500 034 569 640 011 677 701 242 88 × 2 = 0 + 0.831 736 862 659 454 345 703 125 000 069 139 280 023 355 402 485 76;
  • 67) 0.831 736 862 659 454 345 703 125 000 069 139 280 023 355 402 485 76 × 2 = 1 + 0.663 473 725 318 908 691 406 250 000 138 278 560 046 710 804 971 52;
  • 68) 0.663 473 725 318 908 691 406 250 000 138 278 560 046 710 804 971 52 × 2 = 1 + 0.326 947 450 637 817 382 812 500 000 276 557 120 093 421 609 943 04;
  • 69) 0.326 947 450 637 817 382 812 500 000 276 557 120 093 421 609 943 04 × 2 = 0 + 0.653 894 901 275 634 765 625 000 000 553 114 240 186 843 219 886 08;
  • 70) 0.653 894 901 275 634 765 625 000 000 553 114 240 186 843 219 886 08 × 2 = 1 + 0.307 789 802 551 269 531 250 000 001 106 228 480 373 686 439 772 16;
  • 71) 0.307 789 802 551 269 531 250 000 001 106 228 480 373 686 439 772 16 × 2 = 0 + 0.615 579 605 102 539 062 500 000 002 212 456 960 747 372 879 544 32;
  • 72) 0.615 579 605 102 539 062 500 000 002 212 456 960 747 372 879 544 32 × 2 = 1 + 0.231 159 210 205 078 125 000 000 004 424 913 921 494 745 759 088 64;
  • 73) 0.231 159 210 205 078 125 000 000 004 424 913 921 494 745 759 088 64 × 2 = 0 + 0.462 318 420 410 156 250 000 000 008 849 827 842 989 491 518 177 28;
  • 74) 0.462 318 420 410 156 250 000 000 008 849 827 842 989 491 518 177 28 × 2 = 0 + 0.924 636 840 820 312 500 000 000 017 699 655 685 978 983 036 354 56;
  • 75) 0.924 636 840 820 312 500 000 000 017 699 655 685 978 983 036 354 56 × 2 = 1 + 0.849 273 681 640 625 000 000 000 035 399 311 371 957 966 072 709 12;
  • 76) 0.849 273 681 640 625 000 000 000 035 399 311 371 957 966 072 709 12 × 2 = 1 + 0.698 547 363 281 250 000 000 000 070 798 622 743 915 932 145 418 24;
  • 77) 0.698 547 363 281 250 000 000 000 070 798 622 743 915 932 145 418 24 × 2 = 1 + 0.397 094 726 562 500 000 000 000 141 597 245 487 831 864 290 836 48;
  • 78) 0.397 094 726 562 500 000 000 000 141 597 245 487 831 864 290 836 48 × 2 = 0 + 0.794 189 453 125 000 000 000 000 283 194 490 975 663 728 581 672 96;
  • 79) 0.794 189 453 125 000 000 000 000 283 194 490 975 663 728 581 672 96 × 2 = 1 + 0.588 378 906 250 000 000 000 000 566 388 981 951 327 457 163 345 92;
  • 80) 0.588 378 906 250 000 000 000 000 566 388 981 951 327 457 163 345 92 × 2 = 1 + 0.176 757 812 500 000 000 000 001 132 777 963 902 654 914 326 691 84;
  • 81) 0.176 757 812 500 000 000 000 001 132 777 963 902 654 914 326 691 84 × 2 = 0 + 0.353 515 625 000 000 000 000 002 265 555 927 805 309 828 653 383 68;
  • 82) 0.353 515 625 000 000 000 000 002 265 555 927 805 309 828 653 383 68 × 2 = 0 + 0.707 031 250 000 000 000 000 004 531 111 855 610 619 657 306 767 36;
  • 83) 0.707 031 250 000 000 000 000 004 531 111 855 610 619 657 306 767 36 × 2 = 1 + 0.414 062 500 000 000 000 000 009 062 223 711 221 239 314 613 534 72;
  • 84) 0.414 062 500 000 000 000 000 009 062 223 711 221 239 314 613 534 72 × 2 = 0 + 0.828 125 000 000 000 000 000 018 124 447 422 442 478 629 227 069 44;
  • 85) 0.828 125 000 000 000 000 000 018 124 447 422 442 478 629 227 069 44 × 2 = 1 + 0.656 250 000 000 000 000 000 036 248 894 844 884 957 258 454 138 88;
  • 86) 0.656 250 000 000 000 000 000 036 248 894 844 884 957 258 454 138 88 × 2 = 1 + 0.312 500 000 000 000 000 000 072 497 789 689 769 914 516 908 277 76;
  • 87) 0.312 500 000 000 000 000 000 072 497 789 689 769 914 516 908 277 76 × 2 = 0 + 0.625 000 000 000 000 000 000 144 995 579 379 539 829 033 816 555 52;
  • 88) 0.625 000 000 000 000 000 000 144 995 579 379 539 829 033 816 555 52 × 2 = 1 + 0.250 000 000 000 000 000 000 289 991 158 759 079 658 067 633 111 04;
  • 89) 0.250 000 000 000 000 000 000 289 991 158 759 079 658 067 633 111 04 × 2 = 0 + 0.500 000 000 000 000 000 000 579 982 317 518 159 316 135 266 222 08;
  • 90) 0.500 000 000 000 000 000 000 579 982 317 518 159 316 135 266 222 08 × 2 = 1 + 0.000 000 000 000 000 000 001 159 964 635 036 318 632 270 532 444 16;
  • 91) 0.000 000 000 000 000 000 001 159 964 635 036 318 632 270 532 444 16 × 2 = 0 + 0.000 000 000 000 000 000 002 319 929 270 072 637 264 541 064 888 32;

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 09(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 09(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 09(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 09 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 7 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: Reputation: Bridge Between Company's Actions & Market Value. Gemini Notebook YouTube Video of 6.55 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