0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 029 919 2 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 029 919 2(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
0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 029 919 2(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: 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 029 919 2.

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) 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 029 919 2 × 2 = 0 + 0.000 000 000 007 269 999 999 999 999 516 522 114 064 601 356 670 305 976 059 838 4;
  • 2) 0.000 000 000 007 269 999 999 999 999 516 522 114 064 601 356 670 305 976 059 838 4 × 2 = 0 + 0.000 000 000 014 539 999 999 999 999 033 044 228 129 202 713 340 611 952 119 676 8;
  • 3) 0.000 000 000 014 539 999 999 999 999 033 044 228 129 202 713 340 611 952 119 676 8 × 2 = 0 + 0.000 000 000 029 079 999 999 999 998 066 088 456 258 405 426 681 223 904 239 353 6;
  • 4) 0.000 000 000 029 079 999 999 999 998 066 088 456 258 405 426 681 223 904 239 353 6 × 2 = 0 + 0.000 000 000 058 159 999 999 999 996 132 176 912 516 810 853 362 447 808 478 707 2;
  • 5) 0.000 000 000 058 159 999 999 999 996 132 176 912 516 810 853 362 447 808 478 707 2 × 2 = 0 + 0.000 000 000 116 319 999 999 999 992 264 353 825 033 621 706 724 895 616 957 414 4;
  • 6) 0.000 000 000 116 319 999 999 999 992 264 353 825 033 621 706 724 895 616 957 414 4 × 2 = 0 + 0.000 000 000 232 639 999 999 999 984 528 707 650 067 243 413 449 791 233 914 828 8;
  • 7) 0.000 000 000 232 639 999 999 999 984 528 707 650 067 243 413 449 791 233 914 828 8 × 2 = 0 + 0.000 000 000 465 279 999 999 999 969 057 415 300 134 486 826 899 582 467 829 657 6;
  • 8) 0.000 000 000 465 279 999 999 999 969 057 415 300 134 486 826 899 582 467 829 657 6 × 2 = 0 + 0.000 000 000 930 559 999 999 999 938 114 830 600 268 973 653 799 164 935 659 315 2;
  • 9) 0.000 000 000 930 559 999 999 999 938 114 830 600 268 973 653 799 164 935 659 315 2 × 2 = 0 + 0.000 000 001 861 119 999 999 999 876 229 661 200 537 947 307 598 329 871 318 630 4;
  • 10) 0.000 000 001 861 119 999 999 999 876 229 661 200 537 947 307 598 329 871 318 630 4 × 2 = 0 + 0.000 000 003 722 239 999 999 999 752 459 322 401 075 894 615 196 659 742 637 260 8;
  • 11) 0.000 000 003 722 239 999 999 999 752 459 322 401 075 894 615 196 659 742 637 260 8 × 2 = 0 + 0.000 000 007 444 479 999 999 999 504 918 644 802 151 789 230 393 319 485 274 521 6;
  • 12) 0.000 000 007 444 479 999 999 999 504 918 644 802 151 789 230 393 319 485 274 521 6 × 2 = 0 + 0.000 000 014 888 959 999 999 999 009 837 289 604 303 578 460 786 638 970 549 043 2;
  • 13) 0.000 000 014 888 959 999 999 999 009 837 289 604 303 578 460 786 638 970 549 043 2 × 2 = 0 + 0.000 000 029 777 919 999 999 998 019 674 579 208 607 156 921 573 277 941 098 086 4;
  • 14) 0.000 000 029 777 919 999 999 998 019 674 579 208 607 156 921 573 277 941 098 086 4 × 2 = 0 + 0.000 000 059 555 839 999 999 996 039 349 158 417 214 313 843 146 555 882 196 172 8;
  • 15) 0.000 000 059 555 839 999 999 996 039 349 158 417 214 313 843 146 555 882 196 172 8 × 2 = 0 + 0.000 000 119 111 679 999 999 992 078 698 316 834 428 627 686 293 111 764 392 345 6;
  • 16) 0.000 000 119 111 679 999 999 992 078 698 316 834 428 627 686 293 111 764 392 345 6 × 2 = 0 + 0.000 000 238 223 359 999 999 984 157 396 633 668 857 255 372 586 223 528 784 691 2;
  • 17) 0.000 000 238 223 359 999 999 984 157 396 633 668 857 255 372 586 223 528 784 691 2 × 2 = 0 + 0.000 000 476 446 719 999 999 968 314 793 267 337 714 510 745 172 447 057 569 382 4;
  • 18) 0.000 000 476 446 719 999 999 968 314 793 267 337 714 510 745 172 447 057 569 382 4 × 2 = 0 + 0.000 000 952 893 439 999 999 936 629 586 534 675 429 021 490 344 894 115 138 764 8;
  • 19) 0.000 000 952 893 439 999 999 936 629 586 534 675 429 021 490 344 894 115 138 764 8 × 2 = 0 + 0.000 001 905 786 879 999 999 873 259 173 069 350 858 042 980 689 788 230 277 529 6;
  • 20) 0.000 001 905 786 879 999 999 873 259 173 069 350 858 042 980 689 788 230 277 529 6 × 2 = 0 + 0.000 003 811 573 759 999 999 746 518 346 138 701 716 085 961 379 576 460 555 059 2;
  • 21) 0.000 003 811 573 759 999 999 746 518 346 138 701 716 085 961 379 576 460 555 059 2 × 2 = 0 + 0.000 007 623 147 519 999 999 493 036 692 277 403 432 171 922 759 152 921 110 118 4;
  • 22) 0.000 007 623 147 519 999 999 493 036 692 277 403 432 171 922 759 152 921 110 118 4 × 2 = 0 + 0.000 015 246 295 039 999 998 986 073 384 554 806 864 343 845 518 305 842 220 236 8;
  • 23) 0.000 015 246 295 039 999 998 986 073 384 554 806 864 343 845 518 305 842 220 236 8 × 2 = 0 + 0.000 030 492 590 079 999 997 972 146 769 109 613 728 687 691 036 611 684 440 473 6;
  • 24) 0.000 030 492 590 079 999 997 972 146 769 109 613 728 687 691 036 611 684 440 473 6 × 2 = 0 + 0.000 060 985 180 159 999 995 944 293 538 219 227 457 375 382 073 223 368 880 947 2;
  • 25) 0.000 060 985 180 159 999 995 944 293 538 219 227 457 375 382 073 223 368 880 947 2 × 2 = 0 + 0.000 121 970 360 319 999 991 888 587 076 438 454 914 750 764 146 446 737 761 894 4;
  • 26) 0.000 121 970 360 319 999 991 888 587 076 438 454 914 750 764 146 446 737 761 894 4 × 2 = 0 + 0.000 243 940 720 639 999 983 777 174 152 876 909 829 501 528 292 893 475 523 788 8;
  • 27) 0.000 243 940 720 639 999 983 777 174 152 876 909 829 501 528 292 893 475 523 788 8 × 2 = 0 + 0.000 487 881 441 279 999 967 554 348 305 753 819 659 003 056 585 786 951 047 577 6;
  • 28) 0.000 487 881 441 279 999 967 554 348 305 753 819 659 003 056 585 786 951 047 577 6 × 2 = 0 + 0.000 975 762 882 559 999 935 108 696 611 507 639 318 006 113 171 573 902 095 155 2;
  • 29) 0.000 975 762 882 559 999 935 108 696 611 507 639 318 006 113 171 573 902 095 155 2 × 2 = 0 + 0.001 951 525 765 119 999 870 217 393 223 015 278 636 012 226 343 147 804 190 310 4;
  • 30) 0.001 951 525 765 119 999 870 217 393 223 015 278 636 012 226 343 147 804 190 310 4 × 2 = 0 + 0.003 903 051 530 239 999 740 434 786 446 030 557 272 024 452 686 295 608 380 620 8;
  • 31) 0.003 903 051 530 239 999 740 434 786 446 030 557 272 024 452 686 295 608 380 620 8 × 2 = 0 + 0.007 806 103 060 479 999 480 869 572 892 061 114 544 048 905 372 591 216 761 241 6;
  • 32) 0.007 806 103 060 479 999 480 869 572 892 061 114 544 048 905 372 591 216 761 241 6 × 2 = 0 + 0.015 612 206 120 959 998 961 739 145 784 122 229 088 097 810 745 182 433 522 483 2;
  • 33) 0.015 612 206 120 959 998 961 739 145 784 122 229 088 097 810 745 182 433 522 483 2 × 2 = 0 + 0.031 224 412 241 919 997 923 478 291 568 244 458 176 195 621 490 364 867 044 966 4;
  • 34) 0.031 224 412 241 919 997 923 478 291 568 244 458 176 195 621 490 364 867 044 966 4 × 2 = 0 + 0.062 448 824 483 839 995 846 956 583 136 488 916 352 391 242 980 729 734 089 932 8;
  • 35) 0.062 448 824 483 839 995 846 956 583 136 488 916 352 391 242 980 729 734 089 932 8 × 2 = 0 + 0.124 897 648 967 679 991 693 913 166 272 977 832 704 782 485 961 459 468 179 865 6;
  • 36) 0.124 897 648 967 679 991 693 913 166 272 977 832 704 782 485 961 459 468 179 865 6 × 2 = 0 + 0.249 795 297 935 359 983 387 826 332 545 955 665 409 564 971 922 918 936 359 731 2;
  • 37) 0.249 795 297 935 359 983 387 826 332 545 955 665 409 564 971 922 918 936 359 731 2 × 2 = 0 + 0.499 590 595 870 719 966 775 652 665 091 911 330 819 129 943 845 837 872 719 462 4;
  • 38) 0.499 590 595 870 719 966 775 652 665 091 911 330 819 129 943 845 837 872 719 462 4 × 2 = 0 + 0.999 181 191 741 439 933 551 305 330 183 822 661 638 259 887 691 675 745 438 924 8;
  • 39) 0.999 181 191 741 439 933 551 305 330 183 822 661 638 259 887 691 675 745 438 924 8 × 2 = 1 + 0.998 362 383 482 879 867 102 610 660 367 645 323 276 519 775 383 351 490 877 849 6;
  • 40) 0.998 362 383 482 879 867 102 610 660 367 645 323 276 519 775 383 351 490 877 849 6 × 2 = 1 + 0.996 724 766 965 759 734 205 221 320 735 290 646 553 039 550 766 702 981 755 699 2;
  • 41) 0.996 724 766 965 759 734 205 221 320 735 290 646 553 039 550 766 702 981 755 699 2 × 2 = 1 + 0.993 449 533 931 519 468 410 442 641 470 581 293 106 079 101 533 405 963 511 398 4;
  • 42) 0.993 449 533 931 519 468 410 442 641 470 581 293 106 079 101 533 405 963 511 398 4 × 2 = 1 + 0.986 899 067 863 038 936 820 885 282 941 162 586 212 158 203 066 811 927 022 796 8;
  • 43) 0.986 899 067 863 038 936 820 885 282 941 162 586 212 158 203 066 811 927 022 796 8 × 2 = 1 + 0.973 798 135 726 077 873 641 770 565 882 325 172 424 316 406 133 623 854 045 593 6;
  • 44) 0.973 798 135 726 077 873 641 770 565 882 325 172 424 316 406 133 623 854 045 593 6 × 2 = 1 + 0.947 596 271 452 155 747 283 541 131 764 650 344 848 632 812 267 247 708 091 187 2;
  • 45) 0.947 596 271 452 155 747 283 541 131 764 650 344 848 632 812 267 247 708 091 187 2 × 2 = 1 + 0.895 192 542 904 311 494 567 082 263 529 300 689 697 265 624 534 495 416 182 374 4;
  • 46) 0.895 192 542 904 311 494 567 082 263 529 300 689 697 265 624 534 495 416 182 374 4 × 2 = 1 + 0.790 385 085 808 622 989 134 164 527 058 601 379 394 531 249 068 990 832 364 748 8;
  • 47) 0.790 385 085 808 622 989 134 164 527 058 601 379 394 531 249 068 990 832 364 748 8 × 2 = 1 + 0.580 770 171 617 245 978 268 329 054 117 202 758 789 062 498 137 981 664 729 497 6;
  • 48) 0.580 770 171 617 245 978 268 329 054 117 202 758 789 062 498 137 981 664 729 497 6 × 2 = 1 + 0.161 540 343 234 491 956 536 658 108 234 405 517 578 124 996 275 963 329 458 995 2;
  • 49) 0.161 540 343 234 491 956 536 658 108 234 405 517 578 124 996 275 963 329 458 995 2 × 2 = 0 + 0.323 080 686 468 983 913 073 316 216 468 811 035 156 249 992 551 926 658 917 990 4;
  • 50) 0.323 080 686 468 983 913 073 316 216 468 811 035 156 249 992 551 926 658 917 990 4 × 2 = 0 + 0.646 161 372 937 967 826 146 632 432 937 622 070 312 499 985 103 853 317 835 980 8;
  • 51) 0.646 161 372 937 967 826 146 632 432 937 622 070 312 499 985 103 853 317 835 980 8 × 2 = 1 + 0.292 322 745 875 935 652 293 264 865 875 244 140 624 999 970 207 706 635 671 961 6;
  • 52) 0.292 322 745 875 935 652 293 264 865 875 244 140 624 999 970 207 706 635 671 961 6 × 2 = 0 + 0.584 645 491 751 871 304 586 529 731 750 488 281 249 999 940 415 413 271 343 923 2;
  • 53) 0.584 645 491 751 871 304 586 529 731 750 488 281 249 999 940 415 413 271 343 923 2 × 2 = 1 + 0.169 290 983 503 742 609 173 059 463 500 976 562 499 999 880 830 826 542 687 846 4;
  • 54) 0.169 290 983 503 742 609 173 059 463 500 976 562 499 999 880 830 826 542 687 846 4 × 2 = 0 + 0.338 581 967 007 485 218 346 118 927 001 953 124 999 999 761 661 653 085 375 692 8;
  • 55) 0.338 581 967 007 485 218 346 118 927 001 953 124 999 999 761 661 653 085 375 692 8 × 2 = 0 + 0.677 163 934 014 970 436 692 237 854 003 906 249 999 999 523 323 306 170 751 385 6;
  • 56) 0.677 163 934 014 970 436 692 237 854 003 906 249 999 999 523 323 306 170 751 385 6 × 2 = 1 + 0.354 327 868 029 940 873 384 475 708 007 812 499 999 999 046 646 612 341 502 771 2;
  • 57) 0.354 327 868 029 940 873 384 475 708 007 812 499 999 999 046 646 612 341 502 771 2 × 2 = 0 + 0.708 655 736 059 881 746 768 951 416 015 624 999 999 998 093 293 224 683 005 542 4;
  • 58) 0.708 655 736 059 881 746 768 951 416 015 624 999 999 998 093 293 224 683 005 542 4 × 2 = 1 + 0.417 311 472 119 763 493 537 902 832 031 249 999 999 996 186 586 449 366 011 084 8;
  • 59) 0.417 311 472 119 763 493 537 902 832 031 249 999 999 996 186 586 449 366 011 084 8 × 2 = 0 + 0.834 622 944 239 526 987 075 805 664 062 499 999 999 992 373 172 898 732 022 169 6;
  • 60) 0.834 622 944 239 526 987 075 805 664 062 499 999 999 992 373 172 898 732 022 169 6 × 2 = 1 + 0.669 245 888 479 053 974 151 611 328 124 999 999 999 984 746 345 797 464 044 339 2;
  • 61) 0.669 245 888 479 053 974 151 611 328 124 999 999 999 984 746 345 797 464 044 339 2 × 2 = 1 + 0.338 491 776 958 107 948 303 222 656 249 999 999 999 969 492 691 594 928 088 678 4;
  • 62) 0.338 491 776 958 107 948 303 222 656 249 999 999 999 969 492 691 594 928 088 678 4 × 2 = 0 + 0.676 983 553 916 215 896 606 445 312 499 999 999 999 938 985 383 189 856 177 356 8;
  • 63) 0.676 983 553 916 215 896 606 445 312 499 999 999 999 938 985 383 189 856 177 356 8 × 2 = 1 + 0.353 967 107 832 431 793 212 890 624 999 999 999 999 877 970 766 379 712 354 713 6;
  • 64) 0.353 967 107 832 431 793 212 890 624 999 999 999 999 877 970 766 379 712 354 713 6 × 2 = 0 + 0.707 934 215 664 863 586 425 781 249 999 999 999 999 755 941 532 759 424 709 427 2;
  • 65) 0.707 934 215 664 863 586 425 781 249 999 999 999 999 755 941 532 759 424 709 427 2 × 2 = 1 + 0.415 868 431 329 727 172 851 562 499 999 999 999 999 511 883 065 518 849 418 854 4;
  • 66) 0.415 868 431 329 727 172 851 562 499 999 999 999 999 511 883 065 518 849 418 854 4 × 2 = 0 + 0.831 736 862 659 454 345 703 124 999 999 999 999 999 023 766 131 037 698 837 708 8;
  • 67) 0.831 736 862 659 454 345 703 124 999 999 999 999 999 023 766 131 037 698 837 708 8 × 2 = 1 + 0.663 473 725 318 908 691 406 249 999 999 999 999 998 047 532 262 075 397 675 417 6;
  • 68) 0.663 473 725 318 908 691 406 249 999 999 999 999 998 047 532 262 075 397 675 417 6 × 2 = 1 + 0.326 947 450 637 817 382 812 499 999 999 999 999 996 095 064 524 150 795 350 835 2;
  • 69) 0.326 947 450 637 817 382 812 499 999 999 999 999 996 095 064 524 150 795 350 835 2 × 2 = 0 + 0.653 894 901 275 634 765 624 999 999 999 999 999 992 190 129 048 301 590 701 670 4;
  • 70) 0.653 894 901 275 634 765 624 999 999 999 999 999 992 190 129 048 301 590 701 670 4 × 2 = 1 + 0.307 789 802 551 269 531 249 999 999 999 999 999 984 380 258 096 603 181 403 340 8;
  • 71) 0.307 789 802 551 269 531 249 999 999 999 999 999 984 380 258 096 603 181 403 340 8 × 2 = 0 + 0.615 579 605 102 539 062 499 999 999 999 999 999 968 760 516 193 206 362 806 681 6;
  • 72) 0.615 579 605 102 539 062 499 999 999 999 999 999 968 760 516 193 206 362 806 681 6 × 2 = 1 + 0.231 159 210 205 078 124 999 999 999 999 999 999 937 521 032 386 412 725 613 363 2;
  • 73) 0.231 159 210 205 078 124 999 999 999 999 999 999 937 521 032 386 412 725 613 363 2 × 2 = 0 + 0.462 318 420 410 156 249 999 999 999 999 999 999 875 042 064 772 825 451 226 726 4;
  • 74) 0.462 318 420 410 156 249 999 999 999 999 999 999 875 042 064 772 825 451 226 726 4 × 2 = 0 + 0.924 636 840 820 312 499 999 999 999 999 999 999 750 084 129 545 650 902 453 452 8;
  • 75) 0.924 636 840 820 312 499 999 999 999 999 999 999 750 084 129 545 650 902 453 452 8 × 2 = 1 + 0.849 273 681 640 624 999 999 999 999 999 999 999 500 168 259 091 301 804 906 905 6;
  • 76) 0.849 273 681 640 624 999 999 999 999 999 999 999 500 168 259 091 301 804 906 905 6 × 2 = 1 + 0.698 547 363 281 249 999 999 999 999 999 999 999 000 336 518 182 603 609 813 811 2;
  • 77) 0.698 547 363 281 249 999 999 999 999 999 999 999 000 336 518 182 603 609 813 811 2 × 2 = 1 + 0.397 094 726 562 499 999 999 999 999 999 999 998 000 673 036 365 207 219 627 622 4;
  • 78) 0.397 094 726 562 499 999 999 999 999 999 999 998 000 673 036 365 207 219 627 622 4 × 2 = 0 + 0.794 189 453 124 999 999 999 999 999 999 999 996 001 346 072 730 414 439 255 244 8;
  • 79) 0.794 189 453 124 999 999 999 999 999 999 999 996 001 346 072 730 414 439 255 244 8 × 2 = 1 + 0.588 378 906 249 999 999 999 999 999 999 999 992 002 692 145 460 828 878 510 489 6;
  • 80) 0.588 378 906 249 999 999 999 999 999 999 999 992 002 692 145 460 828 878 510 489 6 × 2 = 1 + 0.176 757 812 499 999 999 999 999 999 999 999 984 005 384 290 921 657 757 020 979 2;
  • 81) 0.176 757 812 499 999 999 999 999 999 999 999 984 005 384 290 921 657 757 020 979 2 × 2 = 0 + 0.353 515 624 999 999 999 999 999 999 999 999 968 010 768 581 843 315 514 041 958 4;
  • 82) 0.353 515 624 999 999 999 999 999 999 999 999 968 010 768 581 843 315 514 041 958 4 × 2 = 0 + 0.707 031 249 999 999 999 999 999 999 999 999 936 021 537 163 686 631 028 083 916 8;
  • 83) 0.707 031 249 999 999 999 999 999 999 999 999 936 021 537 163 686 631 028 083 916 8 × 2 = 1 + 0.414 062 499 999 999 999 999 999 999 999 999 872 043 074 327 373 262 056 167 833 6;
  • 84) 0.414 062 499 999 999 999 999 999 999 999 999 872 043 074 327 373 262 056 167 833 6 × 2 = 0 + 0.828 124 999 999 999 999 999 999 999 999 999 744 086 148 654 746 524 112 335 667 2;
  • 85) 0.828 124 999 999 999 999 999 999 999 999 999 744 086 148 654 746 524 112 335 667 2 × 2 = 1 + 0.656 249 999 999 999 999 999 999 999 999 999 488 172 297 309 493 048 224 671 334 4;
  • 86) 0.656 249 999 999 999 999 999 999 999 999 999 488 172 297 309 493 048 224 671 334 4 × 2 = 1 + 0.312 499 999 999 999 999 999 999 999 999 998 976 344 594 618 986 096 449 342 668 8;
  • 87) 0.312 499 999 999 999 999 999 999 999 999 998 976 344 594 618 986 096 449 342 668 8 × 2 = 0 + 0.624 999 999 999 999 999 999 999 999 999 997 952 689 189 237 972 192 898 685 337 6;
  • 88) 0.624 999 999 999 999 999 999 999 999 999 997 952 689 189 237 972 192 898 685 337 6 × 2 = 1 + 0.249 999 999 999 999 999 999 999 999 999 995 905 378 378 475 944 385 797 370 675 2;
  • 89) 0.249 999 999 999 999 999 999 999 999 999 995 905 378 378 475 944 385 797 370 675 2 × 2 = 0 + 0.499 999 999 999 999 999 999 999 999 999 991 810 756 756 951 888 771 594 741 350 4;
  • 90) 0.499 999 999 999 999 999 999 999 999 999 991 810 756 756 951 888 771 594 741 350 4 × 2 = 0 + 0.999 999 999 999 999 999 999 999 999 999 983 621 513 513 903 777 543 189 482 700 8;
  • 91) 0.999 999 999 999 999 999 999 999 999 999 983 621 513 513 903 777 543 189 482 700 8 × 2 = 1 + 0.999 999 999 999 999 999 999 999 999 999 967 243 027 027 807 555 086 378 965 401 6;

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:


0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 029 919 2(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:

0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 029 919 2(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:


0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 029 919 2(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 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 029 919 2 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

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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