0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 029 934 37 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 029 934 37(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
0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 029 934 37(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: 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 029 934 37.

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) 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 029 934 37 × 2 = 0 + 0.000 000 000 007 269 999 999 999 999 516 522 114 064 601 356 670 305 976 059 868 74;
  • 2) 0.000 000 000 007 269 999 999 999 999 516 522 114 064 601 356 670 305 976 059 868 74 × 2 = 0 + 0.000 000 000 014 539 999 999 999 999 033 044 228 129 202 713 340 611 952 119 737 48;
  • 3) 0.000 000 000 014 539 999 999 999 999 033 044 228 129 202 713 340 611 952 119 737 48 × 2 = 0 + 0.000 000 000 029 079 999 999 999 998 066 088 456 258 405 426 681 223 904 239 474 96;
  • 4) 0.000 000 000 029 079 999 999 999 998 066 088 456 258 405 426 681 223 904 239 474 96 × 2 = 0 + 0.000 000 000 058 159 999 999 999 996 132 176 912 516 810 853 362 447 808 478 949 92;
  • 5) 0.000 000 000 058 159 999 999 999 996 132 176 912 516 810 853 362 447 808 478 949 92 × 2 = 0 + 0.000 000 000 116 319 999 999 999 992 264 353 825 033 621 706 724 895 616 957 899 84;
  • 6) 0.000 000 000 116 319 999 999 999 992 264 353 825 033 621 706 724 895 616 957 899 84 × 2 = 0 + 0.000 000 000 232 639 999 999 999 984 528 707 650 067 243 413 449 791 233 915 799 68;
  • 7) 0.000 000 000 232 639 999 999 999 984 528 707 650 067 243 413 449 791 233 915 799 68 × 2 = 0 + 0.000 000 000 465 279 999 999 999 969 057 415 300 134 486 826 899 582 467 831 599 36;
  • 8) 0.000 000 000 465 279 999 999 999 969 057 415 300 134 486 826 899 582 467 831 599 36 × 2 = 0 + 0.000 000 000 930 559 999 999 999 938 114 830 600 268 973 653 799 164 935 663 198 72;
  • 9) 0.000 000 000 930 559 999 999 999 938 114 830 600 268 973 653 799 164 935 663 198 72 × 2 = 0 + 0.000 000 001 861 119 999 999 999 876 229 661 200 537 947 307 598 329 871 326 397 44;
  • 10) 0.000 000 001 861 119 999 999 999 876 229 661 200 537 947 307 598 329 871 326 397 44 × 2 = 0 + 0.000 000 003 722 239 999 999 999 752 459 322 401 075 894 615 196 659 742 652 794 88;
  • 11) 0.000 000 003 722 239 999 999 999 752 459 322 401 075 894 615 196 659 742 652 794 88 × 2 = 0 + 0.000 000 007 444 479 999 999 999 504 918 644 802 151 789 230 393 319 485 305 589 76;
  • 12) 0.000 000 007 444 479 999 999 999 504 918 644 802 151 789 230 393 319 485 305 589 76 × 2 = 0 + 0.000 000 014 888 959 999 999 999 009 837 289 604 303 578 460 786 638 970 611 179 52;
  • 13) 0.000 000 014 888 959 999 999 999 009 837 289 604 303 578 460 786 638 970 611 179 52 × 2 = 0 + 0.000 000 029 777 919 999 999 998 019 674 579 208 607 156 921 573 277 941 222 359 04;
  • 14) 0.000 000 029 777 919 999 999 998 019 674 579 208 607 156 921 573 277 941 222 359 04 × 2 = 0 + 0.000 000 059 555 839 999 999 996 039 349 158 417 214 313 843 146 555 882 444 718 08;
  • 15) 0.000 000 059 555 839 999 999 996 039 349 158 417 214 313 843 146 555 882 444 718 08 × 2 = 0 + 0.000 000 119 111 679 999 999 992 078 698 316 834 428 627 686 293 111 764 889 436 16;
  • 16) 0.000 000 119 111 679 999 999 992 078 698 316 834 428 627 686 293 111 764 889 436 16 × 2 = 0 + 0.000 000 238 223 359 999 999 984 157 396 633 668 857 255 372 586 223 529 778 872 32;
  • 17) 0.000 000 238 223 359 999 999 984 157 396 633 668 857 255 372 586 223 529 778 872 32 × 2 = 0 + 0.000 000 476 446 719 999 999 968 314 793 267 337 714 510 745 172 447 059 557 744 64;
  • 18) 0.000 000 476 446 719 999 999 968 314 793 267 337 714 510 745 172 447 059 557 744 64 × 2 = 0 + 0.000 000 952 893 439 999 999 936 629 586 534 675 429 021 490 344 894 119 115 489 28;
  • 19) 0.000 000 952 893 439 999 999 936 629 586 534 675 429 021 490 344 894 119 115 489 28 × 2 = 0 + 0.000 001 905 786 879 999 999 873 259 173 069 350 858 042 980 689 788 238 230 978 56;
  • 20) 0.000 001 905 786 879 999 999 873 259 173 069 350 858 042 980 689 788 238 230 978 56 × 2 = 0 + 0.000 003 811 573 759 999 999 746 518 346 138 701 716 085 961 379 576 476 461 957 12;
  • 21) 0.000 003 811 573 759 999 999 746 518 346 138 701 716 085 961 379 576 476 461 957 12 × 2 = 0 + 0.000 007 623 147 519 999 999 493 036 692 277 403 432 171 922 759 152 952 923 914 24;
  • 22) 0.000 007 623 147 519 999 999 493 036 692 277 403 432 171 922 759 152 952 923 914 24 × 2 = 0 + 0.000 015 246 295 039 999 998 986 073 384 554 806 864 343 845 518 305 905 847 828 48;
  • 23) 0.000 015 246 295 039 999 998 986 073 384 554 806 864 343 845 518 305 905 847 828 48 × 2 = 0 + 0.000 030 492 590 079 999 997 972 146 769 109 613 728 687 691 036 611 811 695 656 96;
  • 24) 0.000 030 492 590 079 999 997 972 146 769 109 613 728 687 691 036 611 811 695 656 96 × 2 = 0 + 0.000 060 985 180 159 999 995 944 293 538 219 227 457 375 382 073 223 623 391 313 92;
  • 25) 0.000 060 985 180 159 999 995 944 293 538 219 227 457 375 382 073 223 623 391 313 92 × 2 = 0 + 0.000 121 970 360 319 999 991 888 587 076 438 454 914 750 764 146 447 246 782 627 84;
  • 26) 0.000 121 970 360 319 999 991 888 587 076 438 454 914 750 764 146 447 246 782 627 84 × 2 = 0 + 0.000 243 940 720 639 999 983 777 174 152 876 909 829 501 528 292 894 493 565 255 68;
  • 27) 0.000 243 940 720 639 999 983 777 174 152 876 909 829 501 528 292 894 493 565 255 68 × 2 = 0 + 0.000 487 881 441 279 999 967 554 348 305 753 819 659 003 056 585 788 987 130 511 36;
  • 28) 0.000 487 881 441 279 999 967 554 348 305 753 819 659 003 056 585 788 987 130 511 36 × 2 = 0 + 0.000 975 762 882 559 999 935 108 696 611 507 639 318 006 113 171 577 974 261 022 72;
  • 29) 0.000 975 762 882 559 999 935 108 696 611 507 639 318 006 113 171 577 974 261 022 72 × 2 = 0 + 0.001 951 525 765 119 999 870 217 393 223 015 278 636 012 226 343 155 948 522 045 44;
  • 30) 0.001 951 525 765 119 999 870 217 393 223 015 278 636 012 226 343 155 948 522 045 44 × 2 = 0 + 0.003 903 051 530 239 999 740 434 786 446 030 557 272 024 452 686 311 897 044 090 88;
  • 31) 0.003 903 051 530 239 999 740 434 786 446 030 557 272 024 452 686 311 897 044 090 88 × 2 = 0 + 0.007 806 103 060 479 999 480 869 572 892 061 114 544 048 905 372 623 794 088 181 76;
  • 32) 0.007 806 103 060 479 999 480 869 572 892 061 114 544 048 905 372 623 794 088 181 76 × 2 = 0 + 0.015 612 206 120 959 998 961 739 145 784 122 229 088 097 810 745 247 588 176 363 52;
  • 33) 0.015 612 206 120 959 998 961 739 145 784 122 229 088 097 810 745 247 588 176 363 52 × 2 = 0 + 0.031 224 412 241 919 997 923 478 291 568 244 458 176 195 621 490 495 176 352 727 04;
  • 34) 0.031 224 412 241 919 997 923 478 291 568 244 458 176 195 621 490 495 176 352 727 04 × 2 = 0 + 0.062 448 824 483 839 995 846 956 583 136 488 916 352 391 242 980 990 352 705 454 08;
  • 35) 0.062 448 824 483 839 995 846 956 583 136 488 916 352 391 242 980 990 352 705 454 08 × 2 = 0 + 0.124 897 648 967 679 991 693 913 166 272 977 832 704 782 485 961 980 705 410 908 16;
  • 36) 0.124 897 648 967 679 991 693 913 166 272 977 832 704 782 485 961 980 705 410 908 16 × 2 = 0 + 0.249 795 297 935 359 983 387 826 332 545 955 665 409 564 971 923 961 410 821 816 32;
  • 37) 0.249 795 297 935 359 983 387 826 332 545 955 665 409 564 971 923 961 410 821 816 32 × 2 = 0 + 0.499 590 595 870 719 966 775 652 665 091 911 330 819 129 943 847 922 821 643 632 64;
  • 38) 0.499 590 595 870 719 966 775 652 665 091 911 330 819 129 943 847 922 821 643 632 64 × 2 = 0 + 0.999 181 191 741 439 933 551 305 330 183 822 661 638 259 887 695 845 643 287 265 28;
  • 39) 0.999 181 191 741 439 933 551 305 330 183 822 661 638 259 887 695 845 643 287 265 28 × 2 = 1 + 0.998 362 383 482 879 867 102 610 660 367 645 323 276 519 775 391 691 286 574 530 56;
  • 40) 0.998 362 383 482 879 867 102 610 660 367 645 323 276 519 775 391 691 286 574 530 56 × 2 = 1 + 0.996 724 766 965 759 734 205 221 320 735 290 646 553 039 550 783 382 573 149 061 12;
  • 41) 0.996 724 766 965 759 734 205 221 320 735 290 646 553 039 550 783 382 573 149 061 12 × 2 = 1 + 0.993 449 533 931 519 468 410 442 641 470 581 293 106 079 101 566 765 146 298 122 24;
  • 42) 0.993 449 533 931 519 468 410 442 641 470 581 293 106 079 101 566 765 146 298 122 24 × 2 = 1 + 0.986 899 067 863 038 936 820 885 282 941 162 586 212 158 203 133 530 292 596 244 48;
  • 43) 0.986 899 067 863 038 936 820 885 282 941 162 586 212 158 203 133 530 292 596 244 48 × 2 = 1 + 0.973 798 135 726 077 873 641 770 565 882 325 172 424 316 406 267 060 585 192 488 96;
  • 44) 0.973 798 135 726 077 873 641 770 565 882 325 172 424 316 406 267 060 585 192 488 96 × 2 = 1 + 0.947 596 271 452 155 747 283 541 131 764 650 344 848 632 812 534 121 170 384 977 92;
  • 45) 0.947 596 271 452 155 747 283 541 131 764 650 344 848 632 812 534 121 170 384 977 92 × 2 = 1 + 0.895 192 542 904 311 494 567 082 263 529 300 689 697 265 625 068 242 340 769 955 84;
  • 46) 0.895 192 542 904 311 494 567 082 263 529 300 689 697 265 625 068 242 340 769 955 84 × 2 = 1 + 0.790 385 085 808 622 989 134 164 527 058 601 379 394 531 250 136 484 681 539 911 68;
  • 47) 0.790 385 085 808 622 989 134 164 527 058 601 379 394 531 250 136 484 681 539 911 68 × 2 = 1 + 0.580 770 171 617 245 978 268 329 054 117 202 758 789 062 500 272 969 363 079 823 36;
  • 48) 0.580 770 171 617 245 978 268 329 054 117 202 758 789 062 500 272 969 363 079 823 36 × 2 = 1 + 0.161 540 343 234 491 956 536 658 108 234 405 517 578 125 000 545 938 726 159 646 72;
  • 49) 0.161 540 343 234 491 956 536 658 108 234 405 517 578 125 000 545 938 726 159 646 72 × 2 = 0 + 0.323 080 686 468 983 913 073 316 216 468 811 035 156 250 001 091 877 452 319 293 44;
  • 50) 0.323 080 686 468 983 913 073 316 216 468 811 035 156 250 001 091 877 452 319 293 44 × 2 = 0 + 0.646 161 372 937 967 826 146 632 432 937 622 070 312 500 002 183 754 904 638 586 88;
  • 51) 0.646 161 372 937 967 826 146 632 432 937 622 070 312 500 002 183 754 904 638 586 88 × 2 = 1 + 0.292 322 745 875 935 652 293 264 865 875 244 140 625 000 004 367 509 809 277 173 76;
  • 52) 0.292 322 745 875 935 652 293 264 865 875 244 140 625 000 004 367 509 809 277 173 76 × 2 = 0 + 0.584 645 491 751 871 304 586 529 731 750 488 281 250 000 008 735 019 618 554 347 52;
  • 53) 0.584 645 491 751 871 304 586 529 731 750 488 281 250 000 008 735 019 618 554 347 52 × 2 = 1 + 0.169 290 983 503 742 609 173 059 463 500 976 562 500 000 017 470 039 237 108 695 04;
  • 54) 0.169 290 983 503 742 609 173 059 463 500 976 562 500 000 017 470 039 237 108 695 04 × 2 = 0 + 0.338 581 967 007 485 218 346 118 927 001 953 125 000 000 034 940 078 474 217 390 08;
  • 55) 0.338 581 967 007 485 218 346 118 927 001 953 125 000 000 034 940 078 474 217 390 08 × 2 = 0 + 0.677 163 934 014 970 436 692 237 854 003 906 250 000 000 069 880 156 948 434 780 16;
  • 56) 0.677 163 934 014 970 436 692 237 854 003 906 250 000 000 069 880 156 948 434 780 16 × 2 = 1 + 0.354 327 868 029 940 873 384 475 708 007 812 500 000 000 139 760 313 896 869 560 32;
  • 57) 0.354 327 868 029 940 873 384 475 708 007 812 500 000 000 139 760 313 896 869 560 32 × 2 = 0 + 0.708 655 736 059 881 746 768 951 416 015 625 000 000 000 279 520 627 793 739 120 64;
  • 58) 0.708 655 736 059 881 746 768 951 416 015 625 000 000 000 279 520 627 793 739 120 64 × 2 = 1 + 0.417 311 472 119 763 493 537 902 832 031 250 000 000 000 559 041 255 587 478 241 28;
  • 59) 0.417 311 472 119 763 493 537 902 832 031 250 000 000 000 559 041 255 587 478 241 28 × 2 = 0 + 0.834 622 944 239 526 987 075 805 664 062 500 000 000 001 118 082 511 174 956 482 56;
  • 60) 0.834 622 944 239 526 987 075 805 664 062 500 000 000 001 118 082 511 174 956 482 56 × 2 = 1 + 0.669 245 888 479 053 974 151 611 328 125 000 000 000 002 236 165 022 349 912 965 12;
  • 61) 0.669 245 888 479 053 974 151 611 328 125 000 000 000 002 236 165 022 349 912 965 12 × 2 = 1 + 0.338 491 776 958 107 948 303 222 656 250 000 000 000 004 472 330 044 699 825 930 24;
  • 62) 0.338 491 776 958 107 948 303 222 656 250 000 000 000 004 472 330 044 699 825 930 24 × 2 = 0 + 0.676 983 553 916 215 896 606 445 312 500 000 000 000 008 944 660 089 399 651 860 48;
  • 63) 0.676 983 553 916 215 896 606 445 312 500 000 000 000 008 944 660 089 399 651 860 48 × 2 = 1 + 0.353 967 107 832 431 793 212 890 625 000 000 000 000 017 889 320 178 799 303 720 96;
  • 64) 0.353 967 107 832 431 793 212 890 625 000 000 000 000 017 889 320 178 799 303 720 96 × 2 = 0 + 0.707 934 215 664 863 586 425 781 250 000 000 000 000 035 778 640 357 598 607 441 92;
  • 65) 0.707 934 215 664 863 586 425 781 250 000 000 000 000 035 778 640 357 598 607 441 92 × 2 = 1 + 0.415 868 431 329 727 172 851 562 500 000 000 000 000 071 557 280 715 197 214 883 84;
  • 66) 0.415 868 431 329 727 172 851 562 500 000 000 000 000 071 557 280 715 197 214 883 84 × 2 = 0 + 0.831 736 862 659 454 345 703 125 000 000 000 000 000 143 114 561 430 394 429 767 68;
  • 67) 0.831 736 862 659 454 345 703 125 000 000 000 000 000 143 114 561 430 394 429 767 68 × 2 = 1 + 0.663 473 725 318 908 691 406 250 000 000 000 000 000 286 229 122 860 788 859 535 36;
  • 68) 0.663 473 725 318 908 691 406 250 000 000 000 000 000 286 229 122 860 788 859 535 36 × 2 = 1 + 0.326 947 450 637 817 382 812 500 000 000 000 000 000 572 458 245 721 577 719 070 72;
  • 69) 0.326 947 450 637 817 382 812 500 000 000 000 000 000 572 458 245 721 577 719 070 72 × 2 = 0 + 0.653 894 901 275 634 765 625 000 000 000 000 000 001 144 916 491 443 155 438 141 44;
  • 70) 0.653 894 901 275 634 765 625 000 000 000 000 000 001 144 916 491 443 155 438 141 44 × 2 = 1 + 0.307 789 802 551 269 531 250 000 000 000 000 000 002 289 832 982 886 310 876 282 88;
  • 71) 0.307 789 802 551 269 531 250 000 000 000 000 000 002 289 832 982 886 310 876 282 88 × 2 = 0 + 0.615 579 605 102 539 062 500 000 000 000 000 000 004 579 665 965 772 621 752 565 76;
  • 72) 0.615 579 605 102 539 062 500 000 000 000 000 000 004 579 665 965 772 621 752 565 76 × 2 = 1 + 0.231 159 210 205 078 125 000 000 000 000 000 000 009 159 331 931 545 243 505 131 52;
  • 73) 0.231 159 210 205 078 125 000 000 000 000 000 000 009 159 331 931 545 243 505 131 52 × 2 = 0 + 0.462 318 420 410 156 250 000 000 000 000 000 000 018 318 663 863 090 487 010 263 04;
  • 74) 0.462 318 420 410 156 250 000 000 000 000 000 000 018 318 663 863 090 487 010 263 04 × 2 = 0 + 0.924 636 840 820 312 500 000 000 000 000 000 000 036 637 327 726 180 974 020 526 08;
  • 75) 0.924 636 840 820 312 500 000 000 000 000 000 000 036 637 327 726 180 974 020 526 08 × 2 = 1 + 0.849 273 681 640 625 000 000 000 000 000 000 000 073 274 655 452 361 948 041 052 16;
  • 76) 0.849 273 681 640 625 000 000 000 000 000 000 000 073 274 655 452 361 948 041 052 16 × 2 = 1 + 0.698 547 363 281 250 000 000 000 000 000 000 000 146 549 310 904 723 896 082 104 32;
  • 77) 0.698 547 363 281 250 000 000 000 000 000 000 000 146 549 310 904 723 896 082 104 32 × 2 = 1 + 0.397 094 726 562 500 000 000 000 000 000 000 000 293 098 621 809 447 792 164 208 64;
  • 78) 0.397 094 726 562 500 000 000 000 000 000 000 000 293 098 621 809 447 792 164 208 64 × 2 = 0 + 0.794 189 453 125 000 000 000 000 000 000 000 000 586 197 243 618 895 584 328 417 28;
  • 79) 0.794 189 453 125 000 000 000 000 000 000 000 000 586 197 243 618 895 584 328 417 28 × 2 = 1 + 0.588 378 906 250 000 000 000 000 000 000 000 001 172 394 487 237 791 168 656 834 56;
  • 80) 0.588 378 906 250 000 000 000 000 000 000 000 001 172 394 487 237 791 168 656 834 56 × 2 = 1 + 0.176 757 812 500 000 000 000 000 000 000 000 002 344 788 974 475 582 337 313 669 12;
  • 81) 0.176 757 812 500 000 000 000 000 000 000 000 002 344 788 974 475 582 337 313 669 12 × 2 = 0 + 0.353 515 625 000 000 000 000 000 000 000 000 004 689 577 948 951 164 674 627 338 24;
  • 82) 0.353 515 625 000 000 000 000 000 000 000 000 004 689 577 948 951 164 674 627 338 24 × 2 = 0 + 0.707 031 250 000 000 000 000 000 000 000 000 009 379 155 897 902 329 349 254 676 48;
  • 83) 0.707 031 250 000 000 000 000 000 000 000 000 009 379 155 897 902 329 349 254 676 48 × 2 = 1 + 0.414 062 500 000 000 000 000 000 000 000 000 018 758 311 795 804 658 698 509 352 96;
  • 84) 0.414 062 500 000 000 000 000 000 000 000 000 018 758 311 795 804 658 698 509 352 96 × 2 = 0 + 0.828 125 000 000 000 000 000 000 000 000 000 037 516 623 591 609 317 397 018 705 92;
  • 85) 0.828 125 000 000 000 000 000 000 000 000 000 037 516 623 591 609 317 397 018 705 92 × 2 = 1 + 0.656 250 000 000 000 000 000 000 000 000 000 075 033 247 183 218 634 794 037 411 84;
  • 86) 0.656 250 000 000 000 000 000 000 000 000 000 075 033 247 183 218 634 794 037 411 84 × 2 = 1 + 0.312 500 000 000 000 000 000 000 000 000 000 150 066 494 366 437 269 588 074 823 68;
  • 87) 0.312 500 000 000 000 000 000 000 000 000 000 150 066 494 366 437 269 588 074 823 68 × 2 = 0 + 0.625 000 000 000 000 000 000 000 000 000 000 300 132 988 732 874 539 176 149 647 36;
  • 88) 0.625 000 000 000 000 000 000 000 000 000 000 300 132 988 732 874 539 176 149 647 36 × 2 = 1 + 0.250 000 000 000 000 000 000 000 000 000 000 600 265 977 465 749 078 352 299 294 72;
  • 89) 0.250 000 000 000 000 000 000 000 000 000 000 600 265 977 465 749 078 352 299 294 72 × 2 = 0 + 0.500 000 000 000 000 000 000 000 000 000 001 200 531 954 931 498 156 704 598 589 44;
  • 90) 0.500 000 000 000 000 000 000 000 000 000 001 200 531 954 931 498 156 704 598 589 44 × 2 = 1 + 0.000 000 000 000 000 000 000 000 000 000 002 401 063 909 862 996 313 409 197 178 88;
  • 91) 0.000 000 000 000 000 000 000 000 000 000 002 401 063 909 862 996 313 409 197 178 88 × 2 = 0 + 0.000 000 000 000 000 000 000 000 000 000 004 802 127 819 725 992 626 818 394 357 76;

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:


0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 029 934 37(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:

0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 029 934 37(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:


0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 029 934 37(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 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 029 934 37 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

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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