0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 031 37 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 031 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
0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 031 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: 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 031 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) 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 031 37 × 2 = 0 + 0.000 000 000 007 269 999 999 999 999 516 522 114 064 601 356 670 305 976 062 74;
  • 2) 0.000 000 000 007 269 999 999 999 999 516 522 114 064 601 356 670 305 976 062 74 × 2 = 0 + 0.000 000 000 014 539 999 999 999 999 033 044 228 129 202 713 340 611 952 125 48;
  • 3) 0.000 000 000 014 539 999 999 999 999 033 044 228 129 202 713 340 611 952 125 48 × 2 = 0 + 0.000 000 000 029 079 999 999 999 998 066 088 456 258 405 426 681 223 904 250 96;
  • 4) 0.000 000 000 029 079 999 999 999 998 066 088 456 258 405 426 681 223 904 250 96 × 2 = 0 + 0.000 000 000 058 159 999 999 999 996 132 176 912 516 810 853 362 447 808 501 92;
  • 5) 0.000 000 000 058 159 999 999 999 996 132 176 912 516 810 853 362 447 808 501 92 × 2 = 0 + 0.000 000 000 116 319 999 999 999 992 264 353 825 033 621 706 724 895 617 003 84;
  • 6) 0.000 000 000 116 319 999 999 999 992 264 353 825 033 621 706 724 895 617 003 84 × 2 = 0 + 0.000 000 000 232 639 999 999 999 984 528 707 650 067 243 413 449 791 234 007 68;
  • 7) 0.000 000 000 232 639 999 999 999 984 528 707 650 067 243 413 449 791 234 007 68 × 2 = 0 + 0.000 000 000 465 279 999 999 999 969 057 415 300 134 486 826 899 582 468 015 36;
  • 8) 0.000 000 000 465 279 999 999 999 969 057 415 300 134 486 826 899 582 468 015 36 × 2 = 0 + 0.000 000 000 930 559 999 999 999 938 114 830 600 268 973 653 799 164 936 030 72;
  • 9) 0.000 000 000 930 559 999 999 999 938 114 830 600 268 973 653 799 164 936 030 72 × 2 = 0 + 0.000 000 001 861 119 999 999 999 876 229 661 200 537 947 307 598 329 872 061 44;
  • 10) 0.000 000 001 861 119 999 999 999 876 229 661 200 537 947 307 598 329 872 061 44 × 2 = 0 + 0.000 000 003 722 239 999 999 999 752 459 322 401 075 894 615 196 659 744 122 88;
  • 11) 0.000 000 003 722 239 999 999 999 752 459 322 401 075 894 615 196 659 744 122 88 × 2 = 0 + 0.000 000 007 444 479 999 999 999 504 918 644 802 151 789 230 393 319 488 245 76;
  • 12) 0.000 000 007 444 479 999 999 999 504 918 644 802 151 789 230 393 319 488 245 76 × 2 = 0 + 0.000 000 014 888 959 999 999 999 009 837 289 604 303 578 460 786 638 976 491 52;
  • 13) 0.000 000 014 888 959 999 999 999 009 837 289 604 303 578 460 786 638 976 491 52 × 2 = 0 + 0.000 000 029 777 919 999 999 998 019 674 579 208 607 156 921 573 277 952 983 04;
  • 14) 0.000 000 029 777 919 999 999 998 019 674 579 208 607 156 921 573 277 952 983 04 × 2 = 0 + 0.000 000 059 555 839 999 999 996 039 349 158 417 214 313 843 146 555 905 966 08;
  • 15) 0.000 000 059 555 839 999 999 996 039 349 158 417 214 313 843 146 555 905 966 08 × 2 = 0 + 0.000 000 119 111 679 999 999 992 078 698 316 834 428 627 686 293 111 811 932 16;
  • 16) 0.000 000 119 111 679 999 999 992 078 698 316 834 428 627 686 293 111 811 932 16 × 2 = 0 + 0.000 000 238 223 359 999 999 984 157 396 633 668 857 255 372 586 223 623 864 32;
  • 17) 0.000 000 238 223 359 999 999 984 157 396 633 668 857 255 372 586 223 623 864 32 × 2 = 0 + 0.000 000 476 446 719 999 999 968 314 793 267 337 714 510 745 172 447 247 728 64;
  • 18) 0.000 000 476 446 719 999 999 968 314 793 267 337 714 510 745 172 447 247 728 64 × 2 = 0 + 0.000 000 952 893 439 999 999 936 629 586 534 675 429 021 490 344 894 495 457 28;
  • 19) 0.000 000 952 893 439 999 999 936 629 586 534 675 429 021 490 344 894 495 457 28 × 2 = 0 + 0.000 001 905 786 879 999 999 873 259 173 069 350 858 042 980 689 788 990 914 56;
  • 20) 0.000 001 905 786 879 999 999 873 259 173 069 350 858 042 980 689 788 990 914 56 × 2 = 0 + 0.000 003 811 573 759 999 999 746 518 346 138 701 716 085 961 379 577 981 829 12;
  • 21) 0.000 003 811 573 759 999 999 746 518 346 138 701 716 085 961 379 577 981 829 12 × 2 = 0 + 0.000 007 623 147 519 999 999 493 036 692 277 403 432 171 922 759 155 963 658 24;
  • 22) 0.000 007 623 147 519 999 999 493 036 692 277 403 432 171 922 759 155 963 658 24 × 2 = 0 + 0.000 015 246 295 039 999 998 986 073 384 554 806 864 343 845 518 311 927 316 48;
  • 23) 0.000 015 246 295 039 999 998 986 073 384 554 806 864 343 845 518 311 927 316 48 × 2 = 0 + 0.000 030 492 590 079 999 997 972 146 769 109 613 728 687 691 036 623 854 632 96;
  • 24) 0.000 030 492 590 079 999 997 972 146 769 109 613 728 687 691 036 623 854 632 96 × 2 = 0 + 0.000 060 985 180 159 999 995 944 293 538 219 227 457 375 382 073 247 709 265 92;
  • 25) 0.000 060 985 180 159 999 995 944 293 538 219 227 457 375 382 073 247 709 265 92 × 2 = 0 + 0.000 121 970 360 319 999 991 888 587 076 438 454 914 750 764 146 495 418 531 84;
  • 26) 0.000 121 970 360 319 999 991 888 587 076 438 454 914 750 764 146 495 418 531 84 × 2 = 0 + 0.000 243 940 720 639 999 983 777 174 152 876 909 829 501 528 292 990 837 063 68;
  • 27) 0.000 243 940 720 639 999 983 777 174 152 876 909 829 501 528 292 990 837 063 68 × 2 = 0 + 0.000 487 881 441 279 999 967 554 348 305 753 819 659 003 056 585 981 674 127 36;
  • 28) 0.000 487 881 441 279 999 967 554 348 305 753 819 659 003 056 585 981 674 127 36 × 2 = 0 + 0.000 975 762 882 559 999 935 108 696 611 507 639 318 006 113 171 963 348 254 72;
  • 29) 0.000 975 762 882 559 999 935 108 696 611 507 639 318 006 113 171 963 348 254 72 × 2 = 0 + 0.001 951 525 765 119 999 870 217 393 223 015 278 636 012 226 343 926 696 509 44;
  • 30) 0.001 951 525 765 119 999 870 217 393 223 015 278 636 012 226 343 926 696 509 44 × 2 = 0 + 0.003 903 051 530 239 999 740 434 786 446 030 557 272 024 452 687 853 393 018 88;
  • 31) 0.003 903 051 530 239 999 740 434 786 446 030 557 272 024 452 687 853 393 018 88 × 2 = 0 + 0.007 806 103 060 479 999 480 869 572 892 061 114 544 048 905 375 706 786 037 76;
  • 32) 0.007 806 103 060 479 999 480 869 572 892 061 114 544 048 905 375 706 786 037 76 × 2 = 0 + 0.015 612 206 120 959 998 961 739 145 784 122 229 088 097 810 751 413 572 075 52;
  • 33) 0.015 612 206 120 959 998 961 739 145 784 122 229 088 097 810 751 413 572 075 52 × 2 = 0 + 0.031 224 412 241 919 997 923 478 291 568 244 458 176 195 621 502 827 144 151 04;
  • 34) 0.031 224 412 241 919 997 923 478 291 568 244 458 176 195 621 502 827 144 151 04 × 2 = 0 + 0.062 448 824 483 839 995 846 956 583 136 488 916 352 391 243 005 654 288 302 08;
  • 35) 0.062 448 824 483 839 995 846 956 583 136 488 916 352 391 243 005 654 288 302 08 × 2 = 0 + 0.124 897 648 967 679 991 693 913 166 272 977 832 704 782 486 011 308 576 604 16;
  • 36) 0.124 897 648 967 679 991 693 913 166 272 977 832 704 782 486 011 308 576 604 16 × 2 = 0 + 0.249 795 297 935 359 983 387 826 332 545 955 665 409 564 972 022 617 153 208 32;
  • 37) 0.249 795 297 935 359 983 387 826 332 545 955 665 409 564 972 022 617 153 208 32 × 2 = 0 + 0.499 590 595 870 719 966 775 652 665 091 911 330 819 129 944 045 234 306 416 64;
  • 38) 0.499 590 595 870 719 966 775 652 665 091 911 330 819 129 944 045 234 306 416 64 × 2 = 0 + 0.999 181 191 741 439 933 551 305 330 183 822 661 638 259 888 090 468 612 833 28;
  • 39) 0.999 181 191 741 439 933 551 305 330 183 822 661 638 259 888 090 468 612 833 28 × 2 = 1 + 0.998 362 383 482 879 867 102 610 660 367 645 323 276 519 776 180 937 225 666 56;
  • 40) 0.998 362 383 482 879 867 102 610 660 367 645 323 276 519 776 180 937 225 666 56 × 2 = 1 + 0.996 724 766 965 759 734 205 221 320 735 290 646 553 039 552 361 874 451 333 12;
  • 41) 0.996 724 766 965 759 734 205 221 320 735 290 646 553 039 552 361 874 451 333 12 × 2 = 1 + 0.993 449 533 931 519 468 410 442 641 470 581 293 106 079 104 723 748 902 666 24;
  • 42) 0.993 449 533 931 519 468 410 442 641 470 581 293 106 079 104 723 748 902 666 24 × 2 = 1 + 0.986 899 067 863 038 936 820 885 282 941 162 586 212 158 209 447 497 805 332 48;
  • 43) 0.986 899 067 863 038 936 820 885 282 941 162 586 212 158 209 447 497 805 332 48 × 2 = 1 + 0.973 798 135 726 077 873 641 770 565 882 325 172 424 316 418 894 995 610 664 96;
  • 44) 0.973 798 135 726 077 873 641 770 565 882 325 172 424 316 418 894 995 610 664 96 × 2 = 1 + 0.947 596 271 452 155 747 283 541 131 764 650 344 848 632 837 789 991 221 329 92;
  • 45) 0.947 596 271 452 155 747 283 541 131 764 650 344 848 632 837 789 991 221 329 92 × 2 = 1 + 0.895 192 542 904 311 494 567 082 263 529 300 689 697 265 675 579 982 442 659 84;
  • 46) 0.895 192 542 904 311 494 567 082 263 529 300 689 697 265 675 579 982 442 659 84 × 2 = 1 + 0.790 385 085 808 622 989 134 164 527 058 601 379 394 531 351 159 964 885 319 68;
  • 47) 0.790 385 085 808 622 989 134 164 527 058 601 379 394 531 351 159 964 885 319 68 × 2 = 1 + 0.580 770 171 617 245 978 268 329 054 117 202 758 789 062 702 319 929 770 639 36;
  • 48) 0.580 770 171 617 245 978 268 329 054 117 202 758 789 062 702 319 929 770 639 36 × 2 = 1 + 0.161 540 343 234 491 956 536 658 108 234 405 517 578 125 404 639 859 541 278 72;
  • 49) 0.161 540 343 234 491 956 536 658 108 234 405 517 578 125 404 639 859 541 278 72 × 2 = 0 + 0.323 080 686 468 983 913 073 316 216 468 811 035 156 250 809 279 719 082 557 44;
  • 50) 0.323 080 686 468 983 913 073 316 216 468 811 035 156 250 809 279 719 082 557 44 × 2 = 0 + 0.646 161 372 937 967 826 146 632 432 937 622 070 312 501 618 559 438 165 114 88;
  • 51) 0.646 161 372 937 967 826 146 632 432 937 622 070 312 501 618 559 438 165 114 88 × 2 = 1 + 0.292 322 745 875 935 652 293 264 865 875 244 140 625 003 237 118 876 330 229 76;
  • 52) 0.292 322 745 875 935 652 293 264 865 875 244 140 625 003 237 118 876 330 229 76 × 2 = 0 + 0.584 645 491 751 871 304 586 529 731 750 488 281 250 006 474 237 752 660 459 52;
  • 53) 0.584 645 491 751 871 304 586 529 731 750 488 281 250 006 474 237 752 660 459 52 × 2 = 1 + 0.169 290 983 503 742 609 173 059 463 500 976 562 500 012 948 475 505 320 919 04;
  • 54) 0.169 290 983 503 742 609 173 059 463 500 976 562 500 012 948 475 505 320 919 04 × 2 = 0 + 0.338 581 967 007 485 218 346 118 927 001 953 125 000 025 896 951 010 641 838 08;
  • 55) 0.338 581 967 007 485 218 346 118 927 001 953 125 000 025 896 951 010 641 838 08 × 2 = 0 + 0.677 163 934 014 970 436 692 237 854 003 906 250 000 051 793 902 021 283 676 16;
  • 56) 0.677 163 934 014 970 436 692 237 854 003 906 250 000 051 793 902 021 283 676 16 × 2 = 1 + 0.354 327 868 029 940 873 384 475 708 007 812 500 000 103 587 804 042 567 352 32;
  • 57) 0.354 327 868 029 940 873 384 475 708 007 812 500 000 103 587 804 042 567 352 32 × 2 = 0 + 0.708 655 736 059 881 746 768 951 416 015 625 000 000 207 175 608 085 134 704 64;
  • 58) 0.708 655 736 059 881 746 768 951 416 015 625 000 000 207 175 608 085 134 704 64 × 2 = 1 + 0.417 311 472 119 763 493 537 902 832 031 250 000 000 414 351 216 170 269 409 28;
  • 59) 0.417 311 472 119 763 493 537 902 832 031 250 000 000 414 351 216 170 269 409 28 × 2 = 0 + 0.834 622 944 239 526 987 075 805 664 062 500 000 000 828 702 432 340 538 818 56;
  • 60) 0.834 622 944 239 526 987 075 805 664 062 500 000 000 828 702 432 340 538 818 56 × 2 = 1 + 0.669 245 888 479 053 974 151 611 328 125 000 000 001 657 404 864 681 077 637 12;
  • 61) 0.669 245 888 479 053 974 151 611 328 125 000 000 001 657 404 864 681 077 637 12 × 2 = 1 + 0.338 491 776 958 107 948 303 222 656 250 000 000 003 314 809 729 362 155 274 24;
  • 62) 0.338 491 776 958 107 948 303 222 656 250 000 000 003 314 809 729 362 155 274 24 × 2 = 0 + 0.676 983 553 916 215 896 606 445 312 500 000 000 006 629 619 458 724 310 548 48;
  • 63) 0.676 983 553 916 215 896 606 445 312 500 000 000 006 629 619 458 724 310 548 48 × 2 = 1 + 0.353 967 107 832 431 793 212 890 625 000 000 000 013 259 238 917 448 621 096 96;
  • 64) 0.353 967 107 832 431 793 212 890 625 000 000 000 013 259 238 917 448 621 096 96 × 2 = 0 + 0.707 934 215 664 863 586 425 781 250 000 000 000 026 518 477 834 897 242 193 92;
  • 65) 0.707 934 215 664 863 586 425 781 250 000 000 000 026 518 477 834 897 242 193 92 × 2 = 1 + 0.415 868 431 329 727 172 851 562 500 000 000 000 053 036 955 669 794 484 387 84;
  • 66) 0.415 868 431 329 727 172 851 562 500 000 000 000 053 036 955 669 794 484 387 84 × 2 = 0 + 0.831 736 862 659 454 345 703 125 000 000 000 000 106 073 911 339 588 968 775 68;
  • 67) 0.831 736 862 659 454 345 703 125 000 000 000 000 106 073 911 339 588 968 775 68 × 2 = 1 + 0.663 473 725 318 908 691 406 250 000 000 000 000 212 147 822 679 177 937 551 36;
  • 68) 0.663 473 725 318 908 691 406 250 000 000 000 000 212 147 822 679 177 937 551 36 × 2 = 1 + 0.326 947 450 637 817 382 812 500 000 000 000 000 424 295 645 358 355 875 102 72;
  • 69) 0.326 947 450 637 817 382 812 500 000 000 000 000 424 295 645 358 355 875 102 72 × 2 = 0 + 0.653 894 901 275 634 765 625 000 000 000 000 000 848 591 290 716 711 750 205 44;
  • 70) 0.653 894 901 275 634 765 625 000 000 000 000 000 848 591 290 716 711 750 205 44 × 2 = 1 + 0.307 789 802 551 269 531 250 000 000 000 000 001 697 182 581 433 423 500 410 88;
  • 71) 0.307 789 802 551 269 531 250 000 000 000 000 001 697 182 581 433 423 500 410 88 × 2 = 0 + 0.615 579 605 102 539 062 500 000 000 000 000 003 394 365 162 866 847 000 821 76;
  • 72) 0.615 579 605 102 539 062 500 000 000 000 000 003 394 365 162 866 847 000 821 76 × 2 = 1 + 0.231 159 210 205 078 125 000 000 000 000 000 006 788 730 325 733 694 001 643 52;
  • 73) 0.231 159 210 205 078 125 000 000 000 000 000 006 788 730 325 733 694 001 643 52 × 2 = 0 + 0.462 318 420 410 156 250 000 000 000 000 000 013 577 460 651 467 388 003 287 04;
  • 74) 0.462 318 420 410 156 250 000 000 000 000 000 013 577 460 651 467 388 003 287 04 × 2 = 0 + 0.924 636 840 820 312 500 000 000 000 000 000 027 154 921 302 934 776 006 574 08;
  • 75) 0.924 636 840 820 312 500 000 000 000 000 000 027 154 921 302 934 776 006 574 08 × 2 = 1 + 0.849 273 681 640 625 000 000 000 000 000 000 054 309 842 605 869 552 013 148 16;
  • 76) 0.849 273 681 640 625 000 000 000 000 000 000 054 309 842 605 869 552 013 148 16 × 2 = 1 + 0.698 547 363 281 250 000 000 000 000 000 000 108 619 685 211 739 104 026 296 32;
  • 77) 0.698 547 363 281 250 000 000 000 000 000 000 108 619 685 211 739 104 026 296 32 × 2 = 1 + 0.397 094 726 562 500 000 000 000 000 000 000 217 239 370 423 478 208 052 592 64;
  • 78) 0.397 094 726 562 500 000 000 000 000 000 000 217 239 370 423 478 208 052 592 64 × 2 = 0 + 0.794 189 453 125 000 000 000 000 000 000 000 434 478 740 846 956 416 105 185 28;
  • 79) 0.794 189 453 125 000 000 000 000 000 000 000 434 478 740 846 956 416 105 185 28 × 2 = 1 + 0.588 378 906 250 000 000 000 000 000 000 000 868 957 481 693 912 832 210 370 56;
  • 80) 0.588 378 906 250 000 000 000 000 000 000 000 868 957 481 693 912 832 210 370 56 × 2 = 1 + 0.176 757 812 500 000 000 000 000 000 000 001 737 914 963 387 825 664 420 741 12;
  • 81) 0.176 757 812 500 000 000 000 000 000 000 001 737 914 963 387 825 664 420 741 12 × 2 = 0 + 0.353 515 625 000 000 000 000 000 000 000 003 475 829 926 775 651 328 841 482 24;
  • 82) 0.353 515 625 000 000 000 000 000 000 000 003 475 829 926 775 651 328 841 482 24 × 2 = 0 + 0.707 031 250 000 000 000 000 000 000 000 006 951 659 853 551 302 657 682 964 48;
  • 83) 0.707 031 250 000 000 000 000 000 000 000 006 951 659 853 551 302 657 682 964 48 × 2 = 1 + 0.414 062 500 000 000 000 000 000 000 000 013 903 319 707 102 605 315 365 928 96;
  • 84) 0.414 062 500 000 000 000 000 000 000 000 013 903 319 707 102 605 315 365 928 96 × 2 = 0 + 0.828 125 000 000 000 000 000 000 000 000 027 806 639 414 205 210 630 731 857 92;
  • 85) 0.828 125 000 000 000 000 000 000 000 000 027 806 639 414 205 210 630 731 857 92 × 2 = 1 + 0.656 250 000 000 000 000 000 000 000 000 055 613 278 828 410 421 261 463 715 84;
  • 86) 0.656 250 000 000 000 000 000 000 000 000 055 613 278 828 410 421 261 463 715 84 × 2 = 1 + 0.312 500 000 000 000 000 000 000 000 000 111 226 557 656 820 842 522 927 431 68;
  • 87) 0.312 500 000 000 000 000 000 000 000 000 111 226 557 656 820 842 522 927 431 68 × 2 = 0 + 0.625 000 000 000 000 000 000 000 000 000 222 453 115 313 641 685 045 854 863 36;
  • 88) 0.625 000 000 000 000 000 000 000 000 000 222 453 115 313 641 685 045 854 863 36 × 2 = 1 + 0.250 000 000 000 000 000 000 000 000 000 444 906 230 627 283 370 091 709 726 72;
  • 89) 0.250 000 000 000 000 000 000 000 000 000 444 906 230 627 283 370 091 709 726 72 × 2 = 0 + 0.500 000 000 000 000 000 000 000 000 000 889 812 461 254 566 740 183 419 453 44;
  • 90) 0.500 000 000 000 000 000 000 000 000 000 889 812 461 254 566 740 183 419 453 44 × 2 = 1 + 0.000 000 000 000 000 000 000 000 000 001 779 624 922 509 133 480 366 838 906 88;
  • 91) 0.000 000 000 000 000 000 000 000 000 001 779 624 922 509 133 480 366 838 906 88 × 2 = 0 + 0.000 000 000 000 000 000 000 000 000 003 559 249 845 018 266 960 733 677 813 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:


0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 031 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:

0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 031 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:


0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 031 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 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 031 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