0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 023 1 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 023 1(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
0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 023 1(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: 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 023 1.

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) 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 023 1 × 2 = 0 + 0.000 000 000 007 269 999 999 999 999 516 522 114 064 601 356 670 305 976 046 2;
  • 2) 0.000 000 000 007 269 999 999 999 999 516 522 114 064 601 356 670 305 976 046 2 × 2 = 0 + 0.000 000 000 014 539 999 999 999 999 033 044 228 129 202 713 340 611 952 092 4;
  • 3) 0.000 000 000 014 539 999 999 999 999 033 044 228 129 202 713 340 611 952 092 4 × 2 = 0 + 0.000 000 000 029 079 999 999 999 998 066 088 456 258 405 426 681 223 904 184 8;
  • 4) 0.000 000 000 029 079 999 999 999 998 066 088 456 258 405 426 681 223 904 184 8 × 2 = 0 + 0.000 000 000 058 159 999 999 999 996 132 176 912 516 810 853 362 447 808 369 6;
  • 5) 0.000 000 000 058 159 999 999 999 996 132 176 912 516 810 853 362 447 808 369 6 × 2 = 0 + 0.000 000 000 116 319 999 999 999 992 264 353 825 033 621 706 724 895 616 739 2;
  • 6) 0.000 000 000 116 319 999 999 999 992 264 353 825 033 621 706 724 895 616 739 2 × 2 = 0 + 0.000 000 000 232 639 999 999 999 984 528 707 650 067 243 413 449 791 233 478 4;
  • 7) 0.000 000 000 232 639 999 999 999 984 528 707 650 067 243 413 449 791 233 478 4 × 2 = 0 + 0.000 000 000 465 279 999 999 999 969 057 415 300 134 486 826 899 582 466 956 8;
  • 8) 0.000 000 000 465 279 999 999 999 969 057 415 300 134 486 826 899 582 466 956 8 × 2 = 0 + 0.000 000 000 930 559 999 999 999 938 114 830 600 268 973 653 799 164 933 913 6;
  • 9) 0.000 000 000 930 559 999 999 999 938 114 830 600 268 973 653 799 164 933 913 6 × 2 = 0 + 0.000 000 001 861 119 999 999 999 876 229 661 200 537 947 307 598 329 867 827 2;
  • 10) 0.000 000 001 861 119 999 999 999 876 229 661 200 537 947 307 598 329 867 827 2 × 2 = 0 + 0.000 000 003 722 239 999 999 999 752 459 322 401 075 894 615 196 659 735 654 4;
  • 11) 0.000 000 003 722 239 999 999 999 752 459 322 401 075 894 615 196 659 735 654 4 × 2 = 0 + 0.000 000 007 444 479 999 999 999 504 918 644 802 151 789 230 393 319 471 308 8;
  • 12) 0.000 000 007 444 479 999 999 999 504 918 644 802 151 789 230 393 319 471 308 8 × 2 = 0 + 0.000 000 014 888 959 999 999 999 009 837 289 604 303 578 460 786 638 942 617 6;
  • 13) 0.000 000 014 888 959 999 999 999 009 837 289 604 303 578 460 786 638 942 617 6 × 2 = 0 + 0.000 000 029 777 919 999 999 998 019 674 579 208 607 156 921 573 277 885 235 2;
  • 14) 0.000 000 029 777 919 999 999 998 019 674 579 208 607 156 921 573 277 885 235 2 × 2 = 0 + 0.000 000 059 555 839 999 999 996 039 349 158 417 214 313 843 146 555 770 470 4;
  • 15) 0.000 000 059 555 839 999 999 996 039 349 158 417 214 313 843 146 555 770 470 4 × 2 = 0 + 0.000 000 119 111 679 999 999 992 078 698 316 834 428 627 686 293 111 540 940 8;
  • 16) 0.000 000 119 111 679 999 999 992 078 698 316 834 428 627 686 293 111 540 940 8 × 2 = 0 + 0.000 000 238 223 359 999 999 984 157 396 633 668 857 255 372 586 223 081 881 6;
  • 17) 0.000 000 238 223 359 999 999 984 157 396 633 668 857 255 372 586 223 081 881 6 × 2 = 0 + 0.000 000 476 446 719 999 999 968 314 793 267 337 714 510 745 172 446 163 763 2;
  • 18) 0.000 000 476 446 719 999 999 968 314 793 267 337 714 510 745 172 446 163 763 2 × 2 = 0 + 0.000 000 952 893 439 999 999 936 629 586 534 675 429 021 490 344 892 327 526 4;
  • 19) 0.000 000 952 893 439 999 999 936 629 586 534 675 429 021 490 344 892 327 526 4 × 2 = 0 + 0.000 001 905 786 879 999 999 873 259 173 069 350 858 042 980 689 784 655 052 8;
  • 20) 0.000 001 905 786 879 999 999 873 259 173 069 350 858 042 980 689 784 655 052 8 × 2 = 0 + 0.000 003 811 573 759 999 999 746 518 346 138 701 716 085 961 379 569 310 105 6;
  • 21) 0.000 003 811 573 759 999 999 746 518 346 138 701 716 085 961 379 569 310 105 6 × 2 = 0 + 0.000 007 623 147 519 999 999 493 036 692 277 403 432 171 922 759 138 620 211 2;
  • 22) 0.000 007 623 147 519 999 999 493 036 692 277 403 432 171 922 759 138 620 211 2 × 2 = 0 + 0.000 015 246 295 039 999 998 986 073 384 554 806 864 343 845 518 277 240 422 4;
  • 23) 0.000 015 246 295 039 999 998 986 073 384 554 806 864 343 845 518 277 240 422 4 × 2 = 0 + 0.000 030 492 590 079 999 997 972 146 769 109 613 728 687 691 036 554 480 844 8;
  • 24) 0.000 030 492 590 079 999 997 972 146 769 109 613 728 687 691 036 554 480 844 8 × 2 = 0 + 0.000 060 985 180 159 999 995 944 293 538 219 227 457 375 382 073 108 961 689 6;
  • 25) 0.000 060 985 180 159 999 995 944 293 538 219 227 457 375 382 073 108 961 689 6 × 2 = 0 + 0.000 121 970 360 319 999 991 888 587 076 438 454 914 750 764 146 217 923 379 2;
  • 26) 0.000 121 970 360 319 999 991 888 587 076 438 454 914 750 764 146 217 923 379 2 × 2 = 0 + 0.000 243 940 720 639 999 983 777 174 152 876 909 829 501 528 292 435 846 758 4;
  • 27) 0.000 243 940 720 639 999 983 777 174 152 876 909 829 501 528 292 435 846 758 4 × 2 = 0 + 0.000 487 881 441 279 999 967 554 348 305 753 819 659 003 056 584 871 693 516 8;
  • 28) 0.000 487 881 441 279 999 967 554 348 305 753 819 659 003 056 584 871 693 516 8 × 2 = 0 + 0.000 975 762 882 559 999 935 108 696 611 507 639 318 006 113 169 743 387 033 6;
  • 29) 0.000 975 762 882 559 999 935 108 696 611 507 639 318 006 113 169 743 387 033 6 × 2 = 0 + 0.001 951 525 765 119 999 870 217 393 223 015 278 636 012 226 339 486 774 067 2;
  • 30) 0.001 951 525 765 119 999 870 217 393 223 015 278 636 012 226 339 486 774 067 2 × 2 = 0 + 0.003 903 051 530 239 999 740 434 786 446 030 557 272 024 452 678 973 548 134 4;
  • 31) 0.003 903 051 530 239 999 740 434 786 446 030 557 272 024 452 678 973 548 134 4 × 2 = 0 + 0.007 806 103 060 479 999 480 869 572 892 061 114 544 048 905 357 947 096 268 8;
  • 32) 0.007 806 103 060 479 999 480 869 572 892 061 114 544 048 905 357 947 096 268 8 × 2 = 0 + 0.015 612 206 120 959 998 961 739 145 784 122 229 088 097 810 715 894 192 537 6;
  • 33) 0.015 612 206 120 959 998 961 739 145 784 122 229 088 097 810 715 894 192 537 6 × 2 = 0 + 0.031 224 412 241 919 997 923 478 291 568 244 458 176 195 621 431 788 385 075 2;
  • 34) 0.031 224 412 241 919 997 923 478 291 568 244 458 176 195 621 431 788 385 075 2 × 2 = 0 + 0.062 448 824 483 839 995 846 956 583 136 488 916 352 391 242 863 576 770 150 4;
  • 35) 0.062 448 824 483 839 995 846 956 583 136 488 916 352 391 242 863 576 770 150 4 × 2 = 0 + 0.124 897 648 967 679 991 693 913 166 272 977 832 704 782 485 727 153 540 300 8;
  • 36) 0.124 897 648 967 679 991 693 913 166 272 977 832 704 782 485 727 153 540 300 8 × 2 = 0 + 0.249 795 297 935 359 983 387 826 332 545 955 665 409 564 971 454 307 080 601 6;
  • 37) 0.249 795 297 935 359 983 387 826 332 545 955 665 409 564 971 454 307 080 601 6 × 2 = 0 + 0.499 590 595 870 719 966 775 652 665 091 911 330 819 129 942 908 614 161 203 2;
  • 38) 0.499 590 595 870 719 966 775 652 665 091 911 330 819 129 942 908 614 161 203 2 × 2 = 0 + 0.999 181 191 741 439 933 551 305 330 183 822 661 638 259 885 817 228 322 406 4;
  • 39) 0.999 181 191 741 439 933 551 305 330 183 822 661 638 259 885 817 228 322 406 4 × 2 = 1 + 0.998 362 383 482 879 867 102 610 660 367 645 323 276 519 771 634 456 644 812 8;
  • 40) 0.998 362 383 482 879 867 102 610 660 367 645 323 276 519 771 634 456 644 812 8 × 2 = 1 + 0.996 724 766 965 759 734 205 221 320 735 290 646 553 039 543 268 913 289 625 6;
  • 41) 0.996 724 766 965 759 734 205 221 320 735 290 646 553 039 543 268 913 289 625 6 × 2 = 1 + 0.993 449 533 931 519 468 410 442 641 470 581 293 106 079 086 537 826 579 251 2;
  • 42) 0.993 449 533 931 519 468 410 442 641 470 581 293 106 079 086 537 826 579 251 2 × 2 = 1 + 0.986 899 067 863 038 936 820 885 282 941 162 586 212 158 173 075 653 158 502 4;
  • 43) 0.986 899 067 863 038 936 820 885 282 941 162 586 212 158 173 075 653 158 502 4 × 2 = 1 + 0.973 798 135 726 077 873 641 770 565 882 325 172 424 316 346 151 306 317 004 8;
  • 44) 0.973 798 135 726 077 873 641 770 565 882 325 172 424 316 346 151 306 317 004 8 × 2 = 1 + 0.947 596 271 452 155 747 283 541 131 764 650 344 848 632 692 302 612 634 009 6;
  • 45) 0.947 596 271 452 155 747 283 541 131 764 650 344 848 632 692 302 612 634 009 6 × 2 = 1 + 0.895 192 542 904 311 494 567 082 263 529 300 689 697 265 384 605 225 268 019 2;
  • 46) 0.895 192 542 904 311 494 567 082 263 529 300 689 697 265 384 605 225 268 019 2 × 2 = 1 + 0.790 385 085 808 622 989 134 164 527 058 601 379 394 530 769 210 450 536 038 4;
  • 47) 0.790 385 085 808 622 989 134 164 527 058 601 379 394 530 769 210 450 536 038 4 × 2 = 1 + 0.580 770 171 617 245 978 268 329 054 117 202 758 789 061 538 420 901 072 076 8;
  • 48) 0.580 770 171 617 245 978 268 329 054 117 202 758 789 061 538 420 901 072 076 8 × 2 = 1 + 0.161 540 343 234 491 956 536 658 108 234 405 517 578 123 076 841 802 144 153 6;
  • 49) 0.161 540 343 234 491 956 536 658 108 234 405 517 578 123 076 841 802 144 153 6 × 2 = 0 + 0.323 080 686 468 983 913 073 316 216 468 811 035 156 246 153 683 604 288 307 2;
  • 50) 0.323 080 686 468 983 913 073 316 216 468 811 035 156 246 153 683 604 288 307 2 × 2 = 0 + 0.646 161 372 937 967 826 146 632 432 937 622 070 312 492 307 367 208 576 614 4;
  • 51) 0.646 161 372 937 967 826 146 632 432 937 622 070 312 492 307 367 208 576 614 4 × 2 = 1 + 0.292 322 745 875 935 652 293 264 865 875 244 140 624 984 614 734 417 153 228 8;
  • 52) 0.292 322 745 875 935 652 293 264 865 875 244 140 624 984 614 734 417 153 228 8 × 2 = 0 + 0.584 645 491 751 871 304 586 529 731 750 488 281 249 969 229 468 834 306 457 6;
  • 53) 0.584 645 491 751 871 304 586 529 731 750 488 281 249 969 229 468 834 306 457 6 × 2 = 1 + 0.169 290 983 503 742 609 173 059 463 500 976 562 499 938 458 937 668 612 915 2;
  • 54) 0.169 290 983 503 742 609 173 059 463 500 976 562 499 938 458 937 668 612 915 2 × 2 = 0 + 0.338 581 967 007 485 218 346 118 927 001 953 124 999 876 917 875 337 225 830 4;
  • 55) 0.338 581 967 007 485 218 346 118 927 001 953 124 999 876 917 875 337 225 830 4 × 2 = 0 + 0.677 163 934 014 970 436 692 237 854 003 906 249 999 753 835 750 674 451 660 8;
  • 56) 0.677 163 934 014 970 436 692 237 854 003 906 249 999 753 835 750 674 451 660 8 × 2 = 1 + 0.354 327 868 029 940 873 384 475 708 007 812 499 999 507 671 501 348 903 321 6;
  • 57) 0.354 327 868 029 940 873 384 475 708 007 812 499 999 507 671 501 348 903 321 6 × 2 = 0 + 0.708 655 736 059 881 746 768 951 416 015 624 999 999 015 343 002 697 806 643 2;
  • 58) 0.708 655 736 059 881 746 768 951 416 015 624 999 999 015 343 002 697 806 643 2 × 2 = 1 + 0.417 311 472 119 763 493 537 902 832 031 249 999 998 030 686 005 395 613 286 4;
  • 59) 0.417 311 472 119 763 493 537 902 832 031 249 999 998 030 686 005 395 613 286 4 × 2 = 0 + 0.834 622 944 239 526 987 075 805 664 062 499 999 996 061 372 010 791 226 572 8;
  • 60) 0.834 622 944 239 526 987 075 805 664 062 499 999 996 061 372 010 791 226 572 8 × 2 = 1 + 0.669 245 888 479 053 974 151 611 328 124 999 999 992 122 744 021 582 453 145 6;
  • 61) 0.669 245 888 479 053 974 151 611 328 124 999 999 992 122 744 021 582 453 145 6 × 2 = 1 + 0.338 491 776 958 107 948 303 222 656 249 999 999 984 245 488 043 164 906 291 2;
  • 62) 0.338 491 776 958 107 948 303 222 656 249 999 999 984 245 488 043 164 906 291 2 × 2 = 0 + 0.676 983 553 916 215 896 606 445 312 499 999 999 968 490 976 086 329 812 582 4;
  • 63) 0.676 983 553 916 215 896 606 445 312 499 999 999 968 490 976 086 329 812 582 4 × 2 = 1 + 0.353 967 107 832 431 793 212 890 624 999 999 999 936 981 952 172 659 625 164 8;
  • 64) 0.353 967 107 832 431 793 212 890 624 999 999 999 936 981 952 172 659 625 164 8 × 2 = 0 + 0.707 934 215 664 863 586 425 781 249 999 999 999 873 963 904 345 319 250 329 6;
  • 65) 0.707 934 215 664 863 586 425 781 249 999 999 999 873 963 904 345 319 250 329 6 × 2 = 1 + 0.415 868 431 329 727 172 851 562 499 999 999 999 747 927 808 690 638 500 659 2;
  • 66) 0.415 868 431 329 727 172 851 562 499 999 999 999 747 927 808 690 638 500 659 2 × 2 = 0 + 0.831 736 862 659 454 345 703 124 999 999 999 999 495 855 617 381 277 001 318 4;
  • 67) 0.831 736 862 659 454 345 703 124 999 999 999 999 495 855 617 381 277 001 318 4 × 2 = 1 + 0.663 473 725 318 908 691 406 249 999 999 999 998 991 711 234 762 554 002 636 8;
  • 68) 0.663 473 725 318 908 691 406 249 999 999 999 998 991 711 234 762 554 002 636 8 × 2 = 1 + 0.326 947 450 637 817 382 812 499 999 999 999 997 983 422 469 525 108 005 273 6;
  • 69) 0.326 947 450 637 817 382 812 499 999 999 999 997 983 422 469 525 108 005 273 6 × 2 = 0 + 0.653 894 901 275 634 765 624 999 999 999 999 995 966 844 939 050 216 010 547 2;
  • 70) 0.653 894 901 275 634 765 624 999 999 999 999 995 966 844 939 050 216 010 547 2 × 2 = 1 + 0.307 789 802 551 269 531 249 999 999 999 999 991 933 689 878 100 432 021 094 4;
  • 71) 0.307 789 802 551 269 531 249 999 999 999 999 991 933 689 878 100 432 021 094 4 × 2 = 0 + 0.615 579 605 102 539 062 499 999 999 999 999 983 867 379 756 200 864 042 188 8;
  • 72) 0.615 579 605 102 539 062 499 999 999 999 999 983 867 379 756 200 864 042 188 8 × 2 = 1 + 0.231 159 210 205 078 124 999 999 999 999 999 967 734 759 512 401 728 084 377 6;
  • 73) 0.231 159 210 205 078 124 999 999 999 999 999 967 734 759 512 401 728 084 377 6 × 2 = 0 + 0.462 318 420 410 156 249 999 999 999 999 999 935 469 519 024 803 456 168 755 2;
  • 74) 0.462 318 420 410 156 249 999 999 999 999 999 935 469 519 024 803 456 168 755 2 × 2 = 0 + 0.924 636 840 820 312 499 999 999 999 999 999 870 939 038 049 606 912 337 510 4;
  • 75) 0.924 636 840 820 312 499 999 999 999 999 999 870 939 038 049 606 912 337 510 4 × 2 = 1 + 0.849 273 681 640 624 999 999 999 999 999 999 741 878 076 099 213 824 675 020 8;
  • 76) 0.849 273 681 640 624 999 999 999 999 999 999 741 878 076 099 213 824 675 020 8 × 2 = 1 + 0.698 547 363 281 249 999 999 999 999 999 999 483 756 152 198 427 649 350 041 6;
  • 77) 0.698 547 363 281 249 999 999 999 999 999 999 483 756 152 198 427 649 350 041 6 × 2 = 1 + 0.397 094 726 562 499 999 999 999 999 999 998 967 512 304 396 855 298 700 083 2;
  • 78) 0.397 094 726 562 499 999 999 999 999 999 998 967 512 304 396 855 298 700 083 2 × 2 = 0 + 0.794 189 453 124 999 999 999 999 999 999 997 935 024 608 793 710 597 400 166 4;
  • 79) 0.794 189 453 124 999 999 999 999 999 999 997 935 024 608 793 710 597 400 166 4 × 2 = 1 + 0.588 378 906 249 999 999 999 999 999 999 995 870 049 217 587 421 194 800 332 8;
  • 80) 0.588 378 906 249 999 999 999 999 999 999 995 870 049 217 587 421 194 800 332 8 × 2 = 1 + 0.176 757 812 499 999 999 999 999 999 999 991 740 098 435 174 842 389 600 665 6;
  • 81) 0.176 757 812 499 999 999 999 999 999 999 991 740 098 435 174 842 389 600 665 6 × 2 = 0 + 0.353 515 624 999 999 999 999 999 999 999 983 480 196 870 349 684 779 201 331 2;
  • 82) 0.353 515 624 999 999 999 999 999 999 999 983 480 196 870 349 684 779 201 331 2 × 2 = 0 + 0.707 031 249 999 999 999 999 999 999 999 966 960 393 740 699 369 558 402 662 4;
  • 83) 0.707 031 249 999 999 999 999 999 999 999 966 960 393 740 699 369 558 402 662 4 × 2 = 1 + 0.414 062 499 999 999 999 999 999 999 999 933 920 787 481 398 739 116 805 324 8;
  • 84) 0.414 062 499 999 999 999 999 999 999 999 933 920 787 481 398 739 116 805 324 8 × 2 = 0 + 0.828 124 999 999 999 999 999 999 999 999 867 841 574 962 797 478 233 610 649 6;
  • 85) 0.828 124 999 999 999 999 999 999 999 999 867 841 574 962 797 478 233 610 649 6 × 2 = 1 + 0.656 249 999 999 999 999 999 999 999 999 735 683 149 925 594 956 467 221 299 2;
  • 86) 0.656 249 999 999 999 999 999 999 999 999 735 683 149 925 594 956 467 221 299 2 × 2 = 1 + 0.312 499 999 999 999 999 999 999 999 999 471 366 299 851 189 912 934 442 598 4;
  • 87) 0.312 499 999 999 999 999 999 999 999 999 471 366 299 851 189 912 934 442 598 4 × 2 = 0 + 0.624 999 999 999 999 999 999 999 999 998 942 732 599 702 379 825 868 885 196 8;
  • 88) 0.624 999 999 999 999 999 999 999 999 998 942 732 599 702 379 825 868 885 196 8 × 2 = 1 + 0.249 999 999 999 999 999 999 999 999 997 885 465 199 404 759 651 737 770 393 6;
  • 89) 0.249 999 999 999 999 999 999 999 999 997 885 465 199 404 759 651 737 770 393 6 × 2 = 0 + 0.499 999 999 999 999 999 999 999 999 995 770 930 398 809 519 303 475 540 787 2;
  • 90) 0.499 999 999 999 999 999 999 999 999 995 770 930 398 809 519 303 475 540 787 2 × 2 = 0 + 0.999 999 999 999 999 999 999 999 999 991 541 860 797 619 038 606 951 081 574 4;
  • 91) 0.999 999 999 999 999 999 999 999 999 991 541 860 797 619 038 606 951 081 574 4 × 2 = 1 + 0.999 999 999 999 999 999 999 999 999 983 083 721 595 238 077 213 902 163 148 8;

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:


0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 023 1(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:

0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 023 1(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:


0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 023 1(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 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 023 1 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