0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 029 932 430 448 4 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 029 932 430 448 4(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
0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 029 932 430 448 4(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: 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 029 932 430 448 4.

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) 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 029 932 430 448 4 × 2 = 0 + 0.000 000 000 007 269 999 999 999 999 516 522 114 064 601 356 670 305 976 059 864 860 896 8;
  • 2) 0.000 000 000 007 269 999 999 999 999 516 522 114 064 601 356 670 305 976 059 864 860 896 8 × 2 = 0 + 0.000 000 000 014 539 999 999 999 999 033 044 228 129 202 713 340 611 952 119 729 721 793 6;
  • 3) 0.000 000 000 014 539 999 999 999 999 033 044 228 129 202 713 340 611 952 119 729 721 793 6 × 2 = 0 + 0.000 000 000 029 079 999 999 999 998 066 088 456 258 405 426 681 223 904 239 459 443 587 2;
  • 4) 0.000 000 000 029 079 999 999 999 998 066 088 456 258 405 426 681 223 904 239 459 443 587 2 × 2 = 0 + 0.000 000 000 058 159 999 999 999 996 132 176 912 516 810 853 362 447 808 478 918 887 174 4;
  • 5) 0.000 000 000 058 159 999 999 999 996 132 176 912 516 810 853 362 447 808 478 918 887 174 4 × 2 = 0 + 0.000 000 000 116 319 999 999 999 992 264 353 825 033 621 706 724 895 616 957 837 774 348 8;
  • 6) 0.000 000 000 116 319 999 999 999 992 264 353 825 033 621 706 724 895 616 957 837 774 348 8 × 2 = 0 + 0.000 000 000 232 639 999 999 999 984 528 707 650 067 243 413 449 791 233 915 675 548 697 6;
  • 7) 0.000 000 000 232 639 999 999 999 984 528 707 650 067 243 413 449 791 233 915 675 548 697 6 × 2 = 0 + 0.000 000 000 465 279 999 999 999 969 057 415 300 134 486 826 899 582 467 831 351 097 395 2;
  • 8) 0.000 000 000 465 279 999 999 999 969 057 415 300 134 486 826 899 582 467 831 351 097 395 2 × 2 = 0 + 0.000 000 000 930 559 999 999 999 938 114 830 600 268 973 653 799 164 935 662 702 194 790 4;
  • 9) 0.000 000 000 930 559 999 999 999 938 114 830 600 268 973 653 799 164 935 662 702 194 790 4 × 2 = 0 + 0.000 000 001 861 119 999 999 999 876 229 661 200 537 947 307 598 329 871 325 404 389 580 8;
  • 10) 0.000 000 001 861 119 999 999 999 876 229 661 200 537 947 307 598 329 871 325 404 389 580 8 × 2 = 0 + 0.000 000 003 722 239 999 999 999 752 459 322 401 075 894 615 196 659 742 650 808 779 161 6;
  • 11) 0.000 000 003 722 239 999 999 999 752 459 322 401 075 894 615 196 659 742 650 808 779 161 6 × 2 = 0 + 0.000 000 007 444 479 999 999 999 504 918 644 802 151 789 230 393 319 485 301 617 558 323 2;
  • 12) 0.000 000 007 444 479 999 999 999 504 918 644 802 151 789 230 393 319 485 301 617 558 323 2 × 2 = 0 + 0.000 000 014 888 959 999 999 999 009 837 289 604 303 578 460 786 638 970 603 235 116 646 4;
  • 13) 0.000 000 014 888 959 999 999 999 009 837 289 604 303 578 460 786 638 970 603 235 116 646 4 × 2 = 0 + 0.000 000 029 777 919 999 999 998 019 674 579 208 607 156 921 573 277 941 206 470 233 292 8;
  • 14) 0.000 000 029 777 919 999 999 998 019 674 579 208 607 156 921 573 277 941 206 470 233 292 8 × 2 = 0 + 0.000 000 059 555 839 999 999 996 039 349 158 417 214 313 843 146 555 882 412 940 466 585 6;
  • 15) 0.000 000 059 555 839 999 999 996 039 349 158 417 214 313 843 146 555 882 412 940 466 585 6 × 2 = 0 + 0.000 000 119 111 679 999 999 992 078 698 316 834 428 627 686 293 111 764 825 880 933 171 2;
  • 16) 0.000 000 119 111 679 999 999 992 078 698 316 834 428 627 686 293 111 764 825 880 933 171 2 × 2 = 0 + 0.000 000 238 223 359 999 999 984 157 396 633 668 857 255 372 586 223 529 651 761 866 342 4;
  • 17) 0.000 000 238 223 359 999 999 984 157 396 633 668 857 255 372 586 223 529 651 761 866 342 4 × 2 = 0 + 0.000 000 476 446 719 999 999 968 314 793 267 337 714 510 745 172 447 059 303 523 732 684 8;
  • 18) 0.000 000 476 446 719 999 999 968 314 793 267 337 714 510 745 172 447 059 303 523 732 684 8 × 2 = 0 + 0.000 000 952 893 439 999 999 936 629 586 534 675 429 021 490 344 894 118 607 047 465 369 6;
  • 19) 0.000 000 952 893 439 999 999 936 629 586 534 675 429 021 490 344 894 118 607 047 465 369 6 × 2 = 0 + 0.000 001 905 786 879 999 999 873 259 173 069 350 858 042 980 689 788 237 214 094 930 739 2;
  • 20) 0.000 001 905 786 879 999 999 873 259 173 069 350 858 042 980 689 788 237 214 094 930 739 2 × 2 = 0 + 0.000 003 811 573 759 999 999 746 518 346 138 701 716 085 961 379 576 474 428 189 861 478 4;
  • 21) 0.000 003 811 573 759 999 999 746 518 346 138 701 716 085 961 379 576 474 428 189 861 478 4 × 2 = 0 + 0.000 007 623 147 519 999 999 493 036 692 277 403 432 171 922 759 152 948 856 379 722 956 8;
  • 22) 0.000 007 623 147 519 999 999 493 036 692 277 403 432 171 922 759 152 948 856 379 722 956 8 × 2 = 0 + 0.000 015 246 295 039 999 998 986 073 384 554 806 864 343 845 518 305 897 712 759 445 913 6;
  • 23) 0.000 015 246 295 039 999 998 986 073 384 554 806 864 343 845 518 305 897 712 759 445 913 6 × 2 = 0 + 0.000 030 492 590 079 999 997 972 146 769 109 613 728 687 691 036 611 795 425 518 891 827 2;
  • 24) 0.000 030 492 590 079 999 997 972 146 769 109 613 728 687 691 036 611 795 425 518 891 827 2 × 2 = 0 + 0.000 060 985 180 159 999 995 944 293 538 219 227 457 375 382 073 223 590 851 037 783 654 4;
  • 25) 0.000 060 985 180 159 999 995 944 293 538 219 227 457 375 382 073 223 590 851 037 783 654 4 × 2 = 0 + 0.000 121 970 360 319 999 991 888 587 076 438 454 914 750 764 146 447 181 702 075 567 308 8;
  • 26) 0.000 121 970 360 319 999 991 888 587 076 438 454 914 750 764 146 447 181 702 075 567 308 8 × 2 = 0 + 0.000 243 940 720 639 999 983 777 174 152 876 909 829 501 528 292 894 363 404 151 134 617 6;
  • 27) 0.000 243 940 720 639 999 983 777 174 152 876 909 829 501 528 292 894 363 404 151 134 617 6 × 2 = 0 + 0.000 487 881 441 279 999 967 554 348 305 753 819 659 003 056 585 788 726 808 302 269 235 2;
  • 28) 0.000 487 881 441 279 999 967 554 348 305 753 819 659 003 056 585 788 726 808 302 269 235 2 × 2 = 0 + 0.000 975 762 882 559 999 935 108 696 611 507 639 318 006 113 171 577 453 616 604 538 470 4;
  • 29) 0.000 975 762 882 559 999 935 108 696 611 507 639 318 006 113 171 577 453 616 604 538 470 4 × 2 = 0 + 0.001 951 525 765 119 999 870 217 393 223 015 278 636 012 226 343 154 907 233 209 076 940 8;
  • 30) 0.001 951 525 765 119 999 870 217 393 223 015 278 636 012 226 343 154 907 233 209 076 940 8 × 2 = 0 + 0.003 903 051 530 239 999 740 434 786 446 030 557 272 024 452 686 309 814 466 418 153 881 6;
  • 31) 0.003 903 051 530 239 999 740 434 786 446 030 557 272 024 452 686 309 814 466 418 153 881 6 × 2 = 0 + 0.007 806 103 060 479 999 480 869 572 892 061 114 544 048 905 372 619 628 932 836 307 763 2;
  • 32) 0.007 806 103 060 479 999 480 869 572 892 061 114 544 048 905 372 619 628 932 836 307 763 2 × 2 = 0 + 0.015 612 206 120 959 998 961 739 145 784 122 229 088 097 810 745 239 257 865 672 615 526 4;
  • 33) 0.015 612 206 120 959 998 961 739 145 784 122 229 088 097 810 745 239 257 865 672 615 526 4 × 2 = 0 + 0.031 224 412 241 919 997 923 478 291 568 244 458 176 195 621 490 478 515 731 345 231 052 8;
  • 34) 0.031 224 412 241 919 997 923 478 291 568 244 458 176 195 621 490 478 515 731 345 231 052 8 × 2 = 0 + 0.062 448 824 483 839 995 846 956 583 136 488 916 352 391 242 980 957 031 462 690 462 105 6;
  • 35) 0.062 448 824 483 839 995 846 956 583 136 488 916 352 391 242 980 957 031 462 690 462 105 6 × 2 = 0 + 0.124 897 648 967 679 991 693 913 166 272 977 832 704 782 485 961 914 062 925 380 924 211 2;
  • 36) 0.124 897 648 967 679 991 693 913 166 272 977 832 704 782 485 961 914 062 925 380 924 211 2 × 2 = 0 + 0.249 795 297 935 359 983 387 826 332 545 955 665 409 564 971 923 828 125 850 761 848 422 4;
  • 37) 0.249 795 297 935 359 983 387 826 332 545 955 665 409 564 971 923 828 125 850 761 848 422 4 × 2 = 0 + 0.499 590 595 870 719 966 775 652 665 091 911 330 819 129 943 847 656 251 701 523 696 844 8;
  • 38) 0.499 590 595 870 719 966 775 652 665 091 911 330 819 129 943 847 656 251 701 523 696 844 8 × 2 = 0 + 0.999 181 191 741 439 933 551 305 330 183 822 661 638 259 887 695 312 503 403 047 393 689 6;
  • 39) 0.999 181 191 741 439 933 551 305 330 183 822 661 638 259 887 695 312 503 403 047 393 689 6 × 2 = 1 + 0.998 362 383 482 879 867 102 610 660 367 645 323 276 519 775 390 625 006 806 094 787 379 2;
  • 40) 0.998 362 383 482 879 867 102 610 660 367 645 323 276 519 775 390 625 006 806 094 787 379 2 × 2 = 1 + 0.996 724 766 965 759 734 205 221 320 735 290 646 553 039 550 781 250 013 612 189 574 758 4;
  • 41) 0.996 724 766 965 759 734 205 221 320 735 290 646 553 039 550 781 250 013 612 189 574 758 4 × 2 = 1 + 0.993 449 533 931 519 468 410 442 641 470 581 293 106 079 101 562 500 027 224 379 149 516 8;
  • 42) 0.993 449 533 931 519 468 410 442 641 470 581 293 106 079 101 562 500 027 224 379 149 516 8 × 2 = 1 + 0.986 899 067 863 038 936 820 885 282 941 162 586 212 158 203 125 000 054 448 758 299 033 6;
  • 43) 0.986 899 067 863 038 936 820 885 282 941 162 586 212 158 203 125 000 054 448 758 299 033 6 × 2 = 1 + 0.973 798 135 726 077 873 641 770 565 882 325 172 424 316 406 250 000 108 897 516 598 067 2;
  • 44) 0.973 798 135 726 077 873 641 770 565 882 325 172 424 316 406 250 000 108 897 516 598 067 2 × 2 = 1 + 0.947 596 271 452 155 747 283 541 131 764 650 344 848 632 812 500 000 217 795 033 196 134 4;
  • 45) 0.947 596 271 452 155 747 283 541 131 764 650 344 848 632 812 500 000 217 795 033 196 134 4 × 2 = 1 + 0.895 192 542 904 311 494 567 082 263 529 300 689 697 265 625 000 000 435 590 066 392 268 8;
  • 46) 0.895 192 542 904 311 494 567 082 263 529 300 689 697 265 625 000 000 435 590 066 392 268 8 × 2 = 1 + 0.790 385 085 808 622 989 134 164 527 058 601 379 394 531 250 000 000 871 180 132 784 537 6;
  • 47) 0.790 385 085 808 622 989 134 164 527 058 601 379 394 531 250 000 000 871 180 132 784 537 6 × 2 = 1 + 0.580 770 171 617 245 978 268 329 054 117 202 758 789 062 500 000 001 742 360 265 569 075 2;
  • 48) 0.580 770 171 617 245 978 268 329 054 117 202 758 789 062 500 000 001 742 360 265 569 075 2 × 2 = 1 + 0.161 540 343 234 491 956 536 658 108 234 405 517 578 125 000 000 003 484 720 531 138 150 4;
  • 49) 0.161 540 343 234 491 956 536 658 108 234 405 517 578 125 000 000 003 484 720 531 138 150 4 × 2 = 0 + 0.323 080 686 468 983 913 073 316 216 468 811 035 156 250 000 000 006 969 441 062 276 300 8;
  • 50) 0.323 080 686 468 983 913 073 316 216 468 811 035 156 250 000 000 006 969 441 062 276 300 8 × 2 = 0 + 0.646 161 372 937 967 826 146 632 432 937 622 070 312 500 000 000 013 938 882 124 552 601 6;
  • 51) 0.646 161 372 937 967 826 146 632 432 937 622 070 312 500 000 000 013 938 882 124 552 601 6 × 2 = 1 + 0.292 322 745 875 935 652 293 264 865 875 244 140 625 000 000 000 027 877 764 249 105 203 2;
  • 52) 0.292 322 745 875 935 652 293 264 865 875 244 140 625 000 000 000 027 877 764 249 105 203 2 × 2 = 0 + 0.584 645 491 751 871 304 586 529 731 750 488 281 250 000 000 000 055 755 528 498 210 406 4;
  • 53) 0.584 645 491 751 871 304 586 529 731 750 488 281 250 000 000 000 055 755 528 498 210 406 4 × 2 = 1 + 0.169 290 983 503 742 609 173 059 463 500 976 562 500 000 000 000 111 511 056 996 420 812 8;
  • 54) 0.169 290 983 503 742 609 173 059 463 500 976 562 500 000 000 000 111 511 056 996 420 812 8 × 2 = 0 + 0.338 581 967 007 485 218 346 118 927 001 953 125 000 000 000 000 223 022 113 992 841 625 6;
  • 55) 0.338 581 967 007 485 218 346 118 927 001 953 125 000 000 000 000 223 022 113 992 841 625 6 × 2 = 0 + 0.677 163 934 014 970 436 692 237 854 003 906 250 000 000 000 000 446 044 227 985 683 251 2;
  • 56) 0.677 163 934 014 970 436 692 237 854 003 906 250 000 000 000 000 446 044 227 985 683 251 2 × 2 = 1 + 0.354 327 868 029 940 873 384 475 708 007 812 500 000 000 000 000 892 088 455 971 366 502 4;
  • 57) 0.354 327 868 029 940 873 384 475 708 007 812 500 000 000 000 000 892 088 455 971 366 502 4 × 2 = 0 + 0.708 655 736 059 881 746 768 951 416 015 625 000 000 000 000 001 784 176 911 942 733 004 8;
  • 58) 0.708 655 736 059 881 746 768 951 416 015 625 000 000 000 000 001 784 176 911 942 733 004 8 × 2 = 1 + 0.417 311 472 119 763 493 537 902 832 031 250 000 000 000 000 003 568 353 823 885 466 009 6;
  • 59) 0.417 311 472 119 763 493 537 902 832 031 250 000 000 000 000 003 568 353 823 885 466 009 6 × 2 = 0 + 0.834 622 944 239 526 987 075 805 664 062 500 000 000 000 000 007 136 707 647 770 932 019 2;
  • 60) 0.834 622 944 239 526 987 075 805 664 062 500 000 000 000 000 007 136 707 647 770 932 019 2 × 2 = 1 + 0.669 245 888 479 053 974 151 611 328 125 000 000 000 000 000 014 273 415 295 541 864 038 4;
  • 61) 0.669 245 888 479 053 974 151 611 328 125 000 000 000 000 000 014 273 415 295 541 864 038 4 × 2 = 1 + 0.338 491 776 958 107 948 303 222 656 250 000 000 000 000 000 028 546 830 591 083 728 076 8;
  • 62) 0.338 491 776 958 107 948 303 222 656 250 000 000 000 000 000 028 546 830 591 083 728 076 8 × 2 = 0 + 0.676 983 553 916 215 896 606 445 312 500 000 000 000 000 000 057 093 661 182 167 456 153 6;
  • 63) 0.676 983 553 916 215 896 606 445 312 500 000 000 000 000 000 057 093 661 182 167 456 153 6 × 2 = 1 + 0.353 967 107 832 431 793 212 890 625 000 000 000 000 000 000 114 187 322 364 334 912 307 2;
  • 64) 0.353 967 107 832 431 793 212 890 625 000 000 000 000 000 000 114 187 322 364 334 912 307 2 × 2 = 0 + 0.707 934 215 664 863 586 425 781 250 000 000 000 000 000 000 228 374 644 728 669 824 614 4;
  • 65) 0.707 934 215 664 863 586 425 781 250 000 000 000 000 000 000 228 374 644 728 669 824 614 4 × 2 = 1 + 0.415 868 431 329 727 172 851 562 500 000 000 000 000 000 000 456 749 289 457 339 649 228 8;
  • 66) 0.415 868 431 329 727 172 851 562 500 000 000 000 000 000 000 456 749 289 457 339 649 228 8 × 2 = 0 + 0.831 736 862 659 454 345 703 125 000 000 000 000 000 000 000 913 498 578 914 679 298 457 6;
  • 67) 0.831 736 862 659 454 345 703 125 000 000 000 000 000 000 000 913 498 578 914 679 298 457 6 × 2 = 1 + 0.663 473 725 318 908 691 406 250 000 000 000 000 000 000 001 826 997 157 829 358 596 915 2;
  • 68) 0.663 473 725 318 908 691 406 250 000 000 000 000 000 000 001 826 997 157 829 358 596 915 2 × 2 = 1 + 0.326 947 450 637 817 382 812 500 000 000 000 000 000 000 003 653 994 315 658 717 193 830 4;
  • 69) 0.326 947 450 637 817 382 812 500 000 000 000 000 000 000 003 653 994 315 658 717 193 830 4 × 2 = 0 + 0.653 894 901 275 634 765 625 000 000 000 000 000 000 000 007 307 988 631 317 434 387 660 8;
  • 70) 0.653 894 901 275 634 765 625 000 000 000 000 000 000 000 007 307 988 631 317 434 387 660 8 × 2 = 1 + 0.307 789 802 551 269 531 250 000 000 000 000 000 000 000 014 615 977 262 634 868 775 321 6;
  • 71) 0.307 789 802 551 269 531 250 000 000 000 000 000 000 000 014 615 977 262 634 868 775 321 6 × 2 = 0 + 0.615 579 605 102 539 062 500 000 000 000 000 000 000 000 029 231 954 525 269 737 550 643 2;
  • 72) 0.615 579 605 102 539 062 500 000 000 000 000 000 000 000 029 231 954 525 269 737 550 643 2 × 2 = 1 + 0.231 159 210 205 078 125 000 000 000 000 000 000 000 000 058 463 909 050 539 475 101 286 4;
  • 73) 0.231 159 210 205 078 125 000 000 000 000 000 000 000 000 058 463 909 050 539 475 101 286 4 × 2 = 0 + 0.462 318 420 410 156 250 000 000 000 000 000 000 000 000 116 927 818 101 078 950 202 572 8;
  • 74) 0.462 318 420 410 156 250 000 000 000 000 000 000 000 000 116 927 818 101 078 950 202 572 8 × 2 = 0 + 0.924 636 840 820 312 500 000 000 000 000 000 000 000 000 233 855 636 202 157 900 405 145 6;
  • 75) 0.924 636 840 820 312 500 000 000 000 000 000 000 000 000 233 855 636 202 157 900 405 145 6 × 2 = 1 + 0.849 273 681 640 625 000 000 000 000 000 000 000 000 000 467 711 272 404 315 800 810 291 2;
  • 76) 0.849 273 681 640 625 000 000 000 000 000 000 000 000 000 467 711 272 404 315 800 810 291 2 × 2 = 1 + 0.698 547 363 281 250 000 000 000 000 000 000 000 000 000 935 422 544 808 631 601 620 582 4;
  • 77) 0.698 547 363 281 250 000 000 000 000 000 000 000 000 000 935 422 544 808 631 601 620 582 4 × 2 = 1 + 0.397 094 726 562 500 000 000 000 000 000 000 000 000 001 870 845 089 617 263 203 241 164 8;
  • 78) 0.397 094 726 562 500 000 000 000 000 000 000 000 000 001 870 845 089 617 263 203 241 164 8 × 2 = 0 + 0.794 189 453 125 000 000 000 000 000 000 000 000 000 003 741 690 179 234 526 406 482 329 6;
  • 79) 0.794 189 453 125 000 000 000 000 000 000 000 000 000 003 741 690 179 234 526 406 482 329 6 × 2 = 1 + 0.588 378 906 250 000 000 000 000 000 000 000 000 000 007 483 380 358 469 052 812 964 659 2;
  • 80) 0.588 378 906 250 000 000 000 000 000 000 000 000 000 007 483 380 358 469 052 812 964 659 2 × 2 = 1 + 0.176 757 812 500 000 000 000 000 000 000 000 000 000 014 966 760 716 938 105 625 929 318 4;
  • 81) 0.176 757 812 500 000 000 000 000 000 000 000 000 000 014 966 760 716 938 105 625 929 318 4 × 2 = 0 + 0.353 515 625 000 000 000 000 000 000 000 000 000 000 029 933 521 433 876 211 251 858 636 8;
  • 82) 0.353 515 625 000 000 000 000 000 000 000 000 000 000 029 933 521 433 876 211 251 858 636 8 × 2 = 0 + 0.707 031 250 000 000 000 000 000 000 000 000 000 000 059 867 042 867 752 422 503 717 273 6;
  • 83) 0.707 031 250 000 000 000 000 000 000 000 000 000 000 059 867 042 867 752 422 503 717 273 6 × 2 = 1 + 0.414 062 500 000 000 000 000 000 000 000 000 000 000 119 734 085 735 504 845 007 434 547 2;
  • 84) 0.414 062 500 000 000 000 000 000 000 000 000 000 000 119 734 085 735 504 845 007 434 547 2 × 2 = 0 + 0.828 125 000 000 000 000 000 000 000 000 000 000 000 239 468 171 471 009 690 014 869 094 4;
  • 85) 0.828 125 000 000 000 000 000 000 000 000 000 000 000 239 468 171 471 009 690 014 869 094 4 × 2 = 1 + 0.656 250 000 000 000 000 000 000 000 000 000 000 000 478 936 342 942 019 380 029 738 188 8;
  • 86) 0.656 250 000 000 000 000 000 000 000 000 000 000 000 478 936 342 942 019 380 029 738 188 8 × 2 = 1 + 0.312 500 000 000 000 000 000 000 000 000 000 000 000 957 872 685 884 038 760 059 476 377 6;
  • 87) 0.312 500 000 000 000 000 000 000 000 000 000 000 000 957 872 685 884 038 760 059 476 377 6 × 2 = 0 + 0.625 000 000 000 000 000 000 000 000 000 000 000 001 915 745 371 768 077 520 118 952 755 2;
  • 88) 0.625 000 000 000 000 000 000 000 000 000 000 000 001 915 745 371 768 077 520 118 952 755 2 × 2 = 1 + 0.250 000 000 000 000 000 000 000 000 000 000 000 003 831 490 743 536 155 040 237 905 510 4;
  • 89) 0.250 000 000 000 000 000 000 000 000 000 000 000 003 831 490 743 536 155 040 237 905 510 4 × 2 = 0 + 0.500 000 000 000 000 000 000 000 000 000 000 000 007 662 981 487 072 310 080 475 811 020 8;
  • 90) 0.500 000 000 000 000 000 000 000 000 000 000 000 007 662 981 487 072 310 080 475 811 020 8 × 2 = 1 + 0.000 000 000 000 000 000 000 000 000 000 000 000 015 325 962 974 144 620 160 951 622 041 6;
  • 91) 0.000 000 000 000 000 000 000 000 000 000 000 000 015 325 962 974 144 620 160 951 622 041 6 × 2 = 0 + 0.000 000 000 000 000 000 000 000 000 000 000 000 030 651 925 948 289 240 321 903 244 083 2;

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:


0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 029 932 430 448 4(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:

0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 029 932 430 448 4(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:


0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 029 932 430 448 4(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 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 029 932 430 448 4 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