0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 029 932 430 436 019 785 702 228 546 142 574 6 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 029 932 430 436 019 785 702 228 546 142 574 6(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
0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 029 932 430 436 019 785 702 228 546 142 574 6(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: 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 029 932 430 436 019 785 702 228 546 142 574 6.

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) 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 029 932 430 436 019 785 702 228 546 142 574 6 × 2 = 0 + 0.000 000 000 007 269 999 999 999 999 516 522 114 064 601 356 670 305 976 059 864 860 872 039 571 404 457 092 285 149 2;
  • 2) 0.000 000 000 007 269 999 999 999 999 516 522 114 064 601 356 670 305 976 059 864 860 872 039 571 404 457 092 285 149 2 × 2 = 0 + 0.000 000 000 014 539 999 999 999 999 033 044 228 129 202 713 340 611 952 119 729 721 744 079 142 808 914 184 570 298 4;
  • 3) 0.000 000 000 014 539 999 999 999 999 033 044 228 129 202 713 340 611 952 119 729 721 744 079 142 808 914 184 570 298 4 × 2 = 0 + 0.000 000 000 029 079 999 999 999 998 066 088 456 258 405 426 681 223 904 239 459 443 488 158 285 617 828 369 140 596 8;
  • 4) 0.000 000 000 029 079 999 999 999 998 066 088 456 258 405 426 681 223 904 239 459 443 488 158 285 617 828 369 140 596 8 × 2 = 0 + 0.000 000 000 058 159 999 999 999 996 132 176 912 516 810 853 362 447 808 478 918 886 976 316 571 235 656 738 281 193 6;
  • 5) 0.000 000 000 058 159 999 999 999 996 132 176 912 516 810 853 362 447 808 478 918 886 976 316 571 235 656 738 281 193 6 × 2 = 0 + 0.000 000 000 116 319 999 999 999 992 264 353 825 033 621 706 724 895 616 957 837 773 952 633 142 471 313 476 562 387 2;
  • 6) 0.000 000 000 116 319 999 999 999 992 264 353 825 033 621 706 724 895 616 957 837 773 952 633 142 471 313 476 562 387 2 × 2 = 0 + 0.000 000 000 232 639 999 999 999 984 528 707 650 067 243 413 449 791 233 915 675 547 905 266 284 942 626 953 124 774 4;
  • 7) 0.000 000 000 232 639 999 999 999 984 528 707 650 067 243 413 449 791 233 915 675 547 905 266 284 942 626 953 124 774 4 × 2 = 0 + 0.000 000 000 465 279 999 999 999 969 057 415 300 134 486 826 899 582 467 831 351 095 810 532 569 885 253 906 249 548 8;
  • 8) 0.000 000 000 465 279 999 999 999 969 057 415 300 134 486 826 899 582 467 831 351 095 810 532 569 885 253 906 249 548 8 × 2 = 0 + 0.000 000 000 930 559 999 999 999 938 114 830 600 268 973 653 799 164 935 662 702 191 621 065 139 770 507 812 499 097 6;
  • 9) 0.000 000 000 930 559 999 999 999 938 114 830 600 268 973 653 799 164 935 662 702 191 621 065 139 770 507 812 499 097 6 × 2 = 0 + 0.000 000 001 861 119 999 999 999 876 229 661 200 537 947 307 598 329 871 325 404 383 242 130 279 541 015 624 998 195 2;
  • 10) 0.000 000 001 861 119 999 999 999 876 229 661 200 537 947 307 598 329 871 325 404 383 242 130 279 541 015 624 998 195 2 × 2 = 0 + 0.000 000 003 722 239 999 999 999 752 459 322 401 075 894 615 196 659 742 650 808 766 484 260 559 082 031 249 996 390 4;
  • 11) 0.000 000 003 722 239 999 999 999 752 459 322 401 075 894 615 196 659 742 650 808 766 484 260 559 082 031 249 996 390 4 × 2 = 0 + 0.000 000 007 444 479 999 999 999 504 918 644 802 151 789 230 393 319 485 301 617 532 968 521 118 164 062 499 992 780 8;
  • 12) 0.000 000 007 444 479 999 999 999 504 918 644 802 151 789 230 393 319 485 301 617 532 968 521 118 164 062 499 992 780 8 × 2 = 0 + 0.000 000 014 888 959 999 999 999 009 837 289 604 303 578 460 786 638 970 603 235 065 937 042 236 328 124 999 985 561 6;
  • 13) 0.000 000 014 888 959 999 999 999 009 837 289 604 303 578 460 786 638 970 603 235 065 937 042 236 328 124 999 985 561 6 × 2 = 0 + 0.000 000 029 777 919 999 999 998 019 674 579 208 607 156 921 573 277 941 206 470 131 874 084 472 656 249 999 971 123 2;
  • 14) 0.000 000 029 777 919 999 999 998 019 674 579 208 607 156 921 573 277 941 206 470 131 874 084 472 656 249 999 971 123 2 × 2 = 0 + 0.000 000 059 555 839 999 999 996 039 349 158 417 214 313 843 146 555 882 412 940 263 748 168 945 312 499 999 942 246 4;
  • 15) 0.000 000 059 555 839 999 999 996 039 349 158 417 214 313 843 146 555 882 412 940 263 748 168 945 312 499 999 942 246 4 × 2 = 0 + 0.000 000 119 111 679 999 999 992 078 698 316 834 428 627 686 293 111 764 825 880 527 496 337 890 624 999 999 884 492 8;
  • 16) 0.000 000 119 111 679 999 999 992 078 698 316 834 428 627 686 293 111 764 825 880 527 496 337 890 624 999 999 884 492 8 × 2 = 0 + 0.000 000 238 223 359 999 999 984 157 396 633 668 857 255 372 586 223 529 651 761 054 992 675 781 249 999 999 768 985 6;
  • 17) 0.000 000 238 223 359 999 999 984 157 396 633 668 857 255 372 586 223 529 651 761 054 992 675 781 249 999 999 768 985 6 × 2 = 0 + 0.000 000 476 446 719 999 999 968 314 793 267 337 714 510 745 172 447 059 303 522 109 985 351 562 499 999 999 537 971 2;
  • 18) 0.000 000 476 446 719 999 999 968 314 793 267 337 714 510 745 172 447 059 303 522 109 985 351 562 499 999 999 537 971 2 × 2 = 0 + 0.000 000 952 893 439 999 999 936 629 586 534 675 429 021 490 344 894 118 607 044 219 970 703 124 999 999 999 075 942 4;
  • 19) 0.000 000 952 893 439 999 999 936 629 586 534 675 429 021 490 344 894 118 607 044 219 970 703 124 999 999 999 075 942 4 × 2 = 0 + 0.000 001 905 786 879 999 999 873 259 173 069 350 858 042 980 689 788 237 214 088 439 941 406 249 999 999 998 151 884 8;
  • 20) 0.000 001 905 786 879 999 999 873 259 173 069 350 858 042 980 689 788 237 214 088 439 941 406 249 999 999 998 151 884 8 × 2 = 0 + 0.000 003 811 573 759 999 999 746 518 346 138 701 716 085 961 379 576 474 428 176 879 882 812 499 999 999 996 303 769 6;
  • 21) 0.000 003 811 573 759 999 999 746 518 346 138 701 716 085 961 379 576 474 428 176 879 882 812 499 999 999 996 303 769 6 × 2 = 0 + 0.000 007 623 147 519 999 999 493 036 692 277 403 432 171 922 759 152 948 856 353 759 765 624 999 999 999 992 607 539 2;
  • 22) 0.000 007 623 147 519 999 999 493 036 692 277 403 432 171 922 759 152 948 856 353 759 765 624 999 999 999 992 607 539 2 × 2 = 0 + 0.000 015 246 295 039 999 998 986 073 384 554 806 864 343 845 518 305 897 712 707 519 531 249 999 999 999 985 215 078 4;
  • 23) 0.000 015 246 295 039 999 998 986 073 384 554 806 864 343 845 518 305 897 712 707 519 531 249 999 999 999 985 215 078 4 × 2 = 0 + 0.000 030 492 590 079 999 997 972 146 769 109 613 728 687 691 036 611 795 425 415 039 062 499 999 999 999 970 430 156 8;
  • 24) 0.000 030 492 590 079 999 997 972 146 769 109 613 728 687 691 036 611 795 425 415 039 062 499 999 999 999 970 430 156 8 × 2 = 0 + 0.000 060 985 180 159 999 995 944 293 538 219 227 457 375 382 073 223 590 850 830 078 124 999 999 999 999 940 860 313 6;
  • 25) 0.000 060 985 180 159 999 995 944 293 538 219 227 457 375 382 073 223 590 850 830 078 124 999 999 999 999 940 860 313 6 × 2 = 0 + 0.000 121 970 360 319 999 991 888 587 076 438 454 914 750 764 146 447 181 701 660 156 249 999 999 999 999 881 720 627 2;
  • 26) 0.000 121 970 360 319 999 991 888 587 076 438 454 914 750 764 146 447 181 701 660 156 249 999 999 999 999 881 720 627 2 × 2 = 0 + 0.000 243 940 720 639 999 983 777 174 152 876 909 829 501 528 292 894 363 403 320 312 499 999 999 999 999 763 441 254 4;
  • 27) 0.000 243 940 720 639 999 983 777 174 152 876 909 829 501 528 292 894 363 403 320 312 499 999 999 999 999 763 441 254 4 × 2 = 0 + 0.000 487 881 441 279 999 967 554 348 305 753 819 659 003 056 585 788 726 806 640 624 999 999 999 999 999 526 882 508 8;
  • 28) 0.000 487 881 441 279 999 967 554 348 305 753 819 659 003 056 585 788 726 806 640 624 999 999 999 999 999 526 882 508 8 × 2 = 0 + 0.000 975 762 882 559 999 935 108 696 611 507 639 318 006 113 171 577 453 613 281 249 999 999 999 999 999 053 765 017 6;
  • 29) 0.000 975 762 882 559 999 935 108 696 611 507 639 318 006 113 171 577 453 613 281 249 999 999 999 999 999 053 765 017 6 × 2 = 0 + 0.001 951 525 765 119 999 870 217 393 223 015 278 636 012 226 343 154 907 226 562 499 999 999 999 999 998 107 530 035 2;
  • 30) 0.001 951 525 765 119 999 870 217 393 223 015 278 636 012 226 343 154 907 226 562 499 999 999 999 999 998 107 530 035 2 × 2 = 0 + 0.003 903 051 530 239 999 740 434 786 446 030 557 272 024 452 686 309 814 453 124 999 999 999 999 999 996 215 060 070 4;
  • 31) 0.003 903 051 530 239 999 740 434 786 446 030 557 272 024 452 686 309 814 453 124 999 999 999 999 999 996 215 060 070 4 × 2 = 0 + 0.007 806 103 060 479 999 480 869 572 892 061 114 544 048 905 372 619 628 906 249 999 999 999 999 999 992 430 120 140 8;
  • 32) 0.007 806 103 060 479 999 480 869 572 892 061 114 544 048 905 372 619 628 906 249 999 999 999 999 999 992 430 120 140 8 × 2 = 0 + 0.015 612 206 120 959 998 961 739 145 784 122 229 088 097 810 745 239 257 812 499 999 999 999 999 999 984 860 240 281 6;
  • 33) 0.015 612 206 120 959 998 961 739 145 784 122 229 088 097 810 745 239 257 812 499 999 999 999 999 999 984 860 240 281 6 × 2 = 0 + 0.031 224 412 241 919 997 923 478 291 568 244 458 176 195 621 490 478 515 624 999 999 999 999 999 999 969 720 480 563 2;
  • 34) 0.031 224 412 241 919 997 923 478 291 568 244 458 176 195 621 490 478 515 624 999 999 999 999 999 999 969 720 480 563 2 × 2 = 0 + 0.062 448 824 483 839 995 846 956 583 136 488 916 352 391 242 980 957 031 249 999 999 999 999 999 999 939 440 961 126 4;
  • 35) 0.062 448 824 483 839 995 846 956 583 136 488 916 352 391 242 980 957 031 249 999 999 999 999 999 999 939 440 961 126 4 × 2 = 0 + 0.124 897 648 967 679 991 693 913 166 272 977 832 704 782 485 961 914 062 499 999 999 999 999 999 999 878 881 922 252 8;
  • 36) 0.124 897 648 967 679 991 693 913 166 272 977 832 704 782 485 961 914 062 499 999 999 999 999 999 999 878 881 922 252 8 × 2 = 0 + 0.249 795 297 935 359 983 387 826 332 545 955 665 409 564 971 923 828 124 999 999 999 999 999 999 999 757 763 844 505 6;
  • 37) 0.249 795 297 935 359 983 387 826 332 545 955 665 409 564 971 923 828 124 999 999 999 999 999 999 999 757 763 844 505 6 × 2 = 0 + 0.499 590 595 870 719 966 775 652 665 091 911 330 819 129 943 847 656 249 999 999 999 999 999 999 999 515 527 689 011 2;
  • 38) 0.499 590 595 870 719 966 775 652 665 091 911 330 819 129 943 847 656 249 999 999 999 999 999 999 999 515 527 689 011 2 × 2 = 0 + 0.999 181 191 741 439 933 551 305 330 183 822 661 638 259 887 695 312 499 999 999 999 999 999 999 999 031 055 378 022 4;
  • 39) 0.999 181 191 741 439 933 551 305 330 183 822 661 638 259 887 695 312 499 999 999 999 999 999 999 999 031 055 378 022 4 × 2 = 1 + 0.998 362 383 482 879 867 102 610 660 367 645 323 276 519 775 390 624 999 999 999 999 999 999 999 998 062 110 756 044 8;
  • 40) 0.998 362 383 482 879 867 102 610 660 367 645 323 276 519 775 390 624 999 999 999 999 999 999 999 998 062 110 756 044 8 × 2 = 1 + 0.996 724 766 965 759 734 205 221 320 735 290 646 553 039 550 781 249 999 999 999 999 999 999 999 996 124 221 512 089 6;
  • 41) 0.996 724 766 965 759 734 205 221 320 735 290 646 553 039 550 781 249 999 999 999 999 999 999 999 996 124 221 512 089 6 × 2 = 1 + 0.993 449 533 931 519 468 410 442 641 470 581 293 106 079 101 562 499 999 999 999 999 999 999 999 992 248 443 024 179 2;
  • 42) 0.993 449 533 931 519 468 410 442 641 470 581 293 106 079 101 562 499 999 999 999 999 999 999 999 992 248 443 024 179 2 × 2 = 1 + 0.986 899 067 863 038 936 820 885 282 941 162 586 212 158 203 124 999 999 999 999 999 999 999 999 984 496 886 048 358 4;
  • 43) 0.986 899 067 863 038 936 820 885 282 941 162 586 212 158 203 124 999 999 999 999 999 999 999 999 984 496 886 048 358 4 × 2 = 1 + 0.973 798 135 726 077 873 641 770 565 882 325 172 424 316 406 249 999 999 999 999 999 999 999 999 968 993 772 096 716 8;
  • 44) 0.973 798 135 726 077 873 641 770 565 882 325 172 424 316 406 249 999 999 999 999 999 999 999 999 968 993 772 096 716 8 × 2 = 1 + 0.947 596 271 452 155 747 283 541 131 764 650 344 848 632 812 499 999 999 999 999 999 999 999 999 937 987 544 193 433 6;
  • 45) 0.947 596 271 452 155 747 283 541 131 764 650 344 848 632 812 499 999 999 999 999 999 999 999 999 937 987 544 193 433 6 × 2 = 1 + 0.895 192 542 904 311 494 567 082 263 529 300 689 697 265 624 999 999 999 999 999 999 999 999 999 875 975 088 386 867 2;
  • 46) 0.895 192 542 904 311 494 567 082 263 529 300 689 697 265 624 999 999 999 999 999 999 999 999 999 875 975 088 386 867 2 × 2 = 1 + 0.790 385 085 808 622 989 134 164 527 058 601 379 394 531 249 999 999 999 999 999 999 999 999 999 751 950 176 773 734 4;
  • 47) 0.790 385 085 808 622 989 134 164 527 058 601 379 394 531 249 999 999 999 999 999 999 999 999 999 751 950 176 773 734 4 × 2 = 1 + 0.580 770 171 617 245 978 268 329 054 117 202 758 789 062 499 999 999 999 999 999 999 999 999 999 503 900 353 547 468 8;
  • 48) 0.580 770 171 617 245 978 268 329 054 117 202 758 789 062 499 999 999 999 999 999 999 999 999 999 503 900 353 547 468 8 × 2 = 1 + 0.161 540 343 234 491 956 536 658 108 234 405 517 578 124 999 999 999 999 999 999 999 999 999 999 007 800 707 094 937 6;
  • 49) 0.161 540 343 234 491 956 536 658 108 234 405 517 578 124 999 999 999 999 999 999 999 999 999 999 007 800 707 094 937 6 × 2 = 0 + 0.323 080 686 468 983 913 073 316 216 468 811 035 156 249 999 999 999 999 999 999 999 999 999 998 015 601 414 189 875 2;
  • 50) 0.323 080 686 468 983 913 073 316 216 468 811 035 156 249 999 999 999 999 999 999 999 999 999 998 015 601 414 189 875 2 × 2 = 0 + 0.646 161 372 937 967 826 146 632 432 937 622 070 312 499 999 999 999 999 999 999 999 999 999 996 031 202 828 379 750 4;
  • 51) 0.646 161 372 937 967 826 146 632 432 937 622 070 312 499 999 999 999 999 999 999 999 999 999 996 031 202 828 379 750 4 × 2 = 1 + 0.292 322 745 875 935 652 293 264 865 875 244 140 624 999 999 999 999 999 999 999 999 999 999 992 062 405 656 759 500 8;
  • 52) 0.292 322 745 875 935 652 293 264 865 875 244 140 624 999 999 999 999 999 999 999 999 999 999 992 062 405 656 759 500 8 × 2 = 0 + 0.584 645 491 751 871 304 586 529 731 750 488 281 249 999 999 999 999 999 999 999 999 999 999 984 124 811 313 519 001 6;
  • 53) 0.584 645 491 751 871 304 586 529 731 750 488 281 249 999 999 999 999 999 999 999 999 999 999 984 124 811 313 519 001 6 × 2 = 1 + 0.169 290 983 503 742 609 173 059 463 500 976 562 499 999 999 999 999 999 999 999 999 999 999 968 249 622 627 038 003 2;
  • 54) 0.169 290 983 503 742 609 173 059 463 500 976 562 499 999 999 999 999 999 999 999 999 999 999 968 249 622 627 038 003 2 × 2 = 0 + 0.338 581 967 007 485 218 346 118 927 001 953 124 999 999 999 999 999 999 999 999 999 999 999 936 499 245 254 076 006 4;
  • 55) 0.338 581 967 007 485 218 346 118 927 001 953 124 999 999 999 999 999 999 999 999 999 999 999 936 499 245 254 076 006 4 × 2 = 0 + 0.677 163 934 014 970 436 692 237 854 003 906 249 999 999 999 999 999 999 999 999 999 999 999 872 998 490 508 152 012 8;
  • 56) 0.677 163 934 014 970 436 692 237 854 003 906 249 999 999 999 999 999 999 999 999 999 999 999 872 998 490 508 152 012 8 × 2 = 1 + 0.354 327 868 029 940 873 384 475 708 007 812 499 999 999 999 999 999 999 999 999 999 999 999 745 996 981 016 304 025 6;
  • 57) 0.354 327 868 029 940 873 384 475 708 007 812 499 999 999 999 999 999 999 999 999 999 999 999 745 996 981 016 304 025 6 × 2 = 0 + 0.708 655 736 059 881 746 768 951 416 015 624 999 999 999 999 999 999 999 999 999 999 999 999 491 993 962 032 608 051 2;
  • 58) 0.708 655 736 059 881 746 768 951 416 015 624 999 999 999 999 999 999 999 999 999 999 999 999 491 993 962 032 608 051 2 × 2 = 1 + 0.417 311 472 119 763 493 537 902 832 031 249 999 999 999 999 999 999 999 999 999 999 999 998 983 987 924 065 216 102 4;
  • 59) 0.417 311 472 119 763 493 537 902 832 031 249 999 999 999 999 999 999 999 999 999 999 999 998 983 987 924 065 216 102 4 × 2 = 0 + 0.834 622 944 239 526 987 075 805 664 062 499 999 999 999 999 999 999 999 999 999 999 999 997 967 975 848 130 432 204 8;
  • 60) 0.834 622 944 239 526 987 075 805 664 062 499 999 999 999 999 999 999 999 999 999 999 999 997 967 975 848 130 432 204 8 × 2 = 1 + 0.669 245 888 479 053 974 151 611 328 124 999 999 999 999 999 999 999 999 999 999 999 999 995 935 951 696 260 864 409 6;
  • 61) 0.669 245 888 479 053 974 151 611 328 124 999 999 999 999 999 999 999 999 999 999 999 999 995 935 951 696 260 864 409 6 × 2 = 1 + 0.338 491 776 958 107 948 303 222 656 249 999 999 999 999 999 999 999 999 999 999 999 999 991 871 903 392 521 728 819 2;
  • 62) 0.338 491 776 958 107 948 303 222 656 249 999 999 999 999 999 999 999 999 999 999 999 999 991 871 903 392 521 728 819 2 × 2 = 0 + 0.676 983 553 916 215 896 606 445 312 499 999 999 999 999 999 999 999 999 999 999 999 999 983 743 806 785 043 457 638 4;
  • 63) 0.676 983 553 916 215 896 606 445 312 499 999 999 999 999 999 999 999 999 999 999 999 999 983 743 806 785 043 457 638 4 × 2 = 1 + 0.353 967 107 832 431 793 212 890 624 999 999 999 999 999 999 999 999 999 999 999 999 999 967 487 613 570 086 915 276 8;
  • 64) 0.353 967 107 832 431 793 212 890 624 999 999 999 999 999 999 999 999 999 999 999 999 999 967 487 613 570 086 915 276 8 × 2 = 0 + 0.707 934 215 664 863 586 425 781 249 999 999 999 999 999 999 999 999 999 999 999 999 999 934 975 227 140 173 830 553 6;
  • 65) 0.707 934 215 664 863 586 425 781 249 999 999 999 999 999 999 999 999 999 999 999 999 999 934 975 227 140 173 830 553 6 × 2 = 1 + 0.415 868 431 329 727 172 851 562 499 999 999 999 999 999 999 999 999 999 999 999 999 999 869 950 454 280 347 661 107 2;
  • 66) 0.415 868 431 329 727 172 851 562 499 999 999 999 999 999 999 999 999 999 999 999 999 999 869 950 454 280 347 661 107 2 × 2 = 0 + 0.831 736 862 659 454 345 703 124 999 999 999 999 999 999 999 999 999 999 999 999 999 999 739 900 908 560 695 322 214 4;
  • 67) 0.831 736 862 659 454 345 703 124 999 999 999 999 999 999 999 999 999 999 999 999 999 999 739 900 908 560 695 322 214 4 × 2 = 1 + 0.663 473 725 318 908 691 406 249 999 999 999 999 999 999 999 999 999 999 999 999 999 999 479 801 817 121 390 644 428 8;
  • 68) 0.663 473 725 318 908 691 406 249 999 999 999 999 999 999 999 999 999 999 999 999 999 999 479 801 817 121 390 644 428 8 × 2 = 1 + 0.326 947 450 637 817 382 812 499 999 999 999 999 999 999 999 999 999 999 999 999 999 998 959 603 634 242 781 288 857 6;
  • 69) 0.326 947 450 637 817 382 812 499 999 999 999 999 999 999 999 999 999 999 999 999 999 998 959 603 634 242 781 288 857 6 × 2 = 0 + 0.653 894 901 275 634 765 624 999 999 999 999 999 999 999 999 999 999 999 999 999 999 997 919 207 268 485 562 577 715 2;
  • 70) 0.653 894 901 275 634 765 624 999 999 999 999 999 999 999 999 999 999 999 999 999 999 997 919 207 268 485 562 577 715 2 × 2 = 1 + 0.307 789 802 551 269 531 249 999 999 999 999 999 999 999 999 999 999 999 999 999 999 995 838 414 536 971 125 155 430 4;
  • 71) 0.307 789 802 551 269 531 249 999 999 999 999 999 999 999 999 999 999 999 999 999 999 995 838 414 536 971 125 155 430 4 × 2 = 0 + 0.615 579 605 102 539 062 499 999 999 999 999 999 999 999 999 999 999 999 999 999 999 991 676 829 073 942 250 310 860 8;
  • 72) 0.615 579 605 102 539 062 499 999 999 999 999 999 999 999 999 999 999 999 999 999 999 991 676 829 073 942 250 310 860 8 × 2 = 1 + 0.231 159 210 205 078 124 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 983 353 658 147 884 500 621 721 6;
  • 73) 0.231 159 210 205 078 124 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 983 353 658 147 884 500 621 721 6 × 2 = 0 + 0.462 318 420 410 156 249 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 966 707 316 295 769 001 243 443 2;
  • 74) 0.462 318 420 410 156 249 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 966 707 316 295 769 001 243 443 2 × 2 = 0 + 0.924 636 840 820 312 499 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 933 414 632 591 538 002 486 886 4;
  • 75) 0.924 636 840 820 312 499 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 933 414 632 591 538 002 486 886 4 × 2 = 1 + 0.849 273 681 640 624 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 866 829 265 183 076 004 973 772 8;
  • 76) 0.849 273 681 640 624 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 866 829 265 183 076 004 973 772 8 × 2 = 1 + 0.698 547 363 281 249 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 733 658 530 366 152 009 947 545 6;
  • 77) 0.698 547 363 281 249 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 733 658 530 366 152 009 947 545 6 × 2 = 1 + 0.397 094 726 562 499 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 467 317 060 732 304 019 895 091 2;
  • 78) 0.397 094 726 562 499 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 467 317 060 732 304 019 895 091 2 × 2 = 0 + 0.794 189 453 124 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 998 934 634 121 464 608 039 790 182 4;
  • 79) 0.794 189 453 124 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 998 934 634 121 464 608 039 790 182 4 × 2 = 1 + 0.588 378 906 249 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 997 869 268 242 929 216 079 580 364 8;
  • 80) 0.588 378 906 249 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 997 869 268 242 929 216 079 580 364 8 × 2 = 1 + 0.176 757 812 499 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 995 738 536 485 858 432 159 160 729 6;
  • 81) 0.176 757 812 499 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 995 738 536 485 858 432 159 160 729 6 × 2 = 0 + 0.353 515 624 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 991 477 072 971 716 864 318 321 459 2;
  • 82) 0.353 515 624 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 991 477 072 971 716 864 318 321 459 2 × 2 = 0 + 0.707 031 249 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 982 954 145 943 433 728 636 642 918 4;
  • 83) 0.707 031 249 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 982 954 145 943 433 728 636 642 918 4 × 2 = 1 + 0.414 062 499 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 965 908 291 886 867 457 273 285 836 8;
  • 84) 0.414 062 499 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 965 908 291 886 867 457 273 285 836 8 × 2 = 0 + 0.828 124 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 931 816 583 773 734 914 546 571 673 6;
  • 85) 0.828 124 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 931 816 583 773 734 914 546 571 673 6 × 2 = 1 + 0.656 249 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 863 633 167 547 469 829 093 143 347 2;
  • 86) 0.656 249 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 863 633 167 547 469 829 093 143 347 2 × 2 = 1 + 0.312 499 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 727 266 335 094 939 658 186 286 694 4;
  • 87) 0.312 499 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 727 266 335 094 939 658 186 286 694 4 × 2 = 0 + 0.624 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 454 532 670 189 879 316 372 573 388 8;
  • 88) 0.624 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 454 532 670 189 879 316 372 573 388 8 × 2 = 1 + 0.249 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 998 909 065 340 379 758 632 745 146 777 6;
  • 89) 0.249 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 998 909 065 340 379 758 632 745 146 777 6 × 2 = 0 + 0.499 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 997 818 130 680 759 517 265 490 293 555 2;
  • 90) 0.499 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 997 818 130 680 759 517 265 490 293 555 2 × 2 = 0 + 0.999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 995 636 261 361 519 034 530 980 587 110 4;
  • 91) 0.999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 995 636 261 361 519 034 530 980 587 110 4 × 2 = 1 + 0.999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 991 272 522 723 038 069 061 961 174 220 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:


0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 029 932 430 436 019 785 702 228 546 142 574 6(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0011 1111 1111 0010 1001 0101 1010 1011 0101 0011 1011 0010 1101 001(2)

5. Positive number before normalization:

0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 029 932 430 436 019 785 702 228 546 142 574 6(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0011 1111 1111 0010 1001 0101 1010 1011 0101 0011 1011 0010 1101 001(2)

6. Normalize the binary representation of the number.

Shift the decimal mark 39 positions to the right, so that only one non zero digit remains to the left of it:


0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 029 932 430 436 019 785 702 228 546 142 574 6(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0011 1111 1111 0010 1001 0101 1010 1011 0101 0011 1011 0010 1101 001(2) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0011 1111 1111 0010 1001 0101 1010 1011 0101 0011 1011 0010 1101 001(2) × 20 =

1.1111 1111 1001 0100 1010 1101 0101 1010 1001 1101 1001 0110 1001(2) × 2-39


7. Up to this moment, there are the following elements that would feed into the 64 bit double precision IEEE 754 binary floating point representation:

Sign 0 (a positive number)

Exponent (unadjusted): -39

Mantissa (not normalized):
1.1111 1111 1001 0100 1010 1101 0101 1010 1001 1101 1001 0110 1001


8. Adjust the exponent.

Use the 11 bit excess/bias notation:


Exponent (adjusted) =

Exponent (unadjusted) + 2(11-1) - 1 =

-39 + 2(11-1) - 1 =

(-39 + 1 023)(10) =

984(10)


9. Convert the adjusted exponent from the decimal (base 10) to 11 bit binary.

Use the same technique of repeatedly dividing by 2:


  • division = quotient + remainder;
  • 984 ÷ 2 = 492 + 0;
  • 492 ÷ 2 = 246 + 0;
  • 246 ÷ 2 = 123 + 0;
  • 123 ÷ 2 = 61 + 1;
  • 61 ÷ 2 = 30 + 1;
  • 30 ÷ 2 = 15 + 0;
  • 15 ÷ 2 = 7 + 1;
  • 7 ÷ 2 = 3 + 1;
  • 3 ÷ 2 = 1 + 1;
  • 1 ÷ 2 = 0 + 1;

10. Construct the base 2 representation of the adjusted exponent.

Take all the remainders starting from the bottom of the list constructed above.


Exponent (adjusted) =

984(10) =

011 1101 1000(2)


11. Normalize the mantissa.

a) Remove the leading (the leftmost) bit, since it's allways 1, and the decimal point, if the case.

b) Adjust its length to 52 bits, only if necessary (not the case here).


Mantissa (normalized) =

1. 1111 1111 1001 0100 1010 1101 0101 1010 1001 1101 1001 0110 1001 =

1111 1111 1001 0100 1010 1101 0101 1010 1001 1101 1001 0110 1001


12. The three elements that make up the number's 64 bit double precision IEEE 754 binary floating point representation:

Sign (1 bit) =
0 (a positive number)

Exponent (11 bits) =
011 1101 1000

Mantissa (52 bits) =
1111 1111 1001 0100 1010 1101 0101 1010 1001 1101 1001 0110 1001


Decimal number 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 029 932 430 436 019 785 702 228 546 142 574 6 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