0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 029 932 430 436 019 792 3 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 029 932 430 436 019 792 3(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
0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 029 932 430 436 019 792 3(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: 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 029 932 430 436 019 792 3.

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) 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 029 932 430 436 019 792 3 × 2 = 0 + 0.000 000 000 007 269 999 999 999 999 516 522 114 064 601 356 670 305 976 059 864 860 872 039 584 6;
  • 2) 0.000 000 000 007 269 999 999 999 999 516 522 114 064 601 356 670 305 976 059 864 860 872 039 584 6 × 2 = 0 + 0.000 000 000 014 539 999 999 999 999 033 044 228 129 202 713 340 611 952 119 729 721 744 079 169 2;
  • 3) 0.000 000 000 014 539 999 999 999 999 033 044 228 129 202 713 340 611 952 119 729 721 744 079 169 2 × 2 = 0 + 0.000 000 000 029 079 999 999 999 998 066 088 456 258 405 426 681 223 904 239 459 443 488 158 338 4;
  • 4) 0.000 000 000 029 079 999 999 999 998 066 088 456 258 405 426 681 223 904 239 459 443 488 158 338 4 × 2 = 0 + 0.000 000 000 058 159 999 999 999 996 132 176 912 516 810 853 362 447 808 478 918 886 976 316 676 8;
  • 5) 0.000 000 000 058 159 999 999 999 996 132 176 912 516 810 853 362 447 808 478 918 886 976 316 676 8 × 2 = 0 + 0.000 000 000 116 319 999 999 999 992 264 353 825 033 621 706 724 895 616 957 837 773 952 633 353 6;
  • 6) 0.000 000 000 116 319 999 999 999 992 264 353 825 033 621 706 724 895 616 957 837 773 952 633 353 6 × 2 = 0 + 0.000 000 000 232 639 999 999 999 984 528 707 650 067 243 413 449 791 233 915 675 547 905 266 707 2;
  • 7) 0.000 000 000 232 639 999 999 999 984 528 707 650 067 243 413 449 791 233 915 675 547 905 266 707 2 × 2 = 0 + 0.000 000 000 465 279 999 999 999 969 057 415 300 134 486 826 899 582 467 831 351 095 810 533 414 4;
  • 8) 0.000 000 000 465 279 999 999 999 969 057 415 300 134 486 826 899 582 467 831 351 095 810 533 414 4 × 2 = 0 + 0.000 000 000 930 559 999 999 999 938 114 830 600 268 973 653 799 164 935 662 702 191 621 066 828 8;
  • 9) 0.000 000 000 930 559 999 999 999 938 114 830 600 268 973 653 799 164 935 662 702 191 621 066 828 8 × 2 = 0 + 0.000 000 001 861 119 999 999 999 876 229 661 200 537 947 307 598 329 871 325 404 383 242 133 657 6;
  • 10) 0.000 000 001 861 119 999 999 999 876 229 661 200 537 947 307 598 329 871 325 404 383 242 133 657 6 × 2 = 0 + 0.000 000 003 722 239 999 999 999 752 459 322 401 075 894 615 196 659 742 650 808 766 484 267 315 2;
  • 11) 0.000 000 003 722 239 999 999 999 752 459 322 401 075 894 615 196 659 742 650 808 766 484 267 315 2 × 2 = 0 + 0.000 000 007 444 479 999 999 999 504 918 644 802 151 789 230 393 319 485 301 617 532 968 534 630 4;
  • 12) 0.000 000 007 444 479 999 999 999 504 918 644 802 151 789 230 393 319 485 301 617 532 968 534 630 4 × 2 = 0 + 0.000 000 014 888 959 999 999 999 009 837 289 604 303 578 460 786 638 970 603 235 065 937 069 260 8;
  • 13) 0.000 000 014 888 959 999 999 999 009 837 289 604 303 578 460 786 638 970 603 235 065 937 069 260 8 × 2 = 0 + 0.000 000 029 777 919 999 999 998 019 674 579 208 607 156 921 573 277 941 206 470 131 874 138 521 6;
  • 14) 0.000 000 029 777 919 999 999 998 019 674 579 208 607 156 921 573 277 941 206 470 131 874 138 521 6 × 2 = 0 + 0.000 000 059 555 839 999 999 996 039 349 158 417 214 313 843 146 555 882 412 940 263 748 277 043 2;
  • 15) 0.000 000 059 555 839 999 999 996 039 349 158 417 214 313 843 146 555 882 412 940 263 748 277 043 2 × 2 = 0 + 0.000 000 119 111 679 999 999 992 078 698 316 834 428 627 686 293 111 764 825 880 527 496 554 086 4;
  • 16) 0.000 000 119 111 679 999 999 992 078 698 316 834 428 627 686 293 111 764 825 880 527 496 554 086 4 × 2 = 0 + 0.000 000 238 223 359 999 999 984 157 396 633 668 857 255 372 586 223 529 651 761 054 993 108 172 8;
  • 17) 0.000 000 238 223 359 999 999 984 157 396 633 668 857 255 372 586 223 529 651 761 054 993 108 172 8 × 2 = 0 + 0.000 000 476 446 719 999 999 968 314 793 267 337 714 510 745 172 447 059 303 522 109 986 216 345 6;
  • 18) 0.000 000 476 446 719 999 999 968 314 793 267 337 714 510 745 172 447 059 303 522 109 986 216 345 6 × 2 = 0 + 0.000 000 952 893 439 999 999 936 629 586 534 675 429 021 490 344 894 118 607 044 219 972 432 691 2;
  • 19) 0.000 000 952 893 439 999 999 936 629 586 534 675 429 021 490 344 894 118 607 044 219 972 432 691 2 × 2 = 0 + 0.000 001 905 786 879 999 999 873 259 173 069 350 858 042 980 689 788 237 214 088 439 944 865 382 4;
  • 20) 0.000 001 905 786 879 999 999 873 259 173 069 350 858 042 980 689 788 237 214 088 439 944 865 382 4 × 2 = 0 + 0.000 003 811 573 759 999 999 746 518 346 138 701 716 085 961 379 576 474 428 176 879 889 730 764 8;
  • 21) 0.000 003 811 573 759 999 999 746 518 346 138 701 716 085 961 379 576 474 428 176 879 889 730 764 8 × 2 = 0 + 0.000 007 623 147 519 999 999 493 036 692 277 403 432 171 922 759 152 948 856 353 759 779 461 529 6;
  • 22) 0.000 007 623 147 519 999 999 493 036 692 277 403 432 171 922 759 152 948 856 353 759 779 461 529 6 × 2 = 0 + 0.000 015 246 295 039 999 998 986 073 384 554 806 864 343 845 518 305 897 712 707 519 558 923 059 2;
  • 23) 0.000 015 246 295 039 999 998 986 073 384 554 806 864 343 845 518 305 897 712 707 519 558 923 059 2 × 2 = 0 + 0.000 030 492 590 079 999 997 972 146 769 109 613 728 687 691 036 611 795 425 415 039 117 846 118 4;
  • 24) 0.000 030 492 590 079 999 997 972 146 769 109 613 728 687 691 036 611 795 425 415 039 117 846 118 4 × 2 = 0 + 0.000 060 985 180 159 999 995 944 293 538 219 227 457 375 382 073 223 590 850 830 078 235 692 236 8;
  • 25) 0.000 060 985 180 159 999 995 944 293 538 219 227 457 375 382 073 223 590 850 830 078 235 692 236 8 × 2 = 0 + 0.000 121 970 360 319 999 991 888 587 076 438 454 914 750 764 146 447 181 701 660 156 471 384 473 6;
  • 26) 0.000 121 970 360 319 999 991 888 587 076 438 454 914 750 764 146 447 181 701 660 156 471 384 473 6 × 2 = 0 + 0.000 243 940 720 639 999 983 777 174 152 876 909 829 501 528 292 894 363 403 320 312 942 768 947 2;
  • 27) 0.000 243 940 720 639 999 983 777 174 152 876 909 829 501 528 292 894 363 403 320 312 942 768 947 2 × 2 = 0 + 0.000 487 881 441 279 999 967 554 348 305 753 819 659 003 056 585 788 726 806 640 625 885 537 894 4;
  • 28) 0.000 487 881 441 279 999 967 554 348 305 753 819 659 003 056 585 788 726 806 640 625 885 537 894 4 × 2 = 0 + 0.000 975 762 882 559 999 935 108 696 611 507 639 318 006 113 171 577 453 613 281 251 771 075 788 8;
  • 29) 0.000 975 762 882 559 999 935 108 696 611 507 639 318 006 113 171 577 453 613 281 251 771 075 788 8 × 2 = 0 + 0.001 951 525 765 119 999 870 217 393 223 015 278 636 012 226 343 154 907 226 562 503 542 151 577 6;
  • 30) 0.001 951 525 765 119 999 870 217 393 223 015 278 636 012 226 343 154 907 226 562 503 542 151 577 6 × 2 = 0 + 0.003 903 051 530 239 999 740 434 786 446 030 557 272 024 452 686 309 814 453 125 007 084 303 155 2;
  • 31) 0.003 903 051 530 239 999 740 434 786 446 030 557 272 024 452 686 309 814 453 125 007 084 303 155 2 × 2 = 0 + 0.007 806 103 060 479 999 480 869 572 892 061 114 544 048 905 372 619 628 906 250 014 168 606 310 4;
  • 32) 0.007 806 103 060 479 999 480 869 572 892 061 114 544 048 905 372 619 628 906 250 014 168 606 310 4 × 2 = 0 + 0.015 612 206 120 959 998 961 739 145 784 122 229 088 097 810 745 239 257 812 500 028 337 212 620 8;
  • 33) 0.015 612 206 120 959 998 961 739 145 784 122 229 088 097 810 745 239 257 812 500 028 337 212 620 8 × 2 = 0 + 0.031 224 412 241 919 997 923 478 291 568 244 458 176 195 621 490 478 515 625 000 056 674 425 241 6;
  • 34) 0.031 224 412 241 919 997 923 478 291 568 244 458 176 195 621 490 478 515 625 000 056 674 425 241 6 × 2 = 0 + 0.062 448 824 483 839 995 846 956 583 136 488 916 352 391 242 980 957 031 250 000 113 348 850 483 2;
  • 35) 0.062 448 824 483 839 995 846 956 583 136 488 916 352 391 242 980 957 031 250 000 113 348 850 483 2 × 2 = 0 + 0.124 897 648 967 679 991 693 913 166 272 977 832 704 782 485 961 914 062 500 000 226 697 700 966 4;
  • 36) 0.124 897 648 967 679 991 693 913 166 272 977 832 704 782 485 961 914 062 500 000 226 697 700 966 4 × 2 = 0 + 0.249 795 297 935 359 983 387 826 332 545 955 665 409 564 971 923 828 125 000 000 453 395 401 932 8;
  • 37) 0.249 795 297 935 359 983 387 826 332 545 955 665 409 564 971 923 828 125 000 000 453 395 401 932 8 × 2 = 0 + 0.499 590 595 870 719 966 775 652 665 091 911 330 819 129 943 847 656 250 000 000 906 790 803 865 6;
  • 38) 0.499 590 595 870 719 966 775 652 665 091 911 330 819 129 943 847 656 250 000 000 906 790 803 865 6 × 2 = 0 + 0.999 181 191 741 439 933 551 305 330 183 822 661 638 259 887 695 312 500 000 001 813 581 607 731 2;
  • 39) 0.999 181 191 741 439 933 551 305 330 183 822 661 638 259 887 695 312 500 000 001 813 581 607 731 2 × 2 = 1 + 0.998 362 383 482 879 867 102 610 660 367 645 323 276 519 775 390 625 000 000 003 627 163 215 462 4;
  • 40) 0.998 362 383 482 879 867 102 610 660 367 645 323 276 519 775 390 625 000 000 003 627 163 215 462 4 × 2 = 1 + 0.996 724 766 965 759 734 205 221 320 735 290 646 553 039 550 781 250 000 000 007 254 326 430 924 8;
  • 41) 0.996 724 766 965 759 734 205 221 320 735 290 646 553 039 550 781 250 000 000 007 254 326 430 924 8 × 2 = 1 + 0.993 449 533 931 519 468 410 442 641 470 581 293 106 079 101 562 500 000 000 014 508 652 861 849 6;
  • 42) 0.993 449 533 931 519 468 410 442 641 470 581 293 106 079 101 562 500 000 000 014 508 652 861 849 6 × 2 = 1 + 0.986 899 067 863 038 936 820 885 282 941 162 586 212 158 203 125 000 000 000 029 017 305 723 699 2;
  • 43) 0.986 899 067 863 038 936 820 885 282 941 162 586 212 158 203 125 000 000 000 029 017 305 723 699 2 × 2 = 1 + 0.973 798 135 726 077 873 641 770 565 882 325 172 424 316 406 250 000 000 000 058 034 611 447 398 4;
  • 44) 0.973 798 135 726 077 873 641 770 565 882 325 172 424 316 406 250 000 000 000 058 034 611 447 398 4 × 2 = 1 + 0.947 596 271 452 155 747 283 541 131 764 650 344 848 632 812 500 000 000 000 116 069 222 894 796 8;
  • 45) 0.947 596 271 452 155 747 283 541 131 764 650 344 848 632 812 500 000 000 000 116 069 222 894 796 8 × 2 = 1 + 0.895 192 542 904 311 494 567 082 263 529 300 689 697 265 625 000 000 000 000 232 138 445 789 593 6;
  • 46) 0.895 192 542 904 311 494 567 082 263 529 300 689 697 265 625 000 000 000 000 232 138 445 789 593 6 × 2 = 1 + 0.790 385 085 808 622 989 134 164 527 058 601 379 394 531 250 000 000 000 000 464 276 891 579 187 2;
  • 47) 0.790 385 085 808 622 989 134 164 527 058 601 379 394 531 250 000 000 000 000 464 276 891 579 187 2 × 2 = 1 + 0.580 770 171 617 245 978 268 329 054 117 202 758 789 062 500 000 000 000 000 928 553 783 158 374 4;
  • 48) 0.580 770 171 617 245 978 268 329 054 117 202 758 789 062 500 000 000 000 000 928 553 783 158 374 4 × 2 = 1 + 0.161 540 343 234 491 956 536 658 108 234 405 517 578 125 000 000 000 000 001 857 107 566 316 748 8;
  • 49) 0.161 540 343 234 491 956 536 658 108 234 405 517 578 125 000 000 000 000 001 857 107 566 316 748 8 × 2 = 0 + 0.323 080 686 468 983 913 073 316 216 468 811 035 156 250 000 000 000 000 003 714 215 132 633 497 6;
  • 50) 0.323 080 686 468 983 913 073 316 216 468 811 035 156 250 000 000 000 000 003 714 215 132 633 497 6 × 2 = 0 + 0.646 161 372 937 967 826 146 632 432 937 622 070 312 500 000 000 000 000 007 428 430 265 266 995 2;
  • 51) 0.646 161 372 937 967 826 146 632 432 937 622 070 312 500 000 000 000 000 007 428 430 265 266 995 2 × 2 = 1 + 0.292 322 745 875 935 652 293 264 865 875 244 140 625 000 000 000 000 000 014 856 860 530 533 990 4;
  • 52) 0.292 322 745 875 935 652 293 264 865 875 244 140 625 000 000 000 000 000 014 856 860 530 533 990 4 × 2 = 0 + 0.584 645 491 751 871 304 586 529 731 750 488 281 250 000 000 000 000 000 029 713 721 061 067 980 8;
  • 53) 0.584 645 491 751 871 304 586 529 731 750 488 281 250 000 000 000 000 000 029 713 721 061 067 980 8 × 2 = 1 + 0.169 290 983 503 742 609 173 059 463 500 976 562 500 000 000 000 000 000 059 427 442 122 135 961 6;
  • 54) 0.169 290 983 503 742 609 173 059 463 500 976 562 500 000 000 000 000 000 059 427 442 122 135 961 6 × 2 = 0 + 0.338 581 967 007 485 218 346 118 927 001 953 125 000 000 000 000 000 000 118 854 884 244 271 923 2;
  • 55) 0.338 581 967 007 485 218 346 118 927 001 953 125 000 000 000 000 000 000 118 854 884 244 271 923 2 × 2 = 0 + 0.677 163 934 014 970 436 692 237 854 003 906 250 000 000 000 000 000 000 237 709 768 488 543 846 4;
  • 56) 0.677 163 934 014 970 436 692 237 854 003 906 250 000 000 000 000 000 000 237 709 768 488 543 846 4 × 2 = 1 + 0.354 327 868 029 940 873 384 475 708 007 812 500 000 000 000 000 000 000 475 419 536 977 087 692 8;
  • 57) 0.354 327 868 029 940 873 384 475 708 007 812 500 000 000 000 000 000 000 475 419 536 977 087 692 8 × 2 = 0 + 0.708 655 736 059 881 746 768 951 416 015 625 000 000 000 000 000 000 000 950 839 073 954 175 385 6;
  • 58) 0.708 655 736 059 881 746 768 951 416 015 625 000 000 000 000 000 000 000 950 839 073 954 175 385 6 × 2 = 1 + 0.417 311 472 119 763 493 537 902 832 031 250 000 000 000 000 000 000 001 901 678 147 908 350 771 2;
  • 59) 0.417 311 472 119 763 493 537 902 832 031 250 000 000 000 000 000 000 001 901 678 147 908 350 771 2 × 2 = 0 + 0.834 622 944 239 526 987 075 805 664 062 500 000 000 000 000 000 000 003 803 356 295 816 701 542 4;
  • 60) 0.834 622 944 239 526 987 075 805 664 062 500 000 000 000 000 000 000 003 803 356 295 816 701 542 4 × 2 = 1 + 0.669 245 888 479 053 974 151 611 328 125 000 000 000 000 000 000 000 007 606 712 591 633 403 084 8;
  • 61) 0.669 245 888 479 053 974 151 611 328 125 000 000 000 000 000 000 000 007 606 712 591 633 403 084 8 × 2 = 1 + 0.338 491 776 958 107 948 303 222 656 250 000 000 000 000 000 000 000 015 213 425 183 266 806 169 6;
  • 62) 0.338 491 776 958 107 948 303 222 656 250 000 000 000 000 000 000 000 015 213 425 183 266 806 169 6 × 2 = 0 + 0.676 983 553 916 215 896 606 445 312 500 000 000 000 000 000 000 000 030 426 850 366 533 612 339 2;
  • 63) 0.676 983 553 916 215 896 606 445 312 500 000 000 000 000 000 000 000 030 426 850 366 533 612 339 2 × 2 = 1 + 0.353 967 107 832 431 793 212 890 625 000 000 000 000 000 000 000 000 060 853 700 733 067 224 678 4;
  • 64) 0.353 967 107 832 431 793 212 890 625 000 000 000 000 000 000 000 000 060 853 700 733 067 224 678 4 × 2 = 0 + 0.707 934 215 664 863 586 425 781 250 000 000 000 000 000 000 000 000 121 707 401 466 134 449 356 8;
  • 65) 0.707 934 215 664 863 586 425 781 250 000 000 000 000 000 000 000 000 121 707 401 466 134 449 356 8 × 2 = 1 + 0.415 868 431 329 727 172 851 562 500 000 000 000 000 000 000 000 000 243 414 802 932 268 898 713 6;
  • 66) 0.415 868 431 329 727 172 851 562 500 000 000 000 000 000 000 000 000 243 414 802 932 268 898 713 6 × 2 = 0 + 0.831 736 862 659 454 345 703 125 000 000 000 000 000 000 000 000 000 486 829 605 864 537 797 427 2;
  • 67) 0.831 736 862 659 454 345 703 125 000 000 000 000 000 000 000 000 000 486 829 605 864 537 797 427 2 × 2 = 1 + 0.663 473 725 318 908 691 406 250 000 000 000 000 000 000 000 000 000 973 659 211 729 075 594 854 4;
  • 68) 0.663 473 725 318 908 691 406 250 000 000 000 000 000 000 000 000 000 973 659 211 729 075 594 854 4 × 2 = 1 + 0.326 947 450 637 817 382 812 500 000 000 000 000 000 000 000 000 001 947 318 423 458 151 189 708 8;
  • 69) 0.326 947 450 637 817 382 812 500 000 000 000 000 000 000 000 000 001 947 318 423 458 151 189 708 8 × 2 = 0 + 0.653 894 901 275 634 765 625 000 000 000 000 000 000 000 000 000 003 894 636 846 916 302 379 417 6;
  • 70) 0.653 894 901 275 634 765 625 000 000 000 000 000 000 000 000 000 003 894 636 846 916 302 379 417 6 × 2 = 1 + 0.307 789 802 551 269 531 250 000 000 000 000 000 000 000 000 000 007 789 273 693 832 604 758 835 2;
  • 71) 0.307 789 802 551 269 531 250 000 000 000 000 000 000 000 000 000 007 789 273 693 832 604 758 835 2 × 2 = 0 + 0.615 579 605 102 539 062 500 000 000 000 000 000 000 000 000 000 015 578 547 387 665 209 517 670 4;
  • 72) 0.615 579 605 102 539 062 500 000 000 000 000 000 000 000 000 000 015 578 547 387 665 209 517 670 4 × 2 = 1 + 0.231 159 210 205 078 125 000 000 000 000 000 000 000 000 000 000 031 157 094 775 330 419 035 340 8;
  • 73) 0.231 159 210 205 078 125 000 000 000 000 000 000 000 000 000 000 031 157 094 775 330 419 035 340 8 × 2 = 0 + 0.462 318 420 410 156 250 000 000 000 000 000 000 000 000 000 000 062 314 189 550 660 838 070 681 6;
  • 74) 0.462 318 420 410 156 250 000 000 000 000 000 000 000 000 000 000 062 314 189 550 660 838 070 681 6 × 2 = 0 + 0.924 636 840 820 312 500 000 000 000 000 000 000 000 000 000 000 124 628 379 101 321 676 141 363 2;
  • 75) 0.924 636 840 820 312 500 000 000 000 000 000 000 000 000 000 000 124 628 379 101 321 676 141 363 2 × 2 = 1 + 0.849 273 681 640 625 000 000 000 000 000 000 000 000 000 000 000 249 256 758 202 643 352 282 726 4;
  • 76) 0.849 273 681 640 625 000 000 000 000 000 000 000 000 000 000 000 249 256 758 202 643 352 282 726 4 × 2 = 1 + 0.698 547 363 281 250 000 000 000 000 000 000 000 000 000 000 000 498 513 516 405 286 704 565 452 8;
  • 77) 0.698 547 363 281 250 000 000 000 000 000 000 000 000 000 000 000 498 513 516 405 286 704 565 452 8 × 2 = 1 + 0.397 094 726 562 500 000 000 000 000 000 000 000 000 000 000 000 997 027 032 810 573 409 130 905 6;
  • 78) 0.397 094 726 562 500 000 000 000 000 000 000 000 000 000 000 000 997 027 032 810 573 409 130 905 6 × 2 = 0 + 0.794 189 453 125 000 000 000 000 000 000 000 000 000 000 000 001 994 054 065 621 146 818 261 811 2;
  • 79) 0.794 189 453 125 000 000 000 000 000 000 000 000 000 000 000 001 994 054 065 621 146 818 261 811 2 × 2 = 1 + 0.588 378 906 250 000 000 000 000 000 000 000 000 000 000 000 003 988 108 131 242 293 636 523 622 4;
  • 80) 0.588 378 906 250 000 000 000 000 000 000 000 000 000 000 000 003 988 108 131 242 293 636 523 622 4 × 2 = 1 + 0.176 757 812 500 000 000 000 000 000 000 000 000 000 000 000 007 976 216 262 484 587 273 047 244 8;
  • 81) 0.176 757 812 500 000 000 000 000 000 000 000 000 000 000 000 007 976 216 262 484 587 273 047 244 8 × 2 = 0 + 0.353 515 625 000 000 000 000 000 000 000 000 000 000 000 000 015 952 432 524 969 174 546 094 489 6;
  • 82) 0.353 515 625 000 000 000 000 000 000 000 000 000 000 000 000 015 952 432 524 969 174 546 094 489 6 × 2 = 0 + 0.707 031 250 000 000 000 000 000 000 000 000 000 000 000 000 031 904 865 049 938 349 092 188 979 2;
  • 83) 0.707 031 250 000 000 000 000 000 000 000 000 000 000 000 000 031 904 865 049 938 349 092 188 979 2 × 2 = 1 + 0.414 062 500 000 000 000 000 000 000 000 000 000 000 000 000 063 809 730 099 876 698 184 377 958 4;
  • 84) 0.414 062 500 000 000 000 000 000 000 000 000 000 000 000 000 063 809 730 099 876 698 184 377 958 4 × 2 = 0 + 0.828 125 000 000 000 000 000 000 000 000 000 000 000 000 000 127 619 460 199 753 396 368 755 916 8;
  • 85) 0.828 125 000 000 000 000 000 000 000 000 000 000 000 000 000 127 619 460 199 753 396 368 755 916 8 × 2 = 1 + 0.656 250 000 000 000 000 000 000 000 000 000 000 000 000 000 255 238 920 399 506 792 737 511 833 6;
  • 86) 0.656 250 000 000 000 000 000 000 000 000 000 000 000 000 000 255 238 920 399 506 792 737 511 833 6 × 2 = 1 + 0.312 500 000 000 000 000 000 000 000 000 000 000 000 000 000 510 477 840 799 013 585 475 023 667 2;
  • 87) 0.312 500 000 000 000 000 000 000 000 000 000 000 000 000 000 510 477 840 799 013 585 475 023 667 2 × 2 = 0 + 0.625 000 000 000 000 000 000 000 000 000 000 000 000 000 001 020 955 681 598 027 170 950 047 334 4;
  • 88) 0.625 000 000 000 000 000 000 000 000 000 000 000 000 000 001 020 955 681 598 027 170 950 047 334 4 × 2 = 1 + 0.250 000 000 000 000 000 000 000 000 000 000 000 000 000 002 041 911 363 196 054 341 900 094 668 8;
  • 89) 0.250 000 000 000 000 000 000 000 000 000 000 000 000 000 002 041 911 363 196 054 341 900 094 668 8 × 2 = 0 + 0.500 000 000 000 000 000 000 000 000 000 000 000 000 000 004 083 822 726 392 108 683 800 189 337 6;
  • 90) 0.500 000 000 000 000 000 000 000 000 000 000 000 000 000 004 083 822 726 392 108 683 800 189 337 6 × 2 = 1 + 0.000 000 000 000 000 000 000 000 000 000 000 000 000 000 008 167 645 452 784 217 367 600 378 675 2;
  • 91) 0.000 000 000 000 000 000 000 000 000 000 000 000 000 000 008 167 645 452 784 217 367 600 378 675 2 × 2 = 0 + 0.000 000 000 000 000 000 000 000 000 000 000 000 000 000 016 335 290 905 568 434 735 200 757 350 4;

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:


0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 029 932 430 436 019 792 3(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:

0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 029 932 430 436 019 792 3(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:


0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 029 932 430 436 019 792 3(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 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 029 932 430 436 019 792 3 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