0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 029 932 430 212 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 029 932 430 212(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
0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 029 932 430 212(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: 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 029 932 430 212.

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) 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 029 932 430 212 × 2 = 0 + 0.000 000 000 007 269 999 999 999 999 516 522 114 064 601 356 670 305 976 059 864 860 424;
  • 2) 0.000 000 000 007 269 999 999 999 999 516 522 114 064 601 356 670 305 976 059 864 860 424 × 2 = 0 + 0.000 000 000 014 539 999 999 999 999 033 044 228 129 202 713 340 611 952 119 729 720 848;
  • 3) 0.000 000 000 014 539 999 999 999 999 033 044 228 129 202 713 340 611 952 119 729 720 848 × 2 = 0 + 0.000 000 000 029 079 999 999 999 998 066 088 456 258 405 426 681 223 904 239 459 441 696;
  • 4) 0.000 000 000 029 079 999 999 999 998 066 088 456 258 405 426 681 223 904 239 459 441 696 × 2 = 0 + 0.000 000 000 058 159 999 999 999 996 132 176 912 516 810 853 362 447 808 478 918 883 392;
  • 5) 0.000 000 000 058 159 999 999 999 996 132 176 912 516 810 853 362 447 808 478 918 883 392 × 2 = 0 + 0.000 000 000 116 319 999 999 999 992 264 353 825 033 621 706 724 895 616 957 837 766 784;
  • 6) 0.000 000 000 116 319 999 999 999 992 264 353 825 033 621 706 724 895 616 957 837 766 784 × 2 = 0 + 0.000 000 000 232 639 999 999 999 984 528 707 650 067 243 413 449 791 233 915 675 533 568;
  • 7) 0.000 000 000 232 639 999 999 999 984 528 707 650 067 243 413 449 791 233 915 675 533 568 × 2 = 0 + 0.000 000 000 465 279 999 999 999 969 057 415 300 134 486 826 899 582 467 831 351 067 136;
  • 8) 0.000 000 000 465 279 999 999 999 969 057 415 300 134 486 826 899 582 467 831 351 067 136 × 2 = 0 + 0.000 000 000 930 559 999 999 999 938 114 830 600 268 973 653 799 164 935 662 702 134 272;
  • 9) 0.000 000 000 930 559 999 999 999 938 114 830 600 268 973 653 799 164 935 662 702 134 272 × 2 = 0 + 0.000 000 001 861 119 999 999 999 876 229 661 200 537 947 307 598 329 871 325 404 268 544;
  • 10) 0.000 000 001 861 119 999 999 999 876 229 661 200 537 947 307 598 329 871 325 404 268 544 × 2 = 0 + 0.000 000 003 722 239 999 999 999 752 459 322 401 075 894 615 196 659 742 650 808 537 088;
  • 11) 0.000 000 003 722 239 999 999 999 752 459 322 401 075 894 615 196 659 742 650 808 537 088 × 2 = 0 + 0.000 000 007 444 479 999 999 999 504 918 644 802 151 789 230 393 319 485 301 617 074 176;
  • 12) 0.000 000 007 444 479 999 999 999 504 918 644 802 151 789 230 393 319 485 301 617 074 176 × 2 = 0 + 0.000 000 014 888 959 999 999 999 009 837 289 604 303 578 460 786 638 970 603 234 148 352;
  • 13) 0.000 000 014 888 959 999 999 999 009 837 289 604 303 578 460 786 638 970 603 234 148 352 × 2 = 0 + 0.000 000 029 777 919 999 999 998 019 674 579 208 607 156 921 573 277 941 206 468 296 704;
  • 14) 0.000 000 029 777 919 999 999 998 019 674 579 208 607 156 921 573 277 941 206 468 296 704 × 2 = 0 + 0.000 000 059 555 839 999 999 996 039 349 158 417 214 313 843 146 555 882 412 936 593 408;
  • 15) 0.000 000 059 555 839 999 999 996 039 349 158 417 214 313 843 146 555 882 412 936 593 408 × 2 = 0 + 0.000 000 119 111 679 999 999 992 078 698 316 834 428 627 686 293 111 764 825 873 186 816;
  • 16) 0.000 000 119 111 679 999 999 992 078 698 316 834 428 627 686 293 111 764 825 873 186 816 × 2 = 0 + 0.000 000 238 223 359 999 999 984 157 396 633 668 857 255 372 586 223 529 651 746 373 632;
  • 17) 0.000 000 238 223 359 999 999 984 157 396 633 668 857 255 372 586 223 529 651 746 373 632 × 2 = 0 + 0.000 000 476 446 719 999 999 968 314 793 267 337 714 510 745 172 447 059 303 492 747 264;
  • 18) 0.000 000 476 446 719 999 999 968 314 793 267 337 714 510 745 172 447 059 303 492 747 264 × 2 = 0 + 0.000 000 952 893 439 999 999 936 629 586 534 675 429 021 490 344 894 118 606 985 494 528;
  • 19) 0.000 000 952 893 439 999 999 936 629 586 534 675 429 021 490 344 894 118 606 985 494 528 × 2 = 0 + 0.000 001 905 786 879 999 999 873 259 173 069 350 858 042 980 689 788 237 213 970 989 056;
  • 20) 0.000 001 905 786 879 999 999 873 259 173 069 350 858 042 980 689 788 237 213 970 989 056 × 2 = 0 + 0.000 003 811 573 759 999 999 746 518 346 138 701 716 085 961 379 576 474 427 941 978 112;
  • 21) 0.000 003 811 573 759 999 999 746 518 346 138 701 716 085 961 379 576 474 427 941 978 112 × 2 = 0 + 0.000 007 623 147 519 999 999 493 036 692 277 403 432 171 922 759 152 948 855 883 956 224;
  • 22) 0.000 007 623 147 519 999 999 493 036 692 277 403 432 171 922 759 152 948 855 883 956 224 × 2 = 0 + 0.000 015 246 295 039 999 998 986 073 384 554 806 864 343 845 518 305 897 711 767 912 448;
  • 23) 0.000 015 246 295 039 999 998 986 073 384 554 806 864 343 845 518 305 897 711 767 912 448 × 2 = 0 + 0.000 030 492 590 079 999 997 972 146 769 109 613 728 687 691 036 611 795 423 535 824 896;
  • 24) 0.000 030 492 590 079 999 997 972 146 769 109 613 728 687 691 036 611 795 423 535 824 896 × 2 = 0 + 0.000 060 985 180 159 999 995 944 293 538 219 227 457 375 382 073 223 590 847 071 649 792;
  • 25) 0.000 060 985 180 159 999 995 944 293 538 219 227 457 375 382 073 223 590 847 071 649 792 × 2 = 0 + 0.000 121 970 360 319 999 991 888 587 076 438 454 914 750 764 146 447 181 694 143 299 584;
  • 26) 0.000 121 970 360 319 999 991 888 587 076 438 454 914 750 764 146 447 181 694 143 299 584 × 2 = 0 + 0.000 243 940 720 639 999 983 777 174 152 876 909 829 501 528 292 894 363 388 286 599 168;
  • 27) 0.000 243 940 720 639 999 983 777 174 152 876 909 829 501 528 292 894 363 388 286 599 168 × 2 = 0 + 0.000 487 881 441 279 999 967 554 348 305 753 819 659 003 056 585 788 726 776 573 198 336;
  • 28) 0.000 487 881 441 279 999 967 554 348 305 753 819 659 003 056 585 788 726 776 573 198 336 × 2 = 0 + 0.000 975 762 882 559 999 935 108 696 611 507 639 318 006 113 171 577 453 553 146 396 672;
  • 29) 0.000 975 762 882 559 999 935 108 696 611 507 639 318 006 113 171 577 453 553 146 396 672 × 2 = 0 + 0.001 951 525 765 119 999 870 217 393 223 015 278 636 012 226 343 154 907 106 292 793 344;
  • 30) 0.001 951 525 765 119 999 870 217 393 223 015 278 636 012 226 343 154 907 106 292 793 344 × 2 = 0 + 0.003 903 051 530 239 999 740 434 786 446 030 557 272 024 452 686 309 814 212 585 586 688;
  • 31) 0.003 903 051 530 239 999 740 434 786 446 030 557 272 024 452 686 309 814 212 585 586 688 × 2 = 0 + 0.007 806 103 060 479 999 480 869 572 892 061 114 544 048 905 372 619 628 425 171 173 376;
  • 32) 0.007 806 103 060 479 999 480 869 572 892 061 114 544 048 905 372 619 628 425 171 173 376 × 2 = 0 + 0.015 612 206 120 959 998 961 739 145 784 122 229 088 097 810 745 239 256 850 342 346 752;
  • 33) 0.015 612 206 120 959 998 961 739 145 784 122 229 088 097 810 745 239 256 850 342 346 752 × 2 = 0 + 0.031 224 412 241 919 997 923 478 291 568 244 458 176 195 621 490 478 513 700 684 693 504;
  • 34) 0.031 224 412 241 919 997 923 478 291 568 244 458 176 195 621 490 478 513 700 684 693 504 × 2 = 0 + 0.062 448 824 483 839 995 846 956 583 136 488 916 352 391 242 980 957 027 401 369 387 008;
  • 35) 0.062 448 824 483 839 995 846 956 583 136 488 916 352 391 242 980 957 027 401 369 387 008 × 2 = 0 + 0.124 897 648 967 679 991 693 913 166 272 977 832 704 782 485 961 914 054 802 738 774 016;
  • 36) 0.124 897 648 967 679 991 693 913 166 272 977 832 704 782 485 961 914 054 802 738 774 016 × 2 = 0 + 0.249 795 297 935 359 983 387 826 332 545 955 665 409 564 971 923 828 109 605 477 548 032;
  • 37) 0.249 795 297 935 359 983 387 826 332 545 955 665 409 564 971 923 828 109 605 477 548 032 × 2 = 0 + 0.499 590 595 870 719 966 775 652 665 091 911 330 819 129 943 847 656 219 210 955 096 064;
  • 38) 0.499 590 595 870 719 966 775 652 665 091 911 330 819 129 943 847 656 219 210 955 096 064 × 2 = 0 + 0.999 181 191 741 439 933 551 305 330 183 822 661 638 259 887 695 312 438 421 910 192 128;
  • 39) 0.999 181 191 741 439 933 551 305 330 183 822 661 638 259 887 695 312 438 421 910 192 128 × 2 = 1 + 0.998 362 383 482 879 867 102 610 660 367 645 323 276 519 775 390 624 876 843 820 384 256;
  • 40) 0.998 362 383 482 879 867 102 610 660 367 645 323 276 519 775 390 624 876 843 820 384 256 × 2 = 1 + 0.996 724 766 965 759 734 205 221 320 735 290 646 553 039 550 781 249 753 687 640 768 512;
  • 41) 0.996 724 766 965 759 734 205 221 320 735 290 646 553 039 550 781 249 753 687 640 768 512 × 2 = 1 + 0.993 449 533 931 519 468 410 442 641 470 581 293 106 079 101 562 499 507 375 281 537 024;
  • 42) 0.993 449 533 931 519 468 410 442 641 470 581 293 106 079 101 562 499 507 375 281 537 024 × 2 = 1 + 0.986 899 067 863 038 936 820 885 282 941 162 586 212 158 203 124 999 014 750 563 074 048;
  • 43) 0.986 899 067 863 038 936 820 885 282 941 162 586 212 158 203 124 999 014 750 563 074 048 × 2 = 1 + 0.973 798 135 726 077 873 641 770 565 882 325 172 424 316 406 249 998 029 501 126 148 096;
  • 44) 0.973 798 135 726 077 873 641 770 565 882 325 172 424 316 406 249 998 029 501 126 148 096 × 2 = 1 + 0.947 596 271 452 155 747 283 541 131 764 650 344 848 632 812 499 996 059 002 252 296 192;
  • 45) 0.947 596 271 452 155 747 283 541 131 764 650 344 848 632 812 499 996 059 002 252 296 192 × 2 = 1 + 0.895 192 542 904 311 494 567 082 263 529 300 689 697 265 624 999 992 118 004 504 592 384;
  • 46) 0.895 192 542 904 311 494 567 082 263 529 300 689 697 265 624 999 992 118 004 504 592 384 × 2 = 1 + 0.790 385 085 808 622 989 134 164 527 058 601 379 394 531 249 999 984 236 009 009 184 768;
  • 47) 0.790 385 085 808 622 989 134 164 527 058 601 379 394 531 249 999 984 236 009 009 184 768 × 2 = 1 + 0.580 770 171 617 245 978 268 329 054 117 202 758 789 062 499 999 968 472 018 018 369 536;
  • 48) 0.580 770 171 617 245 978 268 329 054 117 202 758 789 062 499 999 968 472 018 018 369 536 × 2 = 1 + 0.161 540 343 234 491 956 536 658 108 234 405 517 578 124 999 999 936 944 036 036 739 072;
  • 49) 0.161 540 343 234 491 956 536 658 108 234 405 517 578 124 999 999 936 944 036 036 739 072 × 2 = 0 + 0.323 080 686 468 983 913 073 316 216 468 811 035 156 249 999 999 873 888 072 073 478 144;
  • 50) 0.323 080 686 468 983 913 073 316 216 468 811 035 156 249 999 999 873 888 072 073 478 144 × 2 = 0 + 0.646 161 372 937 967 826 146 632 432 937 622 070 312 499 999 999 747 776 144 146 956 288;
  • 51) 0.646 161 372 937 967 826 146 632 432 937 622 070 312 499 999 999 747 776 144 146 956 288 × 2 = 1 + 0.292 322 745 875 935 652 293 264 865 875 244 140 624 999 999 999 495 552 288 293 912 576;
  • 52) 0.292 322 745 875 935 652 293 264 865 875 244 140 624 999 999 999 495 552 288 293 912 576 × 2 = 0 + 0.584 645 491 751 871 304 586 529 731 750 488 281 249 999 999 998 991 104 576 587 825 152;
  • 53) 0.584 645 491 751 871 304 586 529 731 750 488 281 249 999 999 998 991 104 576 587 825 152 × 2 = 1 + 0.169 290 983 503 742 609 173 059 463 500 976 562 499 999 999 997 982 209 153 175 650 304;
  • 54) 0.169 290 983 503 742 609 173 059 463 500 976 562 499 999 999 997 982 209 153 175 650 304 × 2 = 0 + 0.338 581 967 007 485 218 346 118 927 001 953 124 999 999 999 995 964 418 306 351 300 608;
  • 55) 0.338 581 967 007 485 218 346 118 927 001 953 124 999 999 999 995 964 418 306 351 300 608 × 2 = 0 + 0.677 163 934 014 970 436 692 237 854 003 906 249 999 999 999 991 928 836 612 702 601 216;
  • 56) 0.677 163 934 014 970 436 692 237 854 003 906 249 999 999 999 991 928 836 612 702 601 216 × 2 = 1 + 0.354 327 868 029 940 873 384 475 708 007 812 499 999 999 999 983 857 673 225 405 202 432;
  • 57) 0.354 327 868 029 940 873 384 475 708 007 812 499 999 999 999 983 857 673 225 405 202 432 × 2 = 0 + 0.708 655 736 059 881 746 768 951 416 015 624 999 999 999 999 967 715 346 450 810 404 864;
  • 58) 0.708 655 736 059 881 746 768 951 416 015 624 999 999 999 999 967 715 346 450 810 404 864 × 2 = 1 + 0.417 311 472 119 763 493 537 902 832 031 249 999 999 999 999 935 430 692 901 620 809 728;
  • 59) 0.417 311 472 119 763 493 537 902 832 031 249 999 999 999 999 935 430 692 901 620 809 728 × 2 = 0 + 0.834 622 944 239 526 987 075 805 664 062 499 999 999 999 999 870 861 385 803 241 619 456;
  • 60) 0.834 622 944 239 526 987 075 805 664 062 499 999 999 999 999 870 861 385 803 241 619 456 × 2 = 1 + 0.669 245 888 479 053 974 151 611 328 124 999 999 999 999 999 741 722 771 606 483 238 912;
  • 61) 0.669 245 888 479 053 974 151 611 328 124 999 999 999 999 999 741 722 771 606 483 238 912 × 2 = 1 + 0.338 491 776 958 107 948 303 222 656 249 999 999 999 999 999 483 445 543 212 966 477 824;
  • 62) 0.338 491 776 958 107 948 303 222 656 249 999 999 999 999 999 483 445 543 212 966 477 824 × 2 = 0 + 0.676 983 553 916 215 896 606 445 312 499 999 999 999 999 998 966 891 086 425 932 955 648;
  • 63) 0.676 983 553 916 215 896 606 445 312 499 999 999 999 999 998 966 891 086 425 932 955 648 × 2 = 1 + 0.353 967 107 832 431 793 212 890 624 999 999 999 999 999 997 933 782 172 851 865 911 296;
  • 64) 0.353 967 107 832 431 793 212 890 624 999 999 999 999 999 997 933 782 172 851 865 911 296 × 2 = 0 + 0.707 934 215 664 863 586 425 781 249 999 999 999 999 999 995 867 564 345 703 731 822 592;
  • 65) 0.707 934 215 664 863 586 425 781 249 999 999 999 999 999 995 867 564 345 703 731 822 592 × 2 = 1 + 0.415 868 431 329 727 172 851 562 499 999 999 999 999 999 991 735 128 691 407 463 645 184;
  • 66) 0.415 868 431 329 727 172 851 562 499 999 999 999 999 999 991 735 128 691 407 463 645 184 × 2 = 0 + 0.831 736 862 659 454 345 703 124 999 999 999 999 999 999 983 470 257 382 814 927 290 368;
  • 67) 0.831 736 862 659 454 345 703 124 999 999 999 999 999 999 983 470 257 382 814 927 290 368 × 2 = 1 + 0.663 473 725 318 908 691 406 249 999 999 999 999 999 999 966 940 514 765 629 854 580 736;
  • 68) 0.663 473 725 318 908 691 406 249 999 999 999 999 999 999 966 940 514 765 629 854 580 736 × 2 = 1 + 0.326 947 450 637 817 382 812 499 999 999 999 999 999 999 933 881 029 531 259 709 161 472;
  • 69) 0.326 947 450 637 817 382 812 499 999 999 999 999 999 999 933 881 029 531 259 709 161 472 × 2 = 0 + 0.653 894 901 275 634 765 624 999 999 999 999 999 999 999 867 762 059 062 519 418 322 944;
  • 70) 0.653 894 901 275 634 765 624 999 999 999 999 999 999 999 867 762 059 062 519 418 322 944 × 2 = 1 + 0.307 789 802 551 269 531 249 999 999 999 999 999 999 999 735 524 118 125 038 836 645 888;
  • 71) 0.307 789 802 551 269 531 249 999 999 999 999 999 999 999 735 524 118 125 038 836 645 888 × 2 = 0 + 0.615 579 605 102 539 062 499 999 999 999 999 999 999 999 471 048 236 250 077 673 291 776;
  • 72) 0.615 579 605 102 539 062 499 999 999 999 999 999 999 999 471 048 236 250 077 673 291 776 × 2 = 1 + 0.231 159 210 205 078 124 999 999 999 999 999 999 999 998 942 096 472 500 155 346 583 552;
  • 73) 0.231 159 210 205 078 124 999 999 999 999 999 999 999 998 942 096 472 500 155 346 583 552 × 2 = 0 + 0.462 318 420 410 156 249 999 999 999 999 999 999 999 997 884 192 945 000 310 693 167 104;
  • 74) 0.462 318 420 410 156 249 999 999 999 999 999 999 999 997 884 192 945 000 310 693 167 104 × 2 = 0 + 0.924 636 840 820 312 499 999 999 999 999 999 999 999 995 768 385 890 000 621 386 334 208;
  • 75) 0.924 636 840 820 312 499 999 999 999 999 999 999 999 995 768 385 890 000 621 386 334 208 × 2 = 1 + 0.849 273 681 640 624 999 999 999 999 999 999 999 999 991 536 771 780 001 242 772 668 416;
  • 76) 0.849 273 681 640 624 999 999 999 999 999 999 999 999 991 536 771 780 001 242 772 668 416 × 2 = 1 + 0.698 547 363 281 249 999 999 999 999 999 999 999 999 983 073 543 560 002 485 545 336 832;
  • 77) 0.698 547 363 281 249 999 999 999 999 999 999 999 999 983 073 543 560 002 485 545 336 832 × 2 = 1 + 0.397 094 726 562 499 999 999 999 999 999 999 999 999 966 147 087 120 004 971 090 673 664;
  • 78) 0.397 094 726 562 499 999 999 999 999 999 999 999 999 966 147 087 120 004 971 090 673 664 × 2 = 0 + 0.794 189 453 124 999 999 999 999 999 999 999 999 999 932 294 174 240 009 942 181 347 328;
  • 79) 0.794 189 453 124 999 999 999 999 999 999 999 999 999 932 294 174 240 009 942 181 347 328 × 2 = 1 + 0.588 378 906 249 999 999 999 999 999 999 999 999 999 864 588 348 480 019 884 362 694 656;
  • 80) 0.588 378 906 249 999 999 999 999 999 999 999 999 999 864 588 348 480 019 884 362 694 656 × 2 = 1 + 0.176 757 812 499 999 999 999 999 999 999 999 999 999 729 176 696 960 039 768 725 389 312;
  • 81) 0.176 757 812 499 999 999 999 999 999 999 999 999 999 729 176 696 960 039 768 725 389 312 × 2 = 0 + 0.353 515 624 999 999 999 999 999 999 999 999 999 999 458 353 393 920 079 537 450 778 624;
  • 82) 0.353 515 624 999 999 999 999 999 999 999 999 999 999 458 353 393 920 079 537 450 778 624 × 2 = 0 + 0.707 031 249 999 999 999 999 999 999 999 999 999 998 916 706 787 840 159 074 901 557 248;
  • 83) 0.707 031 249 999 999 999 999 999 999 999 999 999 998 916 706 787 840 159 074 901 557 248 × 2 = 1 + 0.414 062 499 999 999 999 999 999 999 999 999 999 997 833 413 575 680 318 149 803 114 496;
  • 84) 0.414 062 499 999 999 999 999 999 999 999 999 999 997 833 413 575 680 318 149 803 114 496 × 2 = 0 + 0.828 124 999 999 999 999 999 999 999 999 999 999 995 666 827 151 360 636 299 606 228 992;
  • 85) 0.828 124 999 999 999 999 999 999 999 999 999 999 995 666 827 151 360 636 299 606 228 992 × 2 = 1 + 0.656 249 999 999 999 999 999 999 999 999 999 999 991 333 654 302 721 272 599 212 457 984;
  • 86) 0.656 249 999 999 999 999 999 999 999 999 999 999 991 333 654 302 721 272 599 212 457 984 × 2 = 1 + 0.312 499 999 999 999 999 999 999 999 999 999 999 982 667 308 605 442 545 198 424 915 968;
  • 87) 0.312 499 999 999 999 999 999 999 999 999 999 999 982 667 308 605 442 545 198 424 915 968 × 2 = 0 + 0.624 999 999 999 999 999 999 999 999 999 999 999 965 334 617 210 885 090 396 849 831 936;
  • 88) 0.624 999 999 999 999 999 999 999 999 999 999 999 965 334 617 210 885 090 396 849 831 936 × 2 = 1 + 0.249 999 999 999 999 999 999 999 999 999 999 999 930 669 234 421 770 180 793 699 663 872;
  • 89) 0.249 999 999 999 999 999 999 999 999 999 999 999 930 669 234 421 770 180 793 699 663 872 × 2 = 0 + 0.499 999 999 999 999 999 999 999 999 999 999 999 861 338 468 843 540 361 587 399 327 744;
  • 90) 0.499 999 999 999 999 999 999 999 999 999 999 999 861 338 468 843 540 361 587 399 327 744 × 2 = 0 + 0.999 999 999 999 999 999 999 999 999 999 999 999 722 676 937 687 080 723 174 798 655 488;
  • 91) 0.999 999 999 999 999 999 999 999 999 999 999 999 722 676 937 687 080 723 174 798 655 488 × 2 = 1 + 0.999 999 999 999 999 999 999 999 999 999 999 999 445 353 875 374 161 446 349 597 310 976;

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:


0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 029 932 430 212(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:

0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 029 932 430 212(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:


0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 029 932 430 212(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 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 029 932 430 212 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