0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 029 932 467 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 029 932 467(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
0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 029 932 467(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: 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 029 932 467.

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) 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 029 932 467 × 2 = 0 + 0.000 000 000 007 269 999 999 999 999 516 522 114 064 601 356 670 305 976 059 864 934;
  • 2) 0.000 000 000 007 269 999 999 999 999 516 522 114 064 601 356 670 305 976 059 864 934 × 2 = 0 + 0.000 000 000 014 539 999 999 999 999 033 044 228 129 202 713 340 611 952 119 729 868;
  • 3) 0.000 000 000 014 539 999 999 999 999 033 044 228 129 202 713 340 611 952 119 729 868 × 2 = 0 + 0.000 000 000 029 079 999 999 999 998 066 088 456 258 405 426 681 223 904 239 459 736;
  • 4) 0.000 000 000 029 079 999 999 999 998 066 088 456 258 405 426 681 223 904 239 459 736 × 2 = 0 + 0.000 000 000 058 159 999 999 999 996 132 176 912 516 810 853 362 447 808 478 919 472;
  • 5) 0.000 000 000 058 159 999 999 999 996 132 176 912 516 810 853 362 447 808 478 919 472 × 2 = 0 + 0.000 000 000 116 319 999 999 999 992 264 353 825 033 621 706 724 895 616 957 838 944;
  • 6) 0.000 000 000 116 319 999 999 999 992 264 353 825 033 621 706 724 895 616 957 838 944 × 2 = 0 + 0.000 000 000 232 639 999 999 999 984 528 707 650 067 243 413 449 791 233 915 677 888;
  • 7) 0.000 000 000 232 639 999 999 999 984 528 707 650 067 243 413 449 791 233 915 677 888 × 2 = 0 + 0.000 000 000 465 279 999 999 999 969 057 415 300 134 486 826 899 582 467 831 355 776;
  • 8) 0.000 000 000 465 279 999 999 999 969 057 415 300 134 486 826 899 582 467 831 355 776 × 2 = 0 + 0.000 000 000 930 559 999 999 999 938 114 830 600 268 973 653 799 164 935 662 711 552;
  • 9) 0.000 000 000 930 559 999 999 999 938 114 830 600 268 973 653 799 164 935 662 711 552 × 2 = 0 + 0.000 000 001 861 119 999 999 999 876 229 661 200 537 947 307 598 329 871 325 423 104;
  • 10) 0.000 000 001 861 119 999 999 999 876 229 661 200 537 947 307 598 329 871 325 423 104 × 2 = 0 + 0.000 000 003 722 239 999 999 999 752 459 322 401 075 894 615 196 659 742 650 846 208;
  • 11) 0.000 000 003 722 239 999 999 999 752 459 322 401 075 894 615 196 659 742 650 846 208 × 2 = 0 + 0.000 000 007 444 479 999 999 999 504 918 644 802 151 789 230 393 319 485 301 692 416;
  • 12) 0.000 000 007 444 479 999 999 999 504 918 644 802 151 789 230 393 319 485 301 692 416 × 2 = 0 + 0.000 000 014 888 959 999 999 999 009 837 289 604 303 578 460 786 638 970 603 384 832;
  • 13) 0.000 000 014 888 959 999 999 999 009 837 289 604 303 578 460 786 638 970 603 384 832 × 2 = 0 + 0.000 000 029 777 919 999 999 998 019 674 579 208 607 156 921 573 277 941 206 769 664;
  • 14) 0.000 000 029 777 919 999 999 998 019 674 579 208 607 156 921 573 277 941 206 769 664 × 2 = 0 + 0.000 000 059 555 839 999 999 996 039 349 158 417 214 313 843 146 555 882 413 539 328;
  • 15) 0.000 000 059 555 839 999 999 996 039 349 158 417 214 313 843 146 555 882 413 539 328 × 2 = 0 + 0.000 000 119 111 679 999 999 992 078 698 316 834 428 627 686 293 111 764 827 078 656;
  • 16) 0.000 000 119 111 679 999 999 992 078 698 316 834 428 627 686 293 111 764 827 078 656 × 2 = 0 + 0.000 000 238 223 359 999 999 984 157 396 633 668 857 255 372 586 223 529 654 157 312;
  • 17) 0.000 000 238 223 359 999 999 984 157 396 633 668 857 255 372 586 223 529 654 157 312 × 2 = 0 + 0.000 000 476 446 719 999 999 968 314 793 267 337 714 510 745 172 447 059 308 314 624;
  • 18) 0.000 000 476 446 719 999 999 968 314 793 267 337 714 510 745 172 447 059 308 314 624 × 2 = 0 + 0.000 000 952 893 439 999 999 936 629 586 534 675 429 021 490 344 894 118 616 629 248;
  • 19) 0.000 000 952 893 439 999 999 936 629 586 534 675 429 021 490 344 894 118 616 629 248 × 2 = 0 + 0.000 001 905 786 879 999 999 873 259 173 069 350 858 042 980 689 788 237 233 258 496;
  • 20) 0.000 001 905 786 879 999 999 873 259 173 069 350 858 042 980 689 788 237 233 258 496 × 2 = 0 + 0.000 003 811 573 759 999 999 746 518 346 138 701 716 085 961 379 576 474 466 516 992;
  • 21) 0.000 003 811 573 759 999 999 746 518 346 138 701 716 085 961 379 576 474 466 516 992 × 2 = 0 + 0.000 007 623 147 519 999 999 493 036 692 277 403 432 171 922 759 152 948 933 033 984;
  • 22) 0.000 007 623 147 519 999 999 493 036 692 277 403 432 171 922 759 152 948 933 033 984 × 2 = 0 + 0.000 015 246 295 039 999 998 986 073 384 554 806 864 343 845 518 305 897 866 067 968;
  • 23) 0.000 015 246 295 039 999 998 986 073 384 554 806 864 343 845 518 305 897 866 067 968 × 2 = 0 + 0.000 030 492 590 079 999 997 972 146 769 109 613 728 687 691 036 611 795 732 135 936;
  • 24) 0.000 030 492 590 079 999 997 972 146 769 109 613 728 687 691 036 611 795 732 135 936 × 2 = 0 + 0.000 060 985 180 159 999 995 944 293 538 219 227 457 375 382 073 223 591 464 271 872;
  • 25) 0.000 060 985 180 159 999 995 944 293 538 219 227 457 375 382 073 223 591 464 271 872 × 2 = 0 + 0.000 121 970 360 319 999 991 888 587 076 438 454 914 750 764 146 447 182 928 543 744;
  • 26) 0.000 121 970 360 319 999 991 888 587 076 438 454 914 750 764 146 447 182 928 543 744 × 2 = 0 + 0.000 243 940 720 639 999 983 777 174 152 876 909 829 501 528 292 894 365 857 087 488;
  • 27) 0.000 243 940 720 639 999 983 777 174 152 876 909 829 501 528 292 894 365 857 087 488 × 2 = 0 + 0.000 487 881 441 279 999 967 554 348 305 753 819 659 003 056 585 788 731 714 174 976;
  • 28) 0.000 487 881 441 279 999 967 554 348 305 753 819 659 003 056 585 788 731 714 174 976 × 2 = 0 + 0.000 975 762 882 559 999 935 108 696 611 507 639 318 006 113 171 577 463 428 349 952;
  • 29) 0.000 975 762 882 559 999 935 108 696 611 507 639 318 006 113 171 577 463 428 349 952 × 2 = 0 + 0.001 951 525 765 119 999 870 217 393 223 015 278 636 012 226 343 154 926 856 699 904;
  • 30) 0.001 951 525 765 119 999 870 217 393 223 015 278 636 012 226 343 154 926 856 699 904 × 2 = 0 + 0.003 903 051 530 239 999 740 434 786 446 030 557 272 024 452 686 309 853 713 399 808;
  • 31) 0.003 903 051 530 239 999 740 434 786 446 030 557 272 024 452 686 309 853 713 399 808 × 2 = 0 + 0.007 806 103 060 479 999 480 869 572 892 061 114 544 048 905 372 619 707 426 799 616;
  • 32) 0.007 806 103 060 479 999 480 869 572 892 061 114 544 048 905 372 619 707 426 799 616 × 2 = 0 + 0.015 612 206 120 959 998 961 739 145 784 122 229 088 097 810 745 239 414 853 599 232;
  • 33) 0.015 612 206 120 959 998 961 739 145 784 122 229 088 097 810 745 239 414 853 599 232 × 2 = 0 + 0.031 224 412 241 919 997 923 478 291 568 244 458 176 195 621 490 478 829 707 198 464;
  • 34) 0.031 224 412 241 919 997 923 478 291 568 244 458 176 195 621 490 478 829 707 198 464 × 2 = 0 + 0.062 448 824 483 839 995 846 956 583 136 488 916 352 391 242 980 957 659 414 396 928;
  • 35) 0.062 448 824 483 839 995 846 956 583 136 488 916 352 391 242 980 957 659 414 396 928 × 2 = 0 + 0.124 897 648 967 679 991 693 913 166 272 977 832 704 782 485 961 915 318 828 793 856;
  • 36) 0.124 897 648 967 679 991 693 913 166 272 977 832 704 782 485 961 915 318 828 793 856 × 2 = 0 + 0.249 795 297 935 359 983 387 826 332 545 955 665 409 564 971 923 830 637 657 587 712;
  • 37) 0.249 795 297 935 359 983 387 826 332 545 955 665 409 564 971 923 830 637 657 587 712 × 2 = 0 + 0.499 590 595 870 719 966 775 652 665 091 911 330 819 129 943 847 661 275 315 175 424;
  • 38) 0.499 590 595 870 719 966 775 652 665 091 911 330 819 129 943 847 661 275 315 175 424 × 2 = 0 + 0.999 181 191 741 439 933 551 305 330 183 822 661 638 259 887 695 322 550 630 350 848;
  • 39) 0.999 181 191 741 439 933 551 305 330 183 822 661 638 259 887 695 322 550 630 350 848 × 2 = 1 + 0.998 362 383 482 879 867 102 610 660 367 645 323 276 519 775 390 645 101 260 701 696;
  • 40) 0.998 362 383 482 879 867 102 610 660 367 645 323 276 519 775 390 645 101 260 701 696 × 2 = 1 + 0.996 724 766 965 759 734 205 221 320 735 290 646 553 039 550 781 290 202 521 403 392;
  • 41) 0.996 724 766 965 759 734 205 221 320 735 290 646 553 039 550 781 290 202 521 403 392 × 2 = 1 + 0.993 449 533 931 519 468 410 442 641 470 581 293 106 079 101 562 580 405 042 806 784;
  • 42) 0.993 449 533 931 519 468 410 442 641 470 581 293 106 079 101 562 580 405 042 806 784 × 2 = 1 + 0.986 899 067 863 038 936 820 885 282 941 162 586 212 158 203 125 160 810 085 613 568;
  • 43) 0.986 899 067 863 038 936 820 885 282 941 162 586 212 158 203 125 160 810 085 613 568 × 2 = 1 + 0.973 798 135 726 077 873 641 770 565 882 325 172 424 316 406 250 321 620 171 227 136;
  • 44) 0.973 798 135 726 077 873 641 770 565 882 325 172 424 316 406 250 321 620 171 227 136 × 2 = 1 + 0.947 596 271 452 155 747 283 541 131 764 650 344 848 632 812 500 643 240 342 454 272;
  • 45) 0.947 596 271 452 155 747 283 541 131 764 650 344 848 632 812 500 643 240 342 454 272 × 2 = 1 + 0.895 192 542 904 311 494 567 082 263 529 300 689 697 265 625 001 286 480 684 908 544;
  • 46) 0.895 192 542 904 311 494 567 082 263 529 300 689 697 265 625 001 286 480 684 908 544 × 2 = 1 + 0.790 385 085 808 622 989 134 164 527 058 601 379 394 531 250 002 572 961 369 817 088;
  • 47) 0.790 385 085 808 622 989 134 164 527 058 601 379 394 531 250 002 572 961 369 817 088 × 2 = 1 + 0.580 770 171 617 245 978 268 329 054 117 202 758 789 062 500 005 145 922 739 634 176;
  • 48) 0.580 770 171 617 245 978 268 329 054 117 202 758 789 062 500 005 145 922 739 634 176 × 2 = 1 + 0.161 540 343 234 491 956 536 658 108 234 405 517 578 125 000 010 291 845 479 268 352;
  • 49) 0.161 540 343 234 491 956 536 658 108 234 405 517 578 125 000 010 291 845 479 268 352 × 2 = 0 + 0.323 080 686 468 983 913 073 316 216 468 811 035 156 250 000 020 583 690 958 536 704;
  • 50) 0.323 080 686 468 983 913 073 316 216 468 811 035 156 250 000 020 583 690 958 536 704 × 2 = 0 + 0.646 161 372 937 967 826 146 632 432 937 622 070 312 500 000 041 167 381 917 073 408;
  • 51) 0.646 161 372 937 967 826 146 632 432 937 622 070 312 500 000 041 167 381 917 073 408 × 2 = 1 + 0.292 322 745 875 935 652 293 264 865 875 244 140 625 000 000 082 334 763 834 146 816;
  • 52) 0.292 322 745 875 935 652 293 264 865 875 244 140 625 000 000 082 334 763 834 146 816 × 2 = 0 + 0.584 645 491 751 871 304 586 529 731 750 488 281 250 000 000 164 669 527 668 293 632;
  • 53) 0.584 645 491 751 871 304 586 529 731 750 488 281 250 000 000 164 669 527 668 293 632 × 2 = 1 + 0.169 290 983 503 742 609 173 059 463 500 976 562 500 000 000 329 339 055 336 587 264;
  • 54) 0.169 290 983 503 742 609 173 059 463 500 976 562 500 000 000 329 339 055 336 587 264 × 2 = 0 + 0.338 581 967 007 485 218 346 118 927 001 953 125 000 000 000 658 678 110 673 174 528;
  • 55) 0.338 581 967 007 485 218 346 118 927 001 953 125 000 000 000 658 678 110 673 174 528 × 2 = 0 + 0.677 163 934 014 970 436 692 237 854 003 906 250 000 000 001 317 356 221 346 349 056;
  • 56) 0.677 163 934 014 970 436 692 237 854 003 906 250 000 000 001 317 356 221 346 349 056 × 2 = 1 + 0.354 327 868 029 940 873 384 475 708 007 812 500 000 000 002 634 712 442 692 698 112;
  • 57) 0.354 327 868 029 940 873 384 475 708 007 812 500 000 000 002 634 712 442 692 698 112 × 2 = 0 + 0.708 655 736 059 881 746 768 951 416 015 625 000 000 000 005 269 424 885 385 396 224;
  • 58) 0.708 655 736 059 881 746 768 951 416 015 625 000 000 000 005 269 424 885 385 396 224 × 2 = 1 + 0.417 311 472 119 763 493 537 902 832 031 250 000 000 000 010 538 849 770 770 792 448;
  • 59) 0.417 311 472 119 763 493 537 902 832 031 250 000 000 000 010 538 849 770 770 792 448 × 2 = 0 + 0.834 622 944 239 526 987 075 805 664 062 500 000 000 000 021 077 699 541 541 584 896;
  • 60) 0.834 622 944 239 526 987 075 805 664 062 500 000 000 000 021 077 699 541 541 584 896 × 2 = 1 + 0.669 245 888 479 053 974 151 611 328 125 000 000 000 000 042 155 399 083 083 169 792;
  • 61) 0.669 245 888 479 053 974 151 611 328 125 000 000 000 000 042 155 399 083 083 169 792 × 2 = 1 + 0.338 491 776 958 107 948 303 222 656 250 000 000 000 000 084 310 798 166 166 339 584;
  • 62) 0.338 491 776 958 107 948 303 222 656 250 000 000 000 000 084 310 798 166 166 339 584 × 2 = 0 + 0.676 983 553 916 215 896 606 445 312 500 000 000 000 000 168 621 596 332 332 679 168;
  • 63) 0.676 983 553 916 215 896 606 445 312 500 000 000 000 000 168 621 596 332 332 679 168 × 2 = 1 + 0.353 967 107 832 431 793 212 890 625 000 000 000 000 000 337 243 192 664 665 358 336;
  • 64) 0.353 967 107 832 431 793 212 890 625 000 000 000 000 000 337 243 192 664 665 358 336 × 2 = 0 + 0.707 934 215 664 863 586 425 781 250 000 000 000 000 000 674 486 385 329 330 716 672;
  • 65) 0.707 934 215 664 863 586 425 781 250 000 000 000 000 000 674 486 385 329 330 716 672 × 2 = 1 + 0.415 868 431 329 727 172 851 562 500 000 000 000 000 001 348 972 770 658 661 433 344;
  • 66) 0.415 868 431 329 727 172 851 562 500 000 000 000 000 001 348 972 770 658 661 433 344 × 2 = 0 + 0.831 736 862 659 454 345 703 125 000 000 000 000 000 002 697 945 541 317 322 866 688;
  • 67) 0.831 736 862 659 454 345 703 125 000 000 000 000 000 002 697 945 541 317 322 866 688 × 2 = 1 + 0.663 473 725 318 908 691 406 250 000 000 000 000 000 005 395 891 082 634 645 733 376;
  • 68) 0.663 473 725 318 908 691 406 250 000 000 000 000 000 005 395 891 082 634 645 733 376 × 2 = 1 + 0.326 947 450 637 817 382 812 500 000 000 000 000 000 010 791 782 165 269 291 466 752;
  • 69) 0.326 947 450 637 817 382 812 500 000 000 000 000 000 010 791 782 165 269 291 466 752 × 2 = 0 + 0.653 894 901 275 634 765 625 000 000 000 000 000 000 021 583 564 330 538 582 933 504;
  • 70) 0.653 894 901 275 634 765 625 000 000 000 000 000 000 021 583 564 330 538 582 933 504 × 2 = 1 + 0.307 789 802 551 269 531 250 000 000 000 000 000 000 043 167 128 661 077 165 867 008;
  • 71) 0.307 789 802 551 269 531 250 000 000 000 000 000 000 043 167 128 661 077 165 867 008 × 2 = 0 + 0.615 579 605 102 539 062 500 000 000 000 000 000 000 086 334 257 322 154 331 734 016;
  • 72) 0.615 579 605 102 539 062 500 000 000 000 000 000 000 086 334 257 322 154 331 734 016 × 2 = 1 + 0.231 159 210 205 078 125 000 000 000 000 000 000 000 172 668 514 644 308 663 468 032;
  • 73) 0.231 159 210 205 078 125 000 000 000 000 000 000 000 172 668 514 644 308 663 468 032 × 2 = 0 + 0.462 318 420 410 156 250 000 000 000 000 000 000 000 345 337 029 288 617 326 936 064;
  • 74) 0.462 318 420 410 156 250 000 000 000 000 000 000 000 345 337 029 288 617 326 936 064 × 2 = 0 + 0.924 636 840 820 312 500 000 000 000 000 000 000 000 690 674 058 577 234 653 872 128;
  • 75) 0.924 636 840 820 312 500 000 000 000 000 000 000 000 690 674 058 577 234 653 872 128 × 2 = 1 + 0.849 273 681 640 625 000 000 000 000 000 000 000 001 381 348 117 154 469 307 744 256;
  • 76) 0.849 273 681 640 625 000 000 000 000 000 000 000 001 381 348 117 154 469 307 744 256 × 2 = 1 + 0.698 547 363 281 250 000 000 000 000 000 000 000 002 762 696 234 308 938 615 488 512;
  • 77) 0.698 547 363 281 250 000 000 000 000 000 000 000 002 762 696 234 308 938 615 488 512 × 2 = 1 + 0.397 094 726 562 500 000 000 000 000 000 000 000 005 525 392 468 617 877 230 977 024;
  • 78) 0.397 094 726 562 500 000 000 000 000 000 000 000 005 525 392 468 617 877 230 977 024 × 2 = 0 + 0.794 189 453 125 000 000 000 000 000 000 000 000 011 050 784 937 235 754 461 954 048;
  • 79) 0.794 189 453 125 000 000 000 000 000 000 000 000 011 050 784 937 235 754 461 954 048 × 2 = 1 + 0.588 378 906 250 000 000 000 000 000 000 000 000 022 101 569 874 471 508 923 908 096;
  • 80) 0.588 378 906 250 000 000 000 000 000 000 000 000 022 101 569 874 471 508 923 908 096 × 2 = 1 + 0.176 757 812 500 000 000 000 000 000 000 000 000 044 203 139 748 943 017 847 816 192;
  • 81) 0.176 757 812 500 000 000 000 000 000 000 000 000 044 203 139 748 943 017 847 816 192 × 2 = 0 + 0.353 515 625 000 000 000 000 000 000 000 000 000 088 406 279 497 886 035 695 632 384;
  • 82) 0.353 515 625 000 000 000 000 000 000 000 000 000 088 406 279 497 886 035 695 632 384 × 2 = 0 + 0.707 031 250 000 000 000 000 000 000 000 000 000 176 812 558 995 772 071 391 264 768;
  • 83) 0.707 031 250 000 000 000 000 000 000 000 000 000 176 812 558 995 772 071 391 264 768 × 2 = 1 + 0.414 062 500 000 000 000 000 000 000 000 000 000 353 625 117 991 544 142 782 529 536;
  • 84) 0.414 062 500 000 000 000 000 000 000 000 000 000 353 625 117 991 544 142 782 529 536 × 2 = 0 + 0.828 125 000 000 000 000 000 000 000 000 000 000 707 250 235 983 088 285 565 059 072;
  • 85) 0.828 125 000 000 000 000 000 000 000 000 000 000 707 250 235 983 088 285 565 059 072 × 2 = 1 + 0.656 250 000 000 000 000 000 000 000 000 000 001 414 500 471 966 176 571 130 118 144;
  • 86) 0.656 250 000 000 000 000 000 000 000 000 000 001 414 500 471 966 176 571 130 118 144 × 2 = 1 + 0.312 500 000 000 000 000 000 000 000 000 000 002 829 000 943 932 353 142 260 236 288;
  • 87) 0.312 500 000 000 000 000 000 000 000 000 000 002 829 000 943 932 353 142 260 236 288 × 2 = 0 + 0.625 000 000 000 000 000 000 000 000 000 000 005 658 001 887 864 706 284 520 472 576;
  • 88) 0.625 000 000 000 000 000 000 000 000 000 000 005 658 001 887 864 706 284 520 472 576 × 2 = 1 + 0.250 000 000 000 000 000 000 000 000 000 000 011 316 003 775 729 412 569 040 945 152;
  • 89) 0.250 000 000 000 000 000 000 000 000 000 000 011 316 003 775 729 412 569 040 945 152 × 2 = 0 + 0.500 000 000 000 000 000 000 000 000 000 000 022 632 007 551 458 825 138 081 890 304;
  • 90) 0.500 000 000 000 000 000 000 000 000 000 000 022 632 007 551 458 825 138 081 890 304 × 2 = 1 + 0.000 000 000 000 000 000 000 000 000 000 000 045 264 015 102 917 650 276 163 780 608;
  • 91) 0.000 000 000 000 000 000 000 000 000 000 000 045 264 015 102 917 650 276 163 780 608 × 2 = 0 + 0.000 000 000 000 000 000 000 000 000 000 000 090 528 030 205 835 300 552 327 561 216;

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:


0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 029 932 467(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:

0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 029 932 467(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:


0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 029 932 467(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 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 029 932 467 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