0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 029 932 430 436 019 785 561 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 029 932 430 436 019 785 561(10) to 64 bit double precision IEEE 754 binary floating point representation standard (1 bit for sign, 11 bits for exponent, 52 bits for mantissa)

What are the steps to convert decimal number
0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 029 932 430 436 019 785 561(10) to 64 bit double precision IEEE 754 binary floating point representation (1 bit for sign, 11 bits for exponent, 52 bits for mantissa)

1. First, convert to binary (in base 2) the integer part: 0.
Divide the number repeatedly by 2.

Keep track of each remainder.

We stop when we get a quotient that is equal to zero.


  • division = quotient + remainder;
  • 0 ÷ 2 = 0 + 0;

2. Construct the base 2 representation of the integer part of the number.

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

0(10) =

0(2)


3. Convert to binary (base 2) the fractional part: 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 029 932 430 436 019 785 561.

Multiply it repeatedly by 2.

Keep track of each integer part of the results.

Stop when we get a fractional part that is equal to zero.


  • #) multiplying = integer + fractional part;
  • 1) 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 029 932 430 436 019 785 561 × 2 = 0 + 0.000 000 000 007 269 999 999 999 999 516 522 114 064 601 356 670 305 976 059 864 860 872 039 571 122;
  • 2) 0.000 000 000 007 269 999 999 999 999 516 522 114 064 601 356 670 305 976 059 864 860 872 039 571 122 × 2 = 0 + 0.000 000 000 014 539 999 999 999 999 033 044 228 129 202 713 340 611 952 119 729 721 744 079 142 244;
  • 3) 0.000 000 000 014 539 999 999 999 999 033 044 228 129 202 713 340 611 952 119 729 721 744 079 142 244 × 2 = 0 + 0.000 000 000 029 079 999 999 999 998 066 088 456 258 405 426 681 223 904 239 459 443 488 158 284 488;
  • 4) 0.000 000 000 029 079 999 999 999 998 066 088 456 258 405 426 681 223 904 239 459 443 488 158 284 488 × 2 = 0 + 0.000 000 000 058 159 999 999 999 996 132 176 912 516 810 853 362 447 808 478 918 886 976 316 568 976;
  • 5) 0.000 000 000 058 159 999 999 999 996 132 176 912 516 810 853 362 447 808 478 918 886 976 316 568 976 × 2 = 0 + 0.000 000 000 116 319 999 999 999 992 264 353 825 033 621 706 724 895 616 957 837 773 952 633 137 952;
  • 6) 0.000 000 000 116 319 999 999 999 992 264 353 825 033 621 706 724 895 616 957 837 773 952 633 137 952 × 2 = 0 + 0.000 000 000 232 639 999 999 999 984 528 707 650 067 243 413 449 791 233 915 675 547 905 266 275 904;
  • 7) 0.000 000 000 232 639 999 999 999 984 528 707 650 067 243 413 449 791 233 915 675 547 905 266 275 904 × 2 = 0 + 0.000 000 000 465 279 999 999 999 969 057 415 300 134 486 826 899 582 467 831 351 095 810 532 551 808;
  • 8) 0.000 000 000 465 279 999 999 999 969 057 415 300 134 486 826 899 582 467 831 351 095 810 532 551 808 × 2 = 0 + 0.000 000 000 930 559 999 999 999 938 114 830 600 268 973 653 799 164 935 662 702 191 621 065 103 616;
  • 9) 0.000 000 000 930 559 999 999 999 938 114 830 600 268 973 653 799 164 935 662 702 191 621 065 103 616 × 2 = 0 + 0.000 000 001 861 119 999 999 999 876 229 661 200 537 947 307 598 329 871 325 404 383 242 130 207 232;
  • 10) 0.000 000 001 861 119 999 999 999 876 229 661 200 537 947 307 598 329 871 325 404 383 242 130 207 232 × 2 = 0 + 0.000 000 003 722 239 999 999 999 752 459 322 401 075 894 615 196 659 742 650 808 766 484 260 414 464;
  • 11) 0.000 000 003 722 239 999 999 999 752 459 322 401 075 894 615 196 659 742 650 808 766 484 260 414 464 × 2 = 0 + 0.000 000 007 444 479 999 999 999 504 918 644 802 151 789 230 393 319 485 301 617 532 968 520 828 928;
  • 12) 0.000 000 007 444 479 999 999 999 504 918 644 802 151 789 230 393 319 485 301 617 532 968 520 828 928 × 2 = 0 + 0.000 000 014 888 959 999 999 999 009 837 289 604 303 578 460 786 638 970 603 235 065 937 041 657 856;
  • 13) 0.000 000 014 888 959 999 999 999 009 837 289 604 303 578 460 786 638 970 603 235 065 937 041 657 856 × 2 = 0 + 0.000 000 029 777 919 999 999 998 019 674 579 208 607 156 921 573 277 941 206 470 131 874 083 315 712;
  • 14) 0.000 000 029 777 919 999 999 998 019 674 579 208 607 156 921 573 277 941 206 470 131 874 083 315 712 × 2 = 0 + 0.000 000 059 555 839 999 999 996 039 349 158 417 214 313 843 146 555 882 412 940 263 748 166 631 424;
  • 15) 0.000 000 059 555 839 999 999 996 039 349 158 417 214 313 843 146 555 882 412 940 263 748 166 631 424 × 2 = 0 + 0.000 000 119 111 679 999 999 992 078 698 316 834 428 627 686 293 111 764 825 880 527 496 333 262 848;
  • 16) 0.000 000 119 111 679 999 999 992 078 698 316 834 428 627 686 293 111 764 825 880 527 496 333 262 848 × 2 = 0 + 0.000 000 238 223 359 999 999 984 157 396 633 668 857 255 372 586 223 529 651 761 054 992 666 525 696;
  • 17) 0.000 000 238 223 359 999 999 984 157 396 633 668 857 255 372 586 223 529 651 761 054 992 666 525 696 × 2 = 0 + 0.000 000 476 446 719 999 999 968 314 793 267 337 714 510 745 172 447 059 303 522 109 985 333 051 392;
  • 18) 0.000 000 476 446 719 999 999 968 314 793 267 337 714 510 745 172 447 059 303 522 109 985 333 051 392 × 2 = 0 + 0.000 000 952 893 439 999 999 936 629 586 534 675 429 021 490 344 894 118 607 044 219 970 666 102 784;
  • 19) 0.000 000 952 893 439 999 999 936 629 586 534 675 429 021 490 344 894 118 607 044 219 970 666 102 784 × 2 = 0 + 0.000 001 905 786 879 999 999 873 259 173 069 350 858 042 980 689 788 237 214 088 439 941 332 205 568;
  • 20) 0.000 001 905 786 879 999 999 873 259 173 069 350 858 042 980 689 788 237 214 088 439 941 332 205 568 × 2 = 0 + 0.000 003 811 573 759 999 999 746 518 346 138 701 716 085 961 379 576 474 428 176 879 882 664 411 136;
  • 21) 0.000 003 811 573 759 999 999 746 518 346 138 701 716 085 961 379 576 474 428 176 879 882 664 411 136 × 2 = 0 + 0.000 007 623 147 519 999 999 493 036 692 277 403 432 171 922 759 152 948 856 353 759 765 328 822 272;
  • 22) 0.000 007 623 147 519 999 999 493 036 692 277 403 432 171 922 759 152 948 856 353 759 765 328 822 272 × 2 = 0 + 0.000 015 246 295 039 999 998 986 073 384 554 806 864 343 845 518 305 897 712 707 519 530 657 644 544;
  • 23) 0.000 015 246 295 039 999 998 986 073 384 554 806 864 343 845 518 305 897 712 707 519 530 657 644 544 × 2 = 0 + 0.000 030 492 590 079 999 997 972 146 769 109 613 728 687 691 036 611 795 425 415 039 061 315 289 088;
  • 24) 0.000 030 492 590 079 999 997 972 146 769 109 613 728 687 691 036 611 795 425 415 039 061 315 289 088 × 2 = 0 + 0.000 060 985 180 159 999 995 944 293 538 219 227 457 375 382 073 223 590 850 830 078 122 630 578 176;
  • 25) 0.000 060 985 180 159 999 995 944 293 538 219 227 457 375 382 073 223 590 850 830 078 122 630 578 176 × 2 = 0 + 0.000 121 970 360 319 999 991 888 587 076 438 454 914 750 764 146 447 181 701 660 156 245 261 156 352;
  • 26) 0.000 121 970 360 319 999 991 888 587 076 438 454 914 750 764 146 447 181 701 660 156 245 261 156 352 × 2 = 0 + 0.000 243 940 720 639 999 983 777 174 152 876 909 829 501 528 292 894 363 403 320 312 490 522 312 704;
  • 27) 0.000 243 940 720 639 999 983 777 174 152 876 909 829 501 528 292 894 363 403 320 312 490 522 312 704 × 2 = 0 + 0.000 487 881 441 279 999 967 554 348 305 753 819 659 003 056 585 788 726 806 640 624 981 044 625 408;
  • 28) 0.000 487 881 441 279 999 967 554 348 305 753 819 659 003 056 585 788 726 806 640 624 981 044 625 408 × 2 = 0 + 0.000 975 762 882 559 999 935 108 696 611 507 639 318 006 113 171 577 453 613 281 249 962 089 250 816;
  • 29) 0.000 975 762 882 559 999 935 108 696 611 507 639 318 006 113 171 577 453 613 281 249 962 089 250 816 × 2 = 0 + 0.001 951 525 765 119 999 870 217 393 223 015 278 636 012 226 343 154 907 226 562 499 924 178 501 632;
  • 30) 0.001 951 525 765 119 999 870 217 393 223 015 278 636 012 226 343 154 907 226 562 499 924 178 501 632 × 2 = 0 + 0.003 903 051 530 239 999 740 434 786 446 030 557 272 024 452 686 309 814 453 124 999 848 357 003 264;
  • 31) 0.003 903 051 530 239 999 740 434 786 446 030 557 272 024 452 686 309 814 453 124 999 848 357 003 264 × 2 = 0 + 0.007 806 103 060 479 999 480 869 572 892 061 114 544 048 905 372 619 628 906 249 999 696 714 006 528;
  • 32) 0.007 806 103 060 479 999 480 869 572 892 061 114 544 048 905 372 619 628 906 249 999 696 714 006 528 × 2 = 0 + 0.015 612 206 120 959 998 961 739 145 784 122 229 088 097 810 745 239 257 812 499 999 393 428 013 056;
  • 33) 0.015 612 206 120 959 998 961 739 145 784 122 229 088 097 810 745 239 257 812 499 999 393 428 013 056 × 2 = 0 + 0.031 224 412 241 919 997 923 478 291 568 244 458 176 195 621 490 478 515 624 999 998 786 856 026 112;
  • 34) 0.031 224 412 241 919 997 923 478 291 568 244 458 176 195 621 490 478 515 624 999 998 786 856 026 112 × 2 = 0 + 0.062 448 824 483 839 995 846 956 583 136 488 916 352 391 242 980 957 031 249 999 997 573 712 052 224;
  • 35) 0.062 448 824 483 839 995 846 956 583 136 488 916 352 391 242 980 957 031 249 999 997 573 712 052 224 × 2 = 0 + 0.124 897 648 967 679 991 693 913 166 272 977 832 704 782 485 961 914 062 499 999 995 147 424 104 448;
  • 36) 0.124 897 648 967 679 991 693 913 166 272 977 832 704 782 485 961 914 062 499 999 995 147 424 104 448 × 2 = 0 + 0.249 795 297 935 359 983 387 826 332 545 955 665 409 564 971 923 828 124 999 999 990 294 848 208 896;
  • 37) 0.249 795 297 935 359 983 387 826 332 545 955 665 409 564 971 923 828 124 999 999 990 294 848 208 896 × 2 = 0 + 0.499 590 595 870 719 966 775 652 665 091 911 330 819 129 943 847 656 249 999 999 980 589 696 417 792;
  • 38) 0.499 590 595 870 719 966 775 652 665 091 911 330 819 129 943 847 656 249 999 999 980 589 696 417 792 × 2 = 0 + 0.999 181 191 741 439 933 551 305 330 183 822 661 638 259 887 695 312 499 999 999 961 179 392 835 584;
  • 39) 0.999 181 191 741 439 933 551 305 330 183 822 661 638 259 887 695 312 499 999 999 961 179 392 835 584 × 2 = 1 + 0.998 362 383 482 879 867 102 610 660 367 645 323 276 519 775 390 624 999 999 999 922 358 785 671 168;
  • 40) 0.998 362 383 482 879 867 102 610 660 367 645 323 276 519 775 390 624 999 999 999 922 358 785 671 168 × 2 = 1 + 0.996 724 766 965 759 734 205 221 320 735 290 646 553 039 550 781 249 999 999 999 844 717 571 342 336;
  • 41) 0.996 724 766 965 759 734 205 221 320 735 290 646 553 039 550 781 249 999 999 999 844 717 571 342 336 × 2 = 1 + 0.993 449 533 931 519 468 410 442 641 470 581 293 106 079 101 562 499 999 999 999 689 435 142 684 672;
  • 42) 0.993 449 533 931 519 468 410 442 641 470 581 293 106 079 101 562 499 999 999 999 689 435 142 684 672 × 2 = 1 + 0.986 899 067 863 038 936 820 885 282 941 162 586 212 158 203 124 999 999 999 999 378 870 285 369 344;
  • 43) 0.986 899 067 863 038 936 820 885 282 941 162 586 212 158 203 124 999 999 999 999 378 870 285 369 344 × 2 = 1 + 0.973 798 135 726 077 873 641 770 565 882 325 172 424 316 406 249 999 999 999 998 757 740 570 738 688;
  • 44) 0.973 798 135 726 077 873 641 770 565 882 325 172 424 316 406 249 999 999 999 998 757 740 570 738 688 × 2 = 1 + 0.947 596 271 452 155 747 283 541 131 764 650 344 848 632 812 499 999 999 999 997 515 481 141 477 376;
  • 45) 0.947 596 271 452 155 747 283 541 131 764 650 344 848 632 812 499 999 999 999 997 515 481 141 477 376 × 2 = 1 + 0.895 192 542 904 311 494 567 082 263 529 300 689 697 265 624 999 999 999 999 995 030 962 282 954 752;
  • 46) 0.895 192 542 904 311 494 567 082 263 529 300 689 697 265 624 999 999 999 999 995 030 962 282 954 752 × 2 = 1 + 0.790 385 085 808 622 989 134 164 527 058 601 379 394 531 249 999 999 999 999 990 061 924 565 909 504;
  • 47) 0.790 385 085 808 622 989 134 164 527 058 601 379 394 531 249 999 999 999 999 990 061 924 565 909 504 × 2 = 1 + 0.580 770 171 617 245 978 268 329 054 117 202 758 789 062 499 999 999 999 999 980 123 849 131 819 008;
  • 48) 0.580 770 171 617 245 978 268 329 054 117 202 758 789 062 499 999 999 999 999 980 123 849 131 819 008 × 2 = 1 + 0.161 540 343 234 491 956 536 658 108 234 405 517 578 124 999 999 999 999 999 960 247 698 263 638 016;
  • 49) 0.161 540 343 234 491 956 536 658 108 234 405 517 578 124 999 999 999 999 999 960 247 698 263 638 016 × 2 = 0 + 0.323 080 686 468 983 913 073 316 216 468 811 035 156 249 999 999 999 999 999 920 495 396 527 276 032;
  • 50) 0.323 080 686 468 983 913 073 316 216 468 811 035 156 249 999 999 999 999 999 920 495 396 527 276 032 × 2 = 0 + 0.646 161 372 937 967 826 146 632 432 937 622 070 312 499 999 999 999 999 999 840 990 793 054 552 064;
  • 51) 0.646 161 372 937 967 826 146 632 432 937 622 070 312 499 999 999 999 999 999 840 990 793 054 552 064 × 2 = 1 + 0.292 322 745 875 935 652 293 264 865 875 244 140 624 999 999 999 999 999 999 681 981 586 109 104 128;
  • 52) 0.292 322 745 875 935 652 293 264 865 875 244 140 624 999 999 999 999 999 999 681 981 586 109 104 128 × 2 = 0 + 0.584 645 491 751 871 304 586 529 731 750 488 281 249 999 999 999 999 999 999 363 963 172 218 208 256;
  • 53) 0.584 645 491 751 871 304 586 529 731 750 488 281 249 999 999 999 999 999 999 363 963 172 218 208 256 × 2 = 1 + 0.169 290 983 503 742 609 173 059 463 500 976 562 499 999 999 999 999 999 998 727 926 344 436 416 512;
  • 54) 0.169 290 983 503 742 609 173 059 463 500 976 562 499 999 999 999 999 999 998 727 926 344 436 416 512 × 2 = 0 + 0.338 581 967 007 485 218 346 118 927 001 953 124 999 999 999 999 999 999 997 455 852 688 872 833 024;
  • 55) 0.338 581 967 007 485 218 346 118 927 001 953 124 999 999 999 999 999 999 997 455 852 688 872 833 024 × 2 = 0 + 0.677 163 934 014 970 436 692 237 854 003 906 249 999 999 999 999 999 999 994 911 705 377 745 666 048;
  • 56) 0.677 163 934 014 970 436 692 237 854 003 906 249 999 999 999 999 999 999 994 911 705 377 745 666 048 × 2 = 1 + 0.354 327 868 029 940 873 384 475 708 007 812 499 999 999 999 999 999 999 989 823 410 755 491 332 096;
  • 57) 0.354 327 868 029 940 873 384 475 708 007 812 499 999 999 999 999 999 999 989 823 410 755 491 332 096 × 2 = 0 + 0.708 655 736 059 881 746 768 951 416 015 624 999 999 999 999 999 999 999 979 646 821 510 982 664 192;
  • 58) 0.708 655 736 059 881 746 768 951 416 015 624 999 999 999 999 999 999 999 979 646 821 510 982 664 192 × 2 = 1 + 0.417 311 472 119 763 493 537 902 832 031 249 999 999 999 999 999 999 999 959 293 643 021 965 328 384;
  • 59) 0.417 311 472 119 763 493 537 902 832 031 249 999 999 999 999 999 999 999 959 293 643 021 965 328 384 × 2 = 0 + 0.834 622 944 239 526 987 075 805 664 062 499 999 999 999 999 999 999 999 918 587 286 043 930 656 768;
  • 60) 0.834 622 944 239 526 987 075 805 664 062 499 999 999 999 999 999 999 999 918 587 286 043 930 656 768 × 2 = 1 + 0.669 245 888 479 053 974 151 611 328 124 999 999 999 999 999 999 999 999 837 174 572 087 861 313 536;
  • 61) 0.669 245 888 479 053 974 151 611 328 124 999 999 999 999 999 999 999 999 837 174 572 087 861 313 536 × 2 = 1 + 0.338 491 776 958 107 948 303 222 656 249 999 999 999 999 999 999 999 999 674 349 144 175 722 627 072;
  • 62) 0.338 491 776 958 107 948 303 222 656 249 999 999 999 999 999 999 999 999 674 349 144 175 722 627 072 × 2 = 0 + 0.676 983 553 916 215 896 606 445 312 499 999 999 999 999 999 999 999 999 348 698 288 351 445 254 144;
  • 63) 0.676 983 553 916 215 896 606 445 312 499 999 999 999 999 999 999 999 999 348 698 288 351 445 254 144 × 2 = 1 + 0.353 967 107 832 431 793 212 890 624 999 999 999 999 999 999 999 999 998 697 396 576 702 890 508 288;
  • 64) 0.353 967 107 832 431 793 212 890 624 999 999 999 999 999 999 999 999 998 697 396 576 702 890 508 288 × 2 = 0 + 0.707 934 215 664 863 586 425 781 249 999 999 999 999 999 999 999 999 997 394 793 153 405 781 016 576;
  • 65) 0.707 934 215 664 863 586 425 781 249 999 999 999 999 999 999 999 999 997 394 793 153 405 781 016 576 × 2 = 1 + 0.415 868 431 329 727 172 851 562 499 999 999 999 999 999 999 999 999 994 789 586 306 811 562 033 152;
  • 66) 0.415 868 431 329 727 172 851 562 499 999 999 999 999 999 999 999 999 994 789 586 306 811 562 033 152 × 2 = 0 + 0.831 736 862 659 454 345 703 124 999 999 999 999 999 999 999 999 999 989 579 172 613 623 124 066 304;
  • 67) 0.831 736 862 659 454 345 703 124 999 999 999 999 999 999 999 999 999 989 579 172 613 623 124 066 304 × 2 = 1 + 0.663 473 725 318 908 691 406 249 999 999 999 999 999 999 999 999 999 979 158 345 227 246 248 132 608;
  • 68) 0.663 473 725 318 908 691 406 249 999 999 999 999 999 999 999 999 999 979 158 345 227 246 248 132 608 × 2 = 1 + 0.326 947 450 637 817 382 812 499 999 999 999 999 999 999 999 999 999 958 316 690 454 492 496 265 216;
  • 69) 0.326 947 450 637 817 382 812 499 999 999 999 999 999 999 999 999 999 958 316 690 454 492 496 265 216 × 2 = 0 + 0.653 894 901 275 634 765 624 999 999 999 999 999 999 999 999 999 999 916 633 380 908 984 992 530 432;
  • 70) 0.653 894 901 275 634 765 624 999 999 999 999 999 999 999 999 999 999 916 633 380 908 984 992 530 432 × 2 = 1 + 0.307 789 802 551 269 531 249 999 999 999 999 999 999 999 999 999 999 833 266 761 817 969 985 060 864;
  • 71) 0.307 789 802 551 269 531 249 999 999 999 999 999 999 999 999 999 999 833 266 761 817 969 985 060 864 × 2 = 0 + 0.615 579 605 102 539 062 499 999 999 999 999 999 999 999 999 999 999 666 533 523 635 939 970 121 728;
  • 72) 0.615 579 605 102 539 062 499 999 999 999 999 999 999 999 999 999 999 666 533 523 635 939 970 121 728 × 2 = 1 + 0.231 159 210 205 078 124 999 999 999 999 999 999 999 999 999 999 999 333 067 047 271 879 940 243 456;
  • 73) 0.231 159 210 205 078 124 999 999 999 999 999 999 999 999 999 999 999 333 067 047 271 879 940 243 456 × 2 = 0 + 0.462 318 420 410 156 249 999 999 999 999 999 999 999 999 999 999 998 666 134 094 543 759 880 486 912;
  • 74) 0.462 318 420 410 156 249 999 999 999 999 999 999 999 999 999 999 998 666 134 094 543 759 880 486 912 × 2 = 0 + 0.924 636 840 820 312 499 999 999 999 999 999 999 999 999 999 999 997 332 268 189 087 519 760 973 824;
  • 75) 0.924 636 840 820 312 499 999 999 999 999 999 999 999 999 999 999 997 332 268 189 087 519 760 973 824 × 2 = 1 + 0.849 273 681 640 624 999 999 999 999 999 999 999 999 999 999 999 994 664 536 378 175 039 521 947 648;
  • 76) 0.849 273 681 640 624 999 999 999 999 999 999 999 999 999 999 999 994 664 536 378 175 039 521 947 648 × 2 = 1 + 0.698 547 363 281 249 999 999 999 999 999 999 999 999 999 999 999 989 329 072 756 350 079 043 895 296;
  • 77) 0.698 547 363 281 249 999 999 999 999 999 999 999 999 999 999 999 989 329 072 756 350 079 043 895 296 × 2 = 1 + 0.397 094 726 562 499 999 999 999 999 999 999 999 999 999 999 999 978 658 145 512 700 158 087 790 592;
  • 78) 0.397 094 726 562 499 999 999 999 999 999 999 999 999 999 999 999 978 658 145 512 700 158 087 790 592 × 2 = 0 + 0.794 189 453 124 999 999 999 999 999 999 999 999 999 999 999 999 957 316 291 025 400 316 175 581 184;
  • 79) 0.794 189 453 124 999 999 999 999 999 999 999 999 999 999 999 999 957 316 291 025 400 316 175 581 184 × 2 = 1 + 0.588 378 906 249 999 999 999 999 999 999 999 999 999 999 999 999 914 632 582 050 800 632 351 162 368;
  • 80) 0.588 378 906 249 999 999 999 999 999 999 999 999 999 999 999 999 914 632 582 050 800 632 351 162 368 × 2 = 1 + 0.176 757 812 499 999 999 999 999 999 999 999 999 999 999 999 999 829 265 164 101 601 264 702 324 736;
  • 81) 0.176 757 812 499 999 999 999 999 999 999 999 999 999 999 999 999 829 265 164 101 601 264 702 324 736 × 2 = 0 + 0.353 515 624 999 999 999 999 999 999 999 999 999 999 999 999 999 658 530 328 203 202 529 404 649 472;
  • 82) 0.353 515 624 999 999 999 999 999 999 999 999 999 999 999 999 999 658 530 328 203 202 529 404 649 472 × 2 = 0 + 0.707 031 249 999 999 999 999 999 999 999 999 999 999 999 999 999 317 060 656 406 405 058 809 298 944;
  • 83) 0.707 031 249 999 999 999 999 999 999 999 999 999 999 999 999 999 317 060 656 406 405 058 809 298 944 × 2 = 1 + 0.414 062 499 999 999 999 999 999 999 999 999 999 999 999 999 998 634 121 312 812 810 117 618 597 888;
  • 84) 0.414 062 499 999 999 999 999 999 999 999 999 999 999 999 999 998 634 121 312 812 810 117 618 597 888 × 2 = 0 + 0.828 124 999 999 999 999 999 999 999 999 999 999 999 999 999 997 268 242 625 625 620 235 237 195 776;
  • 85) 0.828 124 999 999 999 999 999 999 999 999 999 999 999 999 999 997 268 242 625 625 620 235 237 195 776 × 2 = 1 + 0.656 249 999 999 999 999 999 999 999 999 999 999 999 999 999 994 536 485 251 251 240 470 474 391 552;
  • 86) 0.656 249 999 999 999 999 999 999 999 999 999 999 999 999 999 994 536 485 251 251 240 470 474 391 552 × 2 = 1 + 0.312 499 999 999 999 999 999 999 999 999 999 999 999 999 999 989 072 970 502 502 480 940 948 783 104;
  • 87) 0.312 499 999 999 999 999 999 999 999 999 999 999 999 999 999 989 072 970 502 502 480 940 948 783 104 × 2 = 0 + 0.624 999 999 999 999 999 999 999 999 999 999 999 999 999 999 978 145 941 005 004 961 881 897 566 208;
  • 88) 0.624 999 999 999 999 999 999 999 999 999 999 999 999 999 999 978 145 941 005 004 961 881 897 566 208 × 2 = 1 + 0.249 999 999 999 999 999 999 999 999 999 999 999 999 999 999 956 291 882 010 009 923 763 795 132 416;
  • 89) 0.249 999 999 999 999 999 999 999 999 999 999 999 999 999 999 956 291 882 010 009 923 763 795 132 416 × 2 = 0 + 0.499 999 999 999 999 999 999 999 999 999 999 999 999 999 999 912 583 764 020 019 847 527 590 264 832;
  • 90) 0.499 999 999 999 999 999 999 999 999 999 999 999 999 999 999 912 583 764 020 019 847 527 590 264 832 × 2 = 0 + 0.999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 825 167 528 040 039 695 055 180 529 664;
  • 91) 0.999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 825 167 528 040 039 695 055 180 529 664 × 2 = 1 + 0.999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 650 335 056 080 079 390 110 361 059 328;

We didn't get any fractional part that was equal to zero. But we had enough iterations (over Mantissa limit) and at least one integer that was different from zero => FULL STOP (Losing precision - the converted number we get in the end will be just a very good approximation of the initial one).


4. Construct the base 2 representation of the fractional part of the number.

Take all the integer parts of the multiplying operations, starting from the top of the constructed list above:


0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 029 932 430 436 019 785 561(10) =

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

5. Positive number before normalization:

0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 029 932 430 436 019 785 561(10) =

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

6. Normalize the binary representation of the number.

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


0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 029 932 430 436 019 785 561(10) =

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

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

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


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

Sign 0 (a positive number)

Exponent (unadjusted): -39

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


8. Adjust the exponent.

Use the 11 bit excess/bias notation:


Exponent (adjusted) =

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

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

(-39 + 1 023)(10) =

984(10)


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

Use the same technique of repeatedly dividing by 2:


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

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

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


Exponent (adjusted) =

984(10) =

011 1101 1000(2)


11. Normalize the mantissa.

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

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


Mantissa (normalized) =

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

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


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

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

Exponent (11 bits) =
011 1101 1000

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


Decimal number 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 029 932 430 436 019 785 561 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