0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 029 932 523 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 029 932 523(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
0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 029 932 523(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: 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 029 932 523.

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) 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 029 932 523 × 2 = 0 + 0.000 000 000 007 269 999 999 999 999 516 522 114 064 601 356 670 305 976 059 865 046;
  • 2) 0.000 000 000 007 269 999 999 999 999 516 522 114 064 601 356 670 305 976 059 865 046 × 2 = 0 + 0.000 000 000 014 539 999 999 999 999 033 044 228 129 202 713 340 611 952 119 730 092;
  • 3) 0.000 000 000 014 539 999 999 999 999 033 044 228 129 202 713 340 611 952 119 730 092 × 2 = 0 + 0.000 000 000 029 079 999 999 999 998 066 088 456 258 405 426 681 223 904 239 460 184;
  • 4) 0.000 000 000 029 079 999 999 999 998 066 088 456 258 405 426 681 223 904 239 460 184 × 2 = 0 + 0.000 000 000 058 159 999 999 999 996 132 176 912 516 810 853 362 447 808 478 920 368;
  • 5) 0.000 000 000 058 159 999 999 999 996 132 176 912 516 810 853 362 447 808 478 920 368 × 2 = 0 + 0.000 000 000 116 319 999 999 999 992 264 353 825 033 621 706 724 895 616 957 840 736;
  • 6) 0.000 000 000 116 319 999 999 999 992 264 353 825 033 621 706 724 895 616 957 840 736 × 2 = 0 + 0.000 000 000 232 639 999 999 999 984 528 707 650 067 243 413 449 791 233 915 681 472;
  • 7) 0.000 000 000 232 639 999 999 999 984 528 707 650 067 243 413 449 791 233 915 681 472 × 2 = 0 + 0.000 000 000 465 279 999 999 999 969 057 415 300 134 486 826 899 582 467 831 362 944;
  • 8) 0.000 000 000 465 279 999 999 999 969 057 415 300 134 486 826 899 582 467 831 362 944 × 2 = 0 + 0.000 000 000 930 559 999 999 999 938 114 830 600 268 973 653 799 164 935 662 725 888;
  • 9) 0.000 000 000 930 559 999 999 999 938 114 830 600 268 973 653 799 164 935 662 725 888 × 2 = 0 + 0.000 000 001 861 119 999 999 999 876 229 661 200 537 947 307 598 329 871 325 451 776;
  • 10) 0.000 000 001 861 119 999 999 999 876 229 661 200 537 947 307 598 329 871 325 451 776 × 2 = 0 + 0.000 000 003 722 239 999 999 999 752 459 322 401 075 894 615 196 659 742 650 903 552;
  • 11) 0.000 000 003 722 239 999 999 999 752 459 322 401 075 894 615 196 659 742 650 903 552 × 2 = 0 + 0.000 000 007 444 479 999 999 999 504 918 644 802 151 789 230 393 319 485 301 807 104;
  • 12) 0.000 000 007 444 479 999 999 999 504 918 644 802 151 789 230 393 319 485 301 807 104 × 2 = 0 + 0.000 000 014 888 959 999 999 999 009 837 289 604 303 578 460 786 638 970 603 614 208;
  • 13) 0.000 000 014 888 959 999 999 999 009 837 289 604 303 578 460 786 638 970 603 614 208 × 2 = 0 + 0.000 000 029 777 919 999 999 998 019 674 579 208 607 156 921 573 277 941 207 228 416;
  • 14) 0.000 000 029 777 919 999 999 998 019 674 579 208 607 156 921 573 277 941 207 228 416 × 2 = 0 + 0.000 000 059 555 839 999 999 996 039 349 158 417 214 313 843 146 555 882 414 456 832;
  • 15) 0.000 000 059 555 839 999 999 996 039 349 158 417 214 313 843 146 555 882 414 456 832 × 2 = 0 + 0.000 000 119 111 679 999 999 992 078 698 316 834 428 627 686 293 111 764 828 913 664;
  • 16) 0.000 000 119 111 679 999 999 992 078 698 316 834 428 627 686 293 111 764 828 913 664 × 2 = 0 + 0.000 000 238 223 359 999 999 984 157 396 633 668 857 255 372 586 223 529 657 827 328;
  • 17) 0.000 000 238 223 359 999 999 984 157 396 633 668 857 255 372 586 223 529 657 827 328 × 2 = 0 + 0.000 000 476 446 719 999 999 968 314 793 267 337 714 510 745 172 447 059 315 654 656;
  • 18) 0.000 000 476 446 719 999 999 968 314 793 267 337 714 510 745 172 447 059 315 654 656 × 2 = 0 + 0.000 000 952 893 439 999 999 936 629 586 534 675 429 021 490 344 894 118 631 309 312;
  • 19) 0.000 000 952 893 439 999 999 936 629 586 534 675 429 021 490 344 894 118 631 309 312 × 2 = 0 + 0.000 001 905 786 879 999 999 873 259 173 069 350 858 042 980 689 788 237 262 618 624;
  • 20) 0.000 001 905 786 879 999 999 873 259 173 069 350 858 042 980 689 788 237 262 618 624 × 2 = 0 + 0.000 003 811 573 759 999 999 746 518 346 138 701 716 085 961 379 576 474 525 237 248;
  • 21) 0.000 003 811 573 759 999 999 746 518 346 138 701 716 085 961 379 576 474 525 237 248 × 2 = 0 + 0.000 007 623 147 519 999 999 493 036 692 277 403 432 171 922 759 152 949 050 474 496;
  • 22) 0.000 007 623 147 519 999 999 493 036 692 277 403 432 171 922 759 152 949 050 474 496 × 2 = 0 + 0.000 015 246 295 039 999 998 986 073 384 554 806 864 343 845 518 305 898 100 948 992;
  • 23) 0.000 015 246 295 039 999 998 986 073 384 554 806 864 343 845 518 305 898 100 948 992 × 2 = 0 + 0.000 030 492 590 079 999 997 972 146 769 109 613 728 687 691 036 611 796 201 897 984;
  • 24) 0.000 030 492 590 079 999 997 972 146 769 109 613 728 687 691 036 611 796 201 897 984 × 2 = 0 + 0.000 060 985 180 159 999 995 944 293 538 219 227 457 375 382 073 223 592 403 795 968;
  • 25) 0.000 060 985 180 159 999 995 944 293 538 219 227 457 375 382 073 223 592 403 795 968 × 2 = 0 + 0.000 121 970 360 319 999 991 888 587 076 438 454 914 750 764 146 447 184 807 591 936;
  • 26) 0.000 121 970 360 319 999 991 888 587 076 438 454 914 750 764 146 447 184 807 591 936 × 2 = 0 + 0.000 243 940 720 639 999 983 777 174 152 876 909 829 501 528 292 894 369 615 183 872;
  • 27) 0.000 243 940 720 639 999 983 777 174 152 876 909 829 501 528 292 894 369 615 183 872 × 2 = 0 + 0.000 487 881 441 279 999 967 554 348 305 753 819 659 003 056 585 788 739 230 367 744;
  • 28) 0.000 487 881 441 279 999 967 554 348 305 753 819 659 003 056 585 788 739 230 367 744 × 2 = 0 + 0.000 975 762 882 559 999 935 108 696 611 507 639 318 006 113 171 577 478 460 735 488;
  • 29) 0.000 975 762 882 559 999 935 108 696 611 507 639 318 006 113 171 577 478 460 735 488 × 2 = 0 + 0.001 951 525 765 119 999 870 217 393 223 015 278 636 012 226 343 154 956 921 470 976;
  • 30) 0.001 951 525 765 119 999 870 217 393 223 015 278 636 012 226 343 154 956 921 470 976 × 2 = 0 + 0.003 903 051 530 239 999 740 434 786 446 030 557 272 024 452 686 309 913 842 941 952;
  • 31) 0.003 903 051 530 239 999 740 434 786 446 030 557 272 024 452 686 309 913 842 941 952 × 2 = 0 + 0.007 806 103 060 479 999 480 869 572 892 061 114 544 048 905 372 619 827 685 883 904;
  • 32) 0.007 806 103 060 479 999 480 869 572 892 061 114 544 048 905 372 619 827 685 883 904 × 2 = 0 + 0.015 612 206 120 959 998 961 739 145 784 122 229 088 097 810 745 239 655 371 767 808;
  • 33) 0.015 612 206 120 959 998 961 739 145 784 122 229 088 097 810 745 239 655 371 767 808 × 2 = 0 + 0.031 224 412 241 919 997 923 478 291 568 244 458 176 195 621 490 479 310 743 535 616;
  • 34) 0.031 224 412 241 919 997 923 478 291 568 244 458 176 195 621 490 479 310 743 535 616 × 2 = 0 + 0.062 448 824 483 839 995 846 956 583 136 488 916 352 391 242 980 958 621 487 071 232;
  • 35) 0.062 448 824 483 839 995 846 956 583 136 488 916 352 391 242 980 958 621 487 071 232 × 2 = 0 + 0.124 897 648 967 679 991 693 913 166 272 977 832 704 782 485 961 917 242 974 142 464;
  • 36) 0.124 897 648 967 679 991 693 913 166 272 977 832 704 782 485 961 917 242 974 142 464 × 2 = 0 + 0.249 795 297 935 359 983 387 826 332 545 955 665 409 564 971 923 834 485 948 284 928;
  • 37) 0.249 795 297 935 359 983 387 826 332 545 955 665 409 564 971 923 834 485 948 284 928 × 2 = 0 + 0.499 590 595 870 719 966 775 652 665 091 911 330 819 129 943 847 668 971 896 569 856;
  • 38) 0.499 590 595 870 719 966 775 652 665 091 911 330 819 129 943 847 668 971 896 569 856 × 2 = 0 + 0.999 181 191 741 439 933 551 305 330 183 822 661 638 259 887 695 337 943 793 139 712;
  • 39) 0.999 181 191 741 439 933 551 305 330 183 822 661 638 259 887 695 337 943 793 139 712 × 2 = 1 + 0.998 362 383 482 879 867 102 610 660 367 645 323 276 519 775 390 675 887 586 279 424;
  • 40) 0.998 362 383 482 879 867 102 610 660 367 645 323 276 519 775 390 675 887 586 279 424 × 2 = 1 + 0.996 724 766 965 759 734 205 221 320 735 290 646 553 039 550 781 351 775 172 558 848;
  • 41) 0.996 724 766 965 759 734 205 221 320 735 290 646 553 039 550 781 351 775 172 558 848 × 2 = 1 + 0.993 449 533 931 519 468 410 442 641 470 581 293 106 079 101 562 703 550 345 117 696;
  • 42) 0.993 449 533 931 519 468 410 442 641 470 581 293 106 079 101 562 703 550 345 117 696 × 2 = 1 + 0.986 899 067 863 038 936 820 885 282 941 162 586 212 158 203 125 407 100 690 235 392;
  • 43) 0.986 899 067 863 038 936 820 885 282 941 162 586 212 158 203 125 407 100 690 235 392 × 2 = 1 + 0.973 798 135 726 077 873 641 770 565 882 325 172 424 316 406 250 814 201 380 470 784;
  • 44) 0.973 798 135 726 077 873 641 770 565 882 325 172 424 316 406 250 814 201 380 470 784 × 2 = 1 + 0.947 596 271 452 155 747 283 541 131 764 650 344 848 632 812 501 628 402 760 941 568;
  • 45) 0.947 596 271 452 155 747 283 541 131 764 650 344 848 632 812 501 628 402 760 941 568 × 2 = 1 + 0.895 192 542 904 311 494 567 082 263 529 300 689 697 265 625 003 256 805 521 883 136;
  • 46) 0.895 192 542 904 311 494 567 082 263 529 300 689 697 265 625 003 256 805 521 883 136 × 2 = 1 + 0.790 385 085 808 622 989 134 164 527 058 601 379 394 531 250 006 513 611 043 766 272;
  • 47) 0.790 385 085 808 622 989 134 164 527 058 601 379 394 531 250 006 513 611 043 766 272 × 2 = 1 + 0.580 770 171 617 245 978 268 329 054 117 202 758 789 062 500 013 027 222 087 532 544;
  • 48) 0.580 770 171 617 245 978 268 329 054 117 202 758 789 062 500 013 027 222 087 532 544 × 2 = 1 + 0.161 540 343 234 491 956 536 658 108 234 405 517 578 125 000 026 054 444 175 065 088;
  • 49) 0.161 540 343 234 491 956 536 658 108 234 405 517 578 125 000 026 054 444 175 065 088 × 2 = 0 + 0.323 080 686 468 983 913 073 316 216 468 811 035 156 250 000 052 108 888 350 130 176;
  • 50) 0.323 080 686 468 983 913 073 316 216 468 811 035 156 250 000 052 108 888 350 130 176 × 2 = 0 + 0.646 161 372 937 967 826 146 632 432 937 622 070 312 500 000 104 217 776 700 260 352;
  • 51) 0.646 161 372 937 967 826 146 632 432 937 622 070 312 500 000 104 217 776 700 260 352 × 2 = 1 + 0.292 322 745 875 935 652 293 264 865 875 244 140 625 000 000 208 435 553 400 520 704;
  • 52) 0.292 322 745 875 935 652 293 264 865 875 244 140 625 000 000 208 435 553 400 520 704 × 2 = 0 + 0.584 645 491 751 871 304 586 529 731 750 488 281 250 000 000 416 871 106 801 041 408;
  • 53) 0.584 645 491 751 871 304 586 529 731 750 488 281 250 000 000 416 871 106 801 041 408 × 2 = 1 + 0.169 290 983 503 742 609 173 059 463 500 976 562 500 000 000 833 742 213 602 082 816;
  • 54) 0.169 290 983 503 742 609 173 059 463 500 976 562 500 000 000 833 742 213 602 082 816 × 2 = 0 + 0.338 581 967 007 485 218 346 118 927 001 953 125 000 000 001 667 484 427 204 165 632;
  • 55) 0.338 581 967 007 485 218 346 118 927 001 953 125 000 000 001 667 484 427 204 165 632 × 2 = 0 + 0.677 163 934 014 970 436 692 237 854 003 906 250 000 000 003 334 968 854 408 331 264;
  • 56) 0.677 163 934 014 970 436 692 237 854 003 906 250 000 000 003 334 968 854 408 331 264 × 2 = 1 + 0.354 327 868 029 940 873 384 475 708 007 812 500 000 000 006 669 937 708 816 662 528;
  • 57) 0.354 327 868 029 940 873 384 475 708 007 812 500 000 000 006 669 937 708 816 662 528 × 2 = 0 + 0.708 655 736 059 881 746 768 951 416 015 625 000 000 000 013 339 875 417 633 325 056;
  • 58) 0.708 655 736 059 881 746 768 951 416 015 625 000 000 000 013 339 875 417 633 325 056 × 2 = 1 + 0.417 311 472 119 763 493 537 902 832 031 250 000 000 000 026 679 750 835 266 650 112;
  • 59) 0.417 311 472 119 763 493 537 902 832 031 250 000 000 000 026 679 750 835 266 650 112 × 2 = 0 + 0.834 622 944 239 526 987 075 805 664 062 500 000 000 000 053 359 501 670 533 300 224;
  • 60) 0.834 622 944 239 526 987 075 805 664 062 500 000 000 000 053 359 501 670 533 300 224 × 2 = 1 + 0.669 245 888 479 053 974 151 611 328 125 000 000 000 000 106 719 003 341 066 600 448;
  • 61) 0.669 245 888 479 053 974 151 611 328 125 000 000 000 000 106 719 003 341 066 600 448 × 2 = 1 + 0.338 491 776 958 107 948 303 222 656 250 000 000 000 000 213 438 006 682 133 200 896;
  • 62) 0.338 491 776 958 107 948 303 222 656 250 000 000 000 000 213 438 006 682 133 200 896 × 2 = 0 + 0.676 983 553 916 215 896 606 445 312 500 000 000 000 000 426 876 013 364 266 401 792;
  • 63) 0.676 983 553 916 215 896 606 445 312 500 000 000 000 000 426 876 013 364 266 401 792 × 2 = 1 + 0.353 967 107 832 431 793 212 890 625 000 000 000 000 000 853 752 026 728 532 803 584;
  • 64) 0.353 967 107 832 431 793 212 890 625 000 000 000 000 000 853 752 026 728 532 803 584 × 2 = 0 + 0.707 934 215 664 863 586 425 781 250 000 000 000 000 001 707 504 053 457 065 607 168;
  • 65) 0.707 934 215 664 863 586 425 781 250 000 000 000 000 001 707 504 053 457 065 607 168 × 2 = 1 + 0.415 868 431 329 727 172 851 562 500 000 000 000 000 003 415 008 106 914 131 214 336;
  • 66) 0.415 868 431 329 727 172 851 562 500 000 000 000 000 003 415 008 106 914 131 214 336 × 2 = 0 + 0.831 736 862 659 454 345 703 125 000 000 000 000 000 006 830 016 213 828 262 428 672;
  • 67) 0.831 736 862 659 454 345 703 125 000 000 000 000 000 006 830 016 213 828 262 428 672 × 2 = 1 + 0.663 473 725 318 908 691 406 250 000 000 000 000 000 013 660 032 427 656 524 857 344;
  • 68) 0.663 473 725 318 908 691 406 250 000 000 000 000 000 013 660 032 427 656 524 857 344 × 2 = 1 + 0.326 947 450 637 817 382 812 500 000 000 000 000 000 027 320 064 855 313 049 714 688;
  • 69) 0.326 947 450 637 817 382 812 500 000 000 000 000 000 027 320 064 855 313 049 714 688 × 2 = 0 + 0.653 894 901 275 634 765 625 000 000 000 000 000 000 054 640 129 710 626 099 429 376;
  • 70) 0.653 894 901 275 634 765 625 000 000 000 000 000 000 054 640 129 710 626 099 429 376 × 2 = 1 + 0.307 789 802 551 269 531 250 000 000 000 000 000 000 109 280 259 421 252 198 858 752;
  • 71) 0.307 789 802 551 269 531 250 000 000 000 000 000 000 109 280 259 421 252 198 858 752 × 2 = 0 + 0.615 579 605 102 539 062 500 000 000 000 000 000 000 218 560 518 842 504 397 717 504;
  • 72) 0.615 579 605 102 539 062 500 000 000 000 000 000 000 218 560 518 842 504 397 717 504 × 2 = 1 + 0.231 159 210 205 078 125 000 000 000 000 000 000 000 437 121 037 685 008 795 435 008;
  • 73) 0.231 159 210 205 078 125 000 000 000 000 000 000 000 437 121 037 685 008 795 435 008 × 2 = 0 + 0.462 318 420 410 156 250 000 000 000 000 000 000 000 874 242 075 370 017 590 870 016;
  • 74) 0.462 318 420 410 156 250 000 000 000 000 000 000 000 874 242 075 370 017 590 870 016 × 2 = 0 + 0.924 636 840 820 312 500 000 000 000 000 000 000 001 748 484 150 740 035 181 740 032;
  • 75) 0.924 636 840 820 312 500 000 000 000 000 000 000 001 748 484 150 740 035 181 740 032 × 2 = 1 + 0.849 273 681 640 625 000 000 000 000 000 000 000 003 496 968 301 480 070 363 480 064;
  • 76) 0.849 273 681 640 625 000 000 000 000 000 000 000 003 496 968 301 480 070 363 480 064 × 2 = 1 + 0.698 547 363 281 250 000 000 000 000 000 000 000 006 993 936 602 960 140 726 960 128;
  • 77) 0.698 547 363 281 250 000 000 000 000 000 000 000 006 993 936 602 960 140 726 960 128 × 2 = 1 + 0.397 094 726 562 500 000 000 000 000 000 000 000 013 987 873 205 920 281 453 920 256;
  • 78) 0.397 094 726 562 500 000 000 000 000 000 000 000 013 987 873 205 920 281 453 920 256 × 2 = 0 + 0.794 189 453 125 000 000 000 000 000 000 000 000 027 975 746 411 840 562 907 840 512;
  • 79) 0.794 189 453 125 000 000 000 000 000 000 000 000 027 975 746 411 840 562 907 840 512 × 2 = 1 + 0.588 378 906 250 000 000 000 000 000 000 000 000 055 951 492 823 681 125 815 681 024;
  • 80) 0.588 378 906 250 000 000 000 000 000 000 000 000 055 951 492 823 681 125 815 681 024 × 2 = 1 + 0.176 757 812 500 000 000 000 000 000 000 000 000 111 902 985 647 362 251 631 362 048;
  • 81) 0.176 757 812 500 000 000 000 000 000 000 000 000 111 902 985 647 362 251 631 362 048 × 2 = 0 + 0.353 515 625 000 000 000 000 000 000 000 000 000 223 805 971 294 724 503 262 724 096;
  • 82) 0.353 515 625 000 000 000 000 000 000 000 000 000 223 805 971 294 724 503 262 724 096 × 2 = 0 + 0.707 031 250 000 000 000 000 000 000 000 000 000 447 611 942 589 449 006 525 448 192;
  • 83) 0.707 031 250 000 000 000 000 000 000 000 000 000 447 611 942 589 449 006 525 448 192 × 2 = 1 + 0.414 062 500 000 000 000 000 000 000 000 000 000 895 223 885 178 898 013 050 896 384;
  • 84) 0.414 062 500 000 000 000 000 000 000 000 000 000 895 223 885 178 898 013 050 896 384 × 2 = 0 + 0.828 125 000 000 000 000 000 000 000 000 000 001 790 447 770 357 796 026 101 792 768;
  • 85) 0.828 125 000 000 000 000 000 000 000 000 000 001 790 447 770 357 796 026 101 792 768 × 2 = 1 + 0.656 250 000 000 000 000 000 000 000 000 000 003 580 895 540 715 592 052 203 585 536;
  • 86) 0.656 250 000 000 000 000 000 000 000 000 000 003 580 895 540 715 592 052 203 585 536 × 2 = 1 + 0.312 500 000 000 000 000 000 000 000 000 000 007 161 791 081 431 184 104 407 171 072;
  • 87) 0.312 500 000 000 000 000 000 000 000 000 000 007 161 791 081 431 184 104 407 171 072 × 2 = 0 + 0.625 000 000 000 000 000 000 000 000 000 000 014 323 582 162 862 368 208 814 342 144;
  • 88) 0.625 000 000 000 000 000 000 000 000 000 000 014 323 582 162 862 368 208 814 342 144 × 2 = 1 + 0.250 000 000 000 000 000 000 000 000 000 000 028 647 164 325 724 736 417 628 684 288;
  • 89) 0.250 000 000 000 000 000 000 000 000 000 000 028 647 164 325 724 736 417 628 684 288 × 2 = 0 + 0.500 000 000 000 000 000 000 000 000 000 000 057 294 328 651 449 472 835 257 368 576;
  • 90) 0.500 000 000 000 000 000 000 000 000 000 000 057 294 328 651 449 472 835 257 368 576 × 2 = 1 + 0.000 000 000 000 000 000 000 000 000 000 000 114 588 657 302 898 945 670 514 737 152;
  • 91) 0.000 000 000 000 000 000 000 000 000 000 000 114 588 657 302 898 945 670 514 737 152 × 2 = 0 + 0.000 000 000 000 000 000 000 000 000 000 000 229 177 314 605 797 891 341 029 474 304;

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:


0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 029 932 523(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:

0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 029 932 523(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:


0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 029 932 523(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 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 029 932 523 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