0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 029 932 437 8 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 029 932 437 8(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
0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 029 932 437 8(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: 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 029 932 437 8.

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) 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 029 932 437 8 × 2 = 0 + 0.000 000 000 007 269 999 999 999 999 516 522 114 064 601 356 670 305 976 059 864 875 6;
  • 2) 0.000 000 000 007 269 999 999 999 999 516 522 114 064 601 356 670 305 976 059 864 875 6 × 2 = 0 + 0.000 000 000 014 539 999 999 999 999 033 044 228 129 202 713 340 611 952 119 729 751 2;
  • 3) 0.000 000 000 014 539 999 999 999 999 033 044 228 129 202 713 340 611 952 119 729 751 2 × 2 = 0 + 0.000 000 000 029 079 999 999 999 998 066 088 456 258 405 426 681 223 904 239 459 502 4;
  • 4) 0.000 000 000 029 079 999 999 999 998 066 088 456 258 405 426 681 223 904 239 459 502 4 × 2 = 0 + 0.000 000 000 058 159 999 999 999 996 132 176 912 516 810 853 362 447 808 478 919 004 8;
  • 5) 0.000 000 000 058 159 999 999 999 996 132 176 912 516 810 853 362 447 808 478 919 004 8 × 2 = 0 + 0.000 000 000 116 319 999 999 999 992 264 353 825 033 621 706 724 895 616 957 838 009 6;
  • 6) 0.000 000 000 116 319 999 999 999 992 264 353 825 033 621 706 724 895 616 957 838 009 6 × 2 = 0 + 0.000 000 000 232 639 999 999 999 984 528 707 650 067 243 413 449 791 233 915 676 019 2;
  • 7) 0.000 000 000 232 639 999 999 999 984 528 707 650 067 243 413 449 791 233 915 676 019 2 × 2 = 0 + 0.000 000 000 465 279 999 999 999 969 057 415 300 134 486 826 899 582 467 831 352 038 4;
  • 8) 0.000 000 000 465 279 999 999 999 969 057 415 300 134 486 826 899 582 467 831 352 038 4 × 2 = 0 + 0.000 000 000 930 559 999 999 999 938 114 830 600 268 973 653 799 164 935 662 704 076 8;
  • 9) 0.000 000 000 930 559 999 999 999 938 114 830 600 268 973 653 799 164 935 662 704 076 8 × 2 = 0 + 0.000 000 001 861 119 999 999 999 876 229 661 200 537 947 307 598 329 871 325 408 153 6;
  • 10) 0.000 000 001 861 119 999 999 999 876 229 661 200 537 947 307 598 329 871 325 408 153 6 × 2 = 0 + 0.000 000 003 722 239 999 999 999 752 459 322 401 075 894 615 196 659 742 650 816 307 2;
  • 11) 0.000 000 003 722 239 999 999 999 752 459 322 401 075 894 615 196 659 742 650 816 307 2 × 2 = 0 + 0.000 000 007 444 479 999 999 999 504 918 644 802 151 789 230 393 319 485 301 632 614 4;
  • 12) 0.000 000 007 444 479 999 999 999 504 918 644 802 151 789 230 393 319 485 301 632 614 4 × 2 = 0 + 0.000 000 014 888 959 999 999 999 009 837 289 604 303 578 460 786 638 970 603 265 228 8;
  • 13) 0.000 000 014 888 959 999 999 999 009 837 289 604 303 578 460 786 638 970 603 265 228 8 × 2 = 0 + 0.000 000 029 777 919 999 999 998 019 674 579 208 607 156 921 573 277 941 206 530 457 6;
  • 14) 0.000 000 029 777 919 999 999 998 019 674 579 208 607 156 921 573 277 941 206 530 457 6 × 2 = 0 + 0.000 000 059 555 839 999 999 996 039 349 158 417 214 313 843 146 555 882 413 060 915 2;
  • 15) 0.000 000 059 555 839 999 999 996 039 349 158 417 214 313 843 146 555 882 413 060 915 2 × 2 = 0 + 0.000 000 119 111 679 999 999 992 078 698 316 834 428 627 686 293 111 764 826 121 830 4;
  • 16) 0.000 000 119 111 679 999 999 992 078 698 316 834 428 627 686 293 111 764 826 121 830 4 × 2 = 0 + 0.000 000 238 223 359 999 999 984 157 396 633 668 857 255 372 586 223 529 652 243 660 8;
  • 17) 0.000 000 238 223 359 999 999 984 157 396 633 668 857 255 372 586 223 529 652 243 660 8 × 2 = 0 + 0.000 000 476 446 719 999 999 968 314 793 267 337 714 510 745 172 447 059 304 487 321 6;
  • 18) 0.000 000 476 446 719 999 999 968 314 793 267 337 714 510 745 172 447 059 304 487 321 6 × 2 = 0 + 0.000 000 952 893 439 999 999 936 629 586 534 675 429 021 490 344 894 118 608 974 643 2;
  • 19) 0.000 000 952 893 439 999 999 936 629 586 534 675 429 021 490 344 894 118 608 974 643 2 × 2 = 0 + 0.000 001 905 786 879 999 999 873 259 173 069 350 858 042 980 689 788 237 217 949 286 4;
  • 20) 0.000 001 905 786 879 999 999 873 259 173 069 350 858 042 980 689 788 237 217 949 286 4 × 2 = 0 + 0.000 003 811 573 759 999 999 746 518 346 138 701 716 085 961 379 576 474 435 898 572 8;
  • 21) 0.000 003 811 573 759 999 999 746 518 346 138 701 716 085 961 379 576 474 435 898 572 8 × 2 = 0 + 0.000 007 623 147 519 999 999 493 036 692 277 403 432 171 922 759 152 948 871 797 145 6;
  • 22) 0.000 007 623 147 519 999 999 493 036 692 277 403 432 171 922 759 152 948 871 797 145 6 × 2 = 0 + 0.000 015 246 295 039 999 998 986 073 384 554 806 864 343 845 518 305 897 743 594 291 2;
  • 23) 0.000 015 246 295 039 999 998 986 073 384 554 806 864 343 845 518 305 897 743 594 291 2 × 2 = 0 + 0.000 030 492 590 079 999 997 972 146 769 109 613 728 687 691 036 611 795 487 188 582 4;
  • 24) 0.000 030 492 590 079 999 997 972 146 769 109 613 728 687 691 036 611 795 487 188 582 4 × 2 = 0 + 0.000 060 985 180 159 999 995 944 293 538 219 227 457 375 382 073 223 590 974 377 164 8;
  • 25) 0.000 060 985 180 159 999 995 944 293 538 219 227 457 375 382 073 223 590 974 377 164 8 × 2 = 0 + 0.000 121 970 360 319 999 991 888 587 076 438 454 914 750 764 146 447 181 948 754 329 6;
  • 26) 0.000 121 970 360 319 999 991 888 587 076 438 454 914 750 764 146 447 181 948 754 329 6 × 2 = 0 + 0.000 243 940 720 639 999 983 777 174 152 876 909 829 501 528 292 894 363 897 508 659 2;
  • 27) 0.000 243 940 720 639 999 983 777 174 152 876 909 829 501 528 292 894 363 897 508 659 2 × 2 = 0 + 0.000 487 881 441 279 999 967 554 348 305 753 819 659 003 056 585 788 727 795 017 318 4;
  • 28) 0.000 487 881 441 279 999 967 554 348 305 753 819 659 003 056 585 788 727 795 017 318 4 × 2 = 0 + 0.000 975 762 882 559 999 935 108 696 611 507 639 318 006 113 171 577 455 590 034 636 8;
  • 29) 0.000 975 762 882 559 999 935 108 696 611 507 639 318 006 113 171 577 455 590 034 636 8 × 2 = 0 + 0.001 951 525 765 119 999 870 217 393 223 015 278 636 012 226 343 154 911 180 069 273 6;
  • 30) 0.001 951 525 765 119 999 870 217 393 223 015 278 636 012 226 343 154 911 180 069 273 6 × 2 = 0 + 0.003 903 051 530 239 999 740 434 786 446 030 557 272 024 452 686 309 822 360 138 547 2;
  • 31) 0.003 903 051 530 239 999 740 434 786 446 030 557 272 024 452 686 309 822 360 138 547 2 × 2 = 0 + 0.007 806 103 060 479 999 480 869 572 892 061 114 544 048 905 372 619 644 720 277 094 4;
  • 32) 0.007 806 103 060 479 999 480 869 572 892 061 114 544 048 905 372 619 644 720 277 094 4 × 2 = 0 + 0.015 612 206 120 959 998 961 739 145 784 122 229 088 097 810 745 239 289 440 554 188 8;
  • 33) 0.015 612 206 120 959 998 961 739 145 784 122 229 088 097 810 745 239 289 440 554 188 8 × 2 = 0 + 0.031 224 412 241 919 997 923 478 291 568 244 458 176 195 621 490 478 578 881 108 377 6;
  • 34) 0.031 224 412 241 919 997 923 478 291 568 244 458 176 195 621 490 478 578 881 108 377 6 × 2 = 0 + 0.062 448 824 483 839 995 846 956 583 136 488 916 352 391 242 980 957 157 762 216 755 2;
  • 35) 0.062 448 824 483 839 995 846 956 583 136 488 916 352 391 242 980 957 157 762 216 755 2 × 2 = 0 + 0.124 897 648 967 679 991 693 913 166 272 977 832 704 782 485 961 914 315 524 433 510 4;
  • 36) 0.124 897 648 967 679 991 693 913 166 272 977 832 704 782 485 961 914 315 524 433 510 4 × 2 = 0 + 0.249 795 297 935 359 983 387 826 332 545 955 665 409 564 971 923 828 631 048 867 020 8;
  • 37) 0.249 795 297 935 359 983 387 826 332 545 955 665 409 564 971 923 828 631 048 867 020 8 × 2 = 0 + 0.499 590 595 870 719 966 775 652 665 091 911 330 819 129 943 847 657 262 097 734 041 6;
  • 38) 0.499 590 595 870 719 966 775 652 665 091 911 330 819 129 943 847 657 262 097 734 041 6 × 2 = 0 + 0.999 181 191 741 439 933 551 305 330 183 822 661 638 259 887 695 314 524 195 468 083 2;
  • 39) 0.999 181 191 741 439 933 551 305 330 183 822 661 638 259 887 695 314 524 195 468 083 2 × 2 = 1 + 0.998 362 383 482 879 867 102 610 660 367 645 323 276 519 775 390 629 048 390 936 166 4;
  • 40) 0.998 362 383 482 879 867 102 610 660 367 645 323 276 519 775 390 629 048 390 936 166 4 × 2 = 1 + 0.996 724 766 965 759 734 205 221 320 735 290 646 553 039 550 781 258 096 781 872 332 8;
  • 41) 0.996 724 766 965 759 734 205 221 320 735 290 646 553 039 550 781 258 096 781 872 332 8 × 2 = 1 + 0.993 449 533 931 519 468 410 442 641 470 581 293 106 079 101 562 516 193 563 744 665 6;
  • 42) 0.993 449 533 931 519 468 410 442 641 470 581 293 106 079 101 562 516 193 563 744 665 6 × 2 = 1 + 0.986 899 067 863 038 936 820 885 282 941 162 586 212 158 203 125 032 387 127 489 331 2;
  • 43) 0.986 899 067 863 038 936 820 885 282 941 162 586 212 158 203 125 032 387 127 489 331 2 × 2 = 1 + 0.973 798 135 726 077 873 641 770 565 882 325 172 424 316 406 250 064 774 254 978 662 4;
  • 44) 0.973 798 135 726 077 873 641 770 565 882 325 172 424 316 406 250 064 774 254 978 662 4 × 2 = 1 + 0.947 596 271 452 155 747 283 541 131 764 650 344 848 632 812 500 129 548 509 957 324 8;
  • 45) 0.947 596 271 452 155 747 283 541 131 764 650 344 848 632 812 500 129 548 509 957 324 8 × 2 = 1 + 0.895 192 542 904 311 494 567 082 263 529 300 689 697 265 625 000 259 097 019 914 649 6;
  • 46) 0.895 192 542 904 311 494 567 082 263 529 300 689 697 265 625 000 259 097 019 914 649 6 × 2 = 1 + 0.790 385 085 808 622 989 134 164 527 058 601 379 394 531 250 000 518 194 039 829 299 2;
  • 47) 0.790 385 085 808 622 989 134 164 527 058 601 379 394 531 250 000 518 194 039 829 299 2 × 2 = 1 + 0.580 770 171 617 245 978 268 329 054 117 202 758 789 062 500 001 036 388 079 658 598 4;
  • 48) 0.580 770 171 617 245 978 268 329 054 117 202 758 789 062 500 001 036 388 079 658 598 4 × 2 = 1 + 0.161 540 343 234 491 956 536 658 108 234 405 517 578 125 000 002 072 776 159 317 196 8;
  • 49) 0.161 540 343 234 491 956 536 658 108 234 405 517 578 125 000 002 072 776 159 317 196 8 × 2 = 0 + 0.323 080 686 468 983 913 073 316 216 468 811 035 156 250 000 004 145 552 318 634 393 6;
  • 50) 0.323 080 686 468 983 913 073 316 216 468 811 035 156 250 000 004 145 552 318 634 393 6 × 2 = 0 + 0.646 161 372 937 967 826 146 632 432 937 622 070 312 500 000 008 291 104 637 268 787 2;
  • 51) 0.646 161 372 937 967 826 146 632 432 937 622 070 312 500 000 008 291 104 637 268 787 2 × 2 = 1 + 0.292 322 745 875 935 652 293 264 865 875 244 140 625 000 000 016 582 209 274 537 574 4;
  • 52) 0.292 322 745 875 935 652 293 264 865 875 244 140 625 000 000 016 582 209 274 537 574 4 × 2 = 0 + 0.584 645 491 751 871 304 586 529 731 750 488 281 250 000 000 033 164 418 549 075 148 8;
  • 53) 0.584 645 491 751 871 304 586 529 731 750 488 281 250 000 000 033 164 418 549 075 148 8 × 2 = 1 + 0.169 290 983 503 742 609 173 059 463 500 976 562 500 000 000 066 328 837 098 150 297 6;
  • 54) 0.169 290 983 503 742 609 173 059 463 500 976 562 500 000 000 066 328 837 098 150 297 6 × 2 = 0 + 0.338 581 967 007 485 218 346 118 927 001 953 125 000 000 000 132 657 674 196 300 595 2;
  • 55) 0.338 581 967 007 485 218 346 118 927 001 953 125 000 000 000 132 657 674 196 300 595 2 × 2 = 0 + 0.677 163 934 014 970 436 692 237 854 003 906 250 000 000 000 265 315 348 392 601 190 4;
  • 56) 0.677 163 934 014 970 436 692 237 854 003 906 250 000 000 000 265 315 348 392 601 190 4 × 2 = 1 + 0.354 327 868 029 940 873 384 475 708 007 812 500 000 000 000 530 630 696 785 202 380 8;
  • 57) 0.354 327 868 029 940 873 384 475 708 007 812 500 000 000 000 530 630 696 785 202 380 8 × 2 = 0 + 0.708 655 736 059 881 746 768 951 416 015 625 000 000 000 001 061 261 393 570 404 761 6;
  • 58) 0.708 655 736 059 881 746 768 951 416 015 625 000 000 000 001 061 261 393 570 404 761 6 × 2 = 1 + 0.417 311 472 119 763 493 537 902 832 031 250 000 000 000 002 122 522 787 140 809 523 2;
  • 59) 0.417 311 472 119 763 493 537 902 832 031 250 000 000 000 002 122 522 787 140 809 523 2 × 2 = 0 + 0.834 622 944 239 526 987 075 805 664 062 500 000 000 000 004 245 045 574 281 619 046 4;
  • 60) 0.834 622 944 239 526 987 075 805 664 062 500 000 000 000 004 245 045 574 281 619 046 4 × 2 = 1 + 0.669 245 888 479 053 974 151 611 328 125 000 000 000 000 008 490 091 148 563 238 092 8;
  • 61) 0.669 245 888 479 053 974 151 611 328 125 000 000 000 000 008 490 091 148 563 238 092 8 × 2 = 1 + 0.338 491 776 958 107 948 303 222 656 250 000 000 000 000 016 980 182 297 126 476 185 6;
  • 62) 0.338 491 776 958 107 948 303 222 656 250 000 000 000 000 016 980 182 297 126 476 185 6 × 2 = 0 + 0.676 983 553 916 215 896 606 445 312 500 000 000 000 000 033 960 364 594 252 952 371 2;
  • 63) 0.676 983 553 916 215 896 606 445 312 500 000 000 000 000 033 960 364 594 252 952 371 2 × 2 = 1 + 0.353 967 107 832 431 793 212 890 625 000 000 000 000 000 067 920 729 188 505 904 742 4;
  • 64) 0.353 967 107 832 431 793 212 890 625 000 000 000 000 000 067 920 729 188 505 904 742 4 × 2 = 0 + 0.707 934 215 664 863 586 425 781 250 000 000 000 000 000 135 841 458 377 011 809 484 8;
  • 65) 0.707 934 215 664 863 586 425 781 250 000 000 000 000 000 135 841 458 377 011 809 484 8 × 2 = 1 + 0.415 868 431 329 727 172 851 562 500 000 000 000 000 000 271 682 916 754 023 618 969 6;
  • 66) 0.415 868 431 329 727 172 851 562 500 000 000 000 000 000 271 682 916 754 023 618 969 6 × 2 = 0 + 0.831 736 862 659 454 345 703 125 000 000 000 000 000 000 543 365 833 508 047 237 939 2;
  • 67) 0.831 736 862 659 454 345 703 125 000 000 000 000 000 000 543 365 833 508 047 237 939 2 × 2 = 1 + 0.663 473 725 318 908 691 406 250 000 000 000 000 000 001 086 731 667 016 094 475 878 4;
  • 68) 0.663 473 725 318 908 691 406 250 000 000 000 000 000 001 086 731 667 016 094 475 878 4 × 2 = 1 + 0.326 947 450 637 817 382 812 500 000 000 000 000 000 002 173 463 334 032 188 951 756 8;
  • 69) 0.326 947 450 637 817 382 812 500 000 000 000 000 000 002 173 463 334 032 188 951 756 8 × 2 = 0 + 0.653 894 901 275 634 765 625 000 000 000 000 000 000 004 346 926 668 064 377 903 513 6;
  • 70) 0.653 894 901 275 634 765 625 000 000 000 000 000 000 004 346 926 668 064 377 903 513 6 × 2 = 1 + 0.307 789 802 551 269 531 250 000 000 000 000 000 000 008 693 853 336 128 755 807 027 2;
  • 71) 0.307 789 802 551 269 531 250 000 000 000 000 000 000 008 693 853 336 128 755 807 027 2 × 2 = 0 + 0.615 579 605 102 539 062 500 000 000 000 000 000 000 017 387 706 672 257 511 614 054 4;
  • 72) 0.615 579 605 102 539 062 500 000 000 000 000 000 000 017 387 706 672 257 511 614 054 4 × 2 = 1 + 0.231 159 210 205 078 125 000 000 000 000 000 000 000 034 775 413 344 515 023 228 108 8;
  • 73) 0.231 159 210 205 078 125 000 000 000 000 000 000 000 034 775 413 344 515 023 228 108 8 × 2 = 0 + 0.462 318 420 410 156 250 000 000 000 000 000 000 000 069 550 826 689 030 046 456 217 6;
  • 74) 0.462 318 420 410 156 250 000 000 000 000 000 000 000 069 550 826 689 030 046 456 217 6 × 2 = 0 + 0.924 636 840 820 312 500 000 000 000 000 000 000 000 139 101 653 378 060 092 912 435 2;
  • 75) 0.924 636 840 820 312 500 000 000 000 000 000 000 000 139 101 653 378 060 092 912 435 2 × 2 = 1 + 0.849 273 681 640 625 000 000 000 000 000 000 000 000 278 203 306 756 120 185 824 870 4;
  • 76) 0.849 273 681 640 625 000 000 000 000 000 000 000 000 278 203 306 756 120 185 824 870 4 × 2 = 1 + 0.698 547 363 281 250 000 000 000 000 000 000 000 000 556 406 613 512 240 371 649 740 8;
  • 77) 0.698 547 363 281 250 000 000 000 000 000 000 000 000 556 406 613 512 240 371 649 740 8 × 2 = 1 + 0.397 094 726 562 500 000 000 000 000 000 000 000 001 112 813 227 024 480 743 299 481 6;
  • 78) 0.397 094 726 562 500 000 000 000 000 000 000 000 001 112 813 227 024 480 743 299 481 6 × 2 = 0 + 0.794 189 453 125 000 000 000 000 000 000 000 000 002 225 626 454 048 961 486 598 963 2;
  • 79) 0.794 189 453 125 000 000 000 000 000 000 000 000 002 225 626 454 048 961 486 598 963 2 × 2 = 1 + 0.588 378 906 250 000 000 000 000 000 000 000 000 004 451 252 908 097 922 973 197 926 4;
  • 80) 0.588 378 906 250 000 000 000 000 000 000 000 000 004 451 252 908 097 922 973 197 926 4 × 2 = 1 + 0.176 757 812 500 000 000 000 000 000 000 000 000 008 902 505 816 195 845 946 395 852 8;
  • 81) 0.176 757 812 500 000 000 000 000 000 000 000 000 008 902 505 816 195 845 946 395 852 8 × 2 = 0 + 0.353 515 625 000 000 000 000 000 000 000 000 000 017 805 011 632 391 691 892 791 705 6;
  • 82) 0.353 515 625 000 000 000 000 000 000 000 000 000 017 805 011 632 391 691 892 791 705 6 × 2 = 0 + 0.707 031 250 000 000 000 000 000 000 000 000 000 035 610 023 264 783 383 785 583 411 2;
  • 83) 0.707 031 250 000 000 000 000 000 000 000 000 000 035 610 023 264 783 383 785 583 411 2 × 2 = 1 + 0.414 062 500 000 000 000 000 000 000 000 000 000 071 220 046 529 566 767 571 166 822 4;
  • 84) 0.414 062 500 000 000 000 000 000 000 000 000 000 071 220 046 529 566 767 571 166 822 4 × 2 = 0 + 0.828 125 000 000 000 000 000 000 000 000 000 000 142 440 093 059 133 535 142 333 644 8;
  • 85) 0.828 125 000 000 000 000 000 000 000 000 000 000 142 440 093 059 133 535 142 333 644 8 × 2 = 1 + 0.656 250 000 000 000 000 000 000 000 000 000 000 284 880 186 118 267 070 284 667 289 6;
  • 86) 0.656 250 000 000 000 000 000 000 000 000 000 000 284 880 186 118 267 070 284 667 289 6 × 2 = 1 + 0.312 500 000 000 000 000 000 000 000 000 000 000 569 760 372 236 534 140 569 334 579 2;
  • 87) 0.312 500 000 000 000 000 000 000 000 000 000 000 569 760 372 236 534 140 569 334 579 2 × 2 = 0 + 0.625 000 000 000 000 000 000 000 000 000 000 001 139 520 744 473 068 281 138 669 158 4;
  • 88) 0.625 000 000 000 000 000 000 000 000 000 000 001 139 520 744 473 068 281 138 669 158 4 × 2 = 1 + 0.250 000 000 000 000 000 000 000 000 000 000 002 279 041 488 946 136 562 277 338 316 8;
  • 89) 0.250 000 000 000 000 000 000 000 000 000 000 002 279 041 488 946 136 562 277 338 316 8 × 2 = 0 + 0.500 000 000 000 000 000 000 000 000 000 000 004 558 082 977 892 273 124 554 676 633 6;
  • 90) 0.500 000 000 000 000 000 000 000 000 000 000 004 558 082 977 892 273 124 554 676 633 6 × 2 = 1 + 0.000 000 000 000 000 000 000 000 000 000 000 009 116 165 955 784 546 249 109 353 267 2;
  • 91) 0.000 000 000 000 000 000 000 000 000 000 000 009 116 165 955 784 546 249 109 353 267 2 × 2 = 0 + 0.000 000 000 000 000 000 000 000 000 000 000 018 232 331 911 569 092 498 218 706 534 4;

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


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

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


0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 029 932 437 8(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:

0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 029 932 437 8(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:


0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 029 932 437 8(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 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 029 932 437 8 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

Share

» Convert decimal 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 029 932 444 9 to 64 bit double precision IEEE 754 binary floating point representation standard

» Calculations Performed by Our Visitors: Decimal Numbers Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard. Data organized on a Monthly Basis

» Month 07, 2026 [July]: Decimal Numbers Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard. Calculations performed during the month of: July - by our visitors

» Improve your social skills: The Attention Economy - The Battlefield For Your Focus. NotebookLM YouTube Video of 7.50 minutes

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