0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 029 978 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 029 978(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
0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 029 978(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: 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 029 978.

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) 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 029 978 × 2 = 0 + 0.000 000 000 007 269 999 999 999 999 516 522 114 064 601 356 670 305 976 059 956;
  • 2) 0.000 000 000 007 269 999 999 999 999 516 522 114 064 601 356 670 305 976 059 956 × 2 = 0 + 0.000 000 000 014 539 999 999 999 999 033 044 228 129 202 713 340 611 952 119 912;
  • 3) 0.000 000 000 014 539 999 999 999 999 033 044 228 129 202 713 340 611 952 119 912 × 2 = 0 + 0.000 000 000 029 079 999 999 999 998 066 088 456 258 405 426 681 223 904 239 824;
  • 4) 0.000 000 000 029 079 999 999 999 998 066 088 456 258 405 426 681 223 904 239 824 × 2 = 0 + 0.000 000 000 058 159 999 999 999 996 132 176 912 516 810 853 362 447 808 479 648;
  • 5) 0.000 000 000 058 159 999 999 999 996 132 176 912 516 810 853 362 447 808 479 648 × 2 = 0 + 0.000 000 000 116 319 999 999 999 992 264 353 825 033 621 706 724 895 616 959 296;
  • 6) 0.000 000 000 116 319 999 999 999 992 264 353 825 033 621 706 724 895 616 959 296 × 2 = 0 + 0.000 000 000 232 639 999 999 999 984 528 707 650 067 243 413 449 791 233 918 592;
  • 7) 0.000 000 000 232 639 999 999 999 984 528 707 650 067 243 413 449 791 233 918 592 × 2 = 0 + 0.000 000 000 465 279 999 999 999 969 057 415 300 134 486 826 899 582 467 837 184;
  • 8) 0.000 000 000 465 279 999 999 999 969 057 415 300 134 486 826 899 582 467 837 184 × 2 = 0 + 0.000 000 000 930 559 999 999 999 938 114 830 600 268 973 653 799 164 935 674 368;
  • 9) 0.000 000 000 930 559 999 999 999 938 114 830 600 268 973 653 799 164 935 674 368 × 2 = 0 + 0.000 000 001 861 119 999 999 999 876 229 661 200 537 947 307 598 329 871 348 736;
  • 10) 0.000 000 001 861 119 999 999 999 876 229 661 200 537 947 307 598 329 871 348 736 × 2 = 0 + 0.000 000 003 722 239 999 999 999 752 459 322 401 075 894 615 196 659 742 697 472;
  • 11) 0.000 000 003 722 239 999 999 999 752 459 322 401 075 894 615 196 659 742 697 472 × 2 = 0 + 0.000 000 007 444 479 999 999 999 504 918 644 802 151 789 230 393 319 485 394 944;
  • 12) 0.000 000 007 444 479 999 999 999 504 918 644 802 151 789 230 393 319 485 394 944 × 2 = 0 + 0.000 000 014 888 959 999 999 999 009 837 289 604 303 578 460 786 638 970 789 888;
  • 13) 0.000 000 014 888 959 999 999 999 009 837 289 604 303 578 460 786 638 970 789 888 × 2 = 0 + 0.000 000 029 777 919 999 999 998 019 674 579 208 607 156 921 573 277 941 579 776;
  • 14) 0.000 000 029 777 919 999 999 998 019 674 579 208 607 156 921 573 277 941 579 776 × 2 = 0 + 0.000 000 059 555 839 999 999 996 039 349 158 417 214 313 843 146 555 883 159 552;
  • 15) 0.000 000 059 555 839 999 999 996 039 349 158 417 214 313 843 146 555 883 159 552 × 2 = 0 + 0.000 000 119 111 679 999 999 992 078 698 316 834 428 627 686 293 111 766 319 104;
  • 16) 0.000 000 119 111 679 999 999 992 078 698 316 834 428 627 686 293 111 766 319 104 × 2 = 0 + 0.000 000 238 223 359 999 999 984 157 396 633 668 857 255 372 586 223 532 638 208;
  • 17) 0.000 000 238 223 359 999 999 984 157 396 633 668 857 255 372 586 223 532 638 208 × 2 = 0 + 0.000 000 476 446 719 999 999 968 314 793 267 337 714 510 745 172 447 065 276 416;
  • 18) 0.000 000 476 446 719 999 999 968 314 793 267 337 714 510 745 172 447 065 276 416 × 2 = 0 + 0.000 000 952 893 439 999 999 936 629 586 534 675 429 021 490 344 894 130 552 832;
  • 19) 0.000 000 952 893 439 999 999 936 629 586 534 675 429 021 490 344 894 130 552 832 × 2 = 0 + 0.000 001 905 786 879 999 999 873 259 173 069 350 858 042 980 689 788 261 105 664;
  • 20) 0.000 001 905 786 879 999 999 873 259 173 069 350 858 042 980 689 788 261 105 664 × 2 = 0 + 0.000 003 811 573 759 999 999 746 518 346 138 701 716 085 961 379 576 522 211 328;
  • 21) 0.000 003 811 573 759 999 999 746 518 346 138 701 716 085 961 379 576 522 211 328 × 2 = 0 + 0.000 007 623 147 519 999 999 493 036 692 277 403 432 171 922 759 153 044 422 656;
  • 22) 0.000 007 623 147 519 999 999 493 036 692 277 403 432 171 922 759 153 044 422 656 × 2 = 0 + 0.000 015 246 295 039 999 998 986 073 384 554 806 864 343 845 518 306 088 845 312;
  • 23) 0.000 015 246 295 039 999 998 986 073 384 554 806 864 343 845 518 306 088 845 312 × 2 = 0 + 0.000 030 492 590 079 999 997 972 146 769 109 613 728 687 691 036 612 177 690 624;
  • 24) 0.000 030 492 590 079 999 997 972 146 769 109 613 728 687 691 036 612 177 690 624 × 2 = 0 + 0.000 060 985 180 159 999 995 944 293 538 219 227 457 375 382 073 224 355 381 248;
  • 25) 0.000 060 985 180 159 999 995 944 293 538 219 227 457 375 382 073 224 355 381 248 × 2 = 0 + 0.000 121 970 360 319 999 991 888 587 076 438 454 914 750 764 146 448 710 762 496;
  • 26) 0.000 121 970 360 319 999 991 888 587 076 438 454 914 750 764 146 448 710 762 496 × 2 = 0 + 0.000 243 940 720 639 999 983 777 174 152 876 909 829 501 528 292 897 421 524 992;
  • 27) 0.000 243 940 720 639 999 983 777 174 152 876 909 829 501 528 292 897 421 524 992 × 2 = 0 + 0.000 487 881 441 279 999 967 554 348 305 753 819 659 003 056 585 794 843 049 984;
  • 28) 0.000 487 881 441 279 999 967 554 348 305 753 819 659 003 056 585 794 843 049 984 × 2 = 0 + 0.000 975 762 882 559 999 935 108 696 611 507 639 318 006 113 171 589 686 099 968;
  • 29) 0.000 975 762 882 559 999 935 108 696 611 507 639 318 006 113 171 589 686 099 968 × 2 = 0 + 0.001 951 525 765 119 999 870 217 393 223 015 278 636 012 226 343 179 372 199 936;
  • 30) 0.001 951 525 765 119 999 870 217 393 223 015 278 636 012 226 343 179 372 199 936 × 2 = 0 + 0.003 903 051 530 239 999 740 434 786 446 030 557 272 024 452 686 358 744 399 872;
  • 31) 0.003 903 051 530 239 999 740 434 786 446 030 557 272 024 452 686 358 744 399 872 × 2 = 0 + 0.007 806 103 060 479 999 480 869 572 892 061 114 544 048 905 372 717 488 799 744;
  • 32) 0.007 806 103 060 479 999 480 869 572 892 061 114 544 048 905 372 717 488 799 744 × 2 = 0 + 0.015 612 206 120 959 998 961 739 145 784 122 229 088 097 810 745 434 977 599 488;
  • 33) 0.015 612 206 120 959 998 961 739 145 784 122 229 088 097 810 745 434 977 599 488 × 2 = 0 + 0.031 224 412 241 919 997 923 478 291 568 244 458 176 195 621 490 869 955 198 976;
  • 34) 0.031 224 412 241 919 997 923 478 291 568 244 458 176 195 621 490 869 955 198 976 × 2 = 0 + 0.062 448 824 483 839 995 846 956 583 136 488 916 352 391 242 981 739 910 397 952;
  • 35) 0.062 448 824 483 839 995 846 956 583 136 488 916 352 391 242 981 739 910 397 952 × 2 = 0 + 0.124 897 648 967 679 991 693 913 166 272 977 832 704 782 485 963 479 820 795 904;
  • 36) 0.124 897 648 967 679 991 693 913 166 272 977 832 704 782 485 963 479 820 795 904 × 2 = 0 + 0.249 795 297 935 359 983 387 826 332 545 955 665 409 564 971 926 959 641 591 808;
  • 37) 0.249 795 297 935 359 983 387 826 332 545 955 665 409 564 971 926 959 641 591 808 × 2 = 0 + 0.499 590 595 870 719 966 775 652 665 091 911 330 819 129 943 853 919 283 183 616;
  • 38) 0.499 590 595 870 719 966 775 652 665 091 911 330 819 129 943 853 919 283 183 616 × 2 = 0 + 0.999 181 191 741 439 933 551 305 330 183 822 661 638 259 887 707 838 566 367 232;
  • 39) 0.999 181 191 741 439 933 551 305 330 183 822 661 638 259 887 707 838 566 367 232 × 2 = 1 + 0.998 362 383 482 879 867 102 610 660 367 645 323 276 519 775 415 677 132 734 464;
  • 40) 0.998 362 383 482 879 867 102 610 660 367 645 323 276 519 775 415 677 132 734 464 × 2 = 1 + 0.996 724 766 965 759 734 205 221 320 735 290 646 553 039 550 831 354 265 468 928;
  • 41) 0.996 724 766 965 759 734 205 221 320 735 290 646 553 039 550 831 354 265 468 928 × 2 = 1 + 0.993 449 533 931 519 468 410 442 641 470 581 293 106 079 101 662 708 530 937 856;
  • 42) 0.993 449 533 931 519 468 410 442 641 470 581 293 106 079 101 662 708 530 937 856 × 2 = 1 + 0.986 899 067 863 038 936 820 885 282 941 162 586 212 158 203 325 417 061 875 712;
  • 43) 0.986 899 067 863 038 936 820 885 282 941 162 586 212 158 203 325 417 061 875 712 × 2 = 1 + 0.973 798 135 726 077 873 641 770 565 882 325 172 424 316 406 650 834 123 751 424;
  • 44) 0.973 798 135 726 077 873 641 770 565 882 325 172 424 316 406 650 834 123 751 424 × 2 = 1 + 0.947 596 271 452 155 747 283 541 131 764 650 344 848 632 813 301 668 247 502 848;
  • 45) 0.947 596 271 452 155 747 283 541 131 764 650 344 848 632 813 301 668 247 502 848 × 2 = 1 + 0.895 192 542 904 311 494 567 082 263 529 300 689 697 265 626 603 336 495 005 696;
  • 46) 0.895 192 542 904 311 494 567 082 263 529 300 689 697 265 626 603 336 495 005 696 × 2 = 1 + 0.790 385 085 808 622 989 134 164 527 058 601 379 394 531 253 206 672 990 011 392;
  • 47) 0.790 385 085 808 622 989 134 164 527 058 601 379 394 531 253 206 672 990 011 392 × 2 = 1 + 0.580 770 171 617 245 978 268 329 054 117 202 758 789 062 506 413 345 980 022 784;
  • 48) 0.580 770 171 617 245 978 268 329 054 117 202 758 789 062 506 413 345 980 022 784 × 2 = 1 + 0.161 540 343 234 491 956 536 658 108 234 405 517 578 125 012 826 691 960 045 568;
  • 49) 0.161 540 343 234 491 956 536 658 108 234 405 517 578 125 012 826 691 960 045 568 × 2 = 0 + 0.323 080 686 468 983 913 073 316 216 468 811 035 156 250 025 653 383 920 091 136;
  • 50) 0.323 080 686 468 983 913 073 316 216 468 811 035 156 250 025 653 383 920 091 136 × 2 = 0 + 0.646 161 372 937 967 826 146 632 432 937 622 070 312 500 051 306 767 840 182 272;
  • 51) 0.646 161 372 937 967 826 146 632 432 937 622 070 312 500 051 306 767 840 182 272 × 2 = 1 + 0.292 322 745 875 935 652 293 264 865 875 244 140 625 000 102 613 535 680 364 544;
  • 52) 0.292 322 745 875 935 652 293 264 865 875 244 140 625 000 102 613 535 680 364 544 × 2 = 0 + 0.584 645 491 751 871 304 586 529 731 750 488 281 250 000 205 227 071 360 729 088;
  • 53) 0.584 645 491 751 871 304 586 529 731 750 488 281 250 000 205 227 071 360 729 088 × 2 = 1 + 0.169 290 983 503 742 609 173 059 463 500 976 562 500 000 410 454 142 721 458 176;
  • 54) 0.169 290 983 503 742 609 173 059 463 500 976 562 500 000 410 454 142 721 458 176 × 2 = 0 + 0.338 581 967 007 485 218 346 118 927 001 953 125 000 000 820 908 285 442 916 352;
  • 55) 0.338 581 967 007 485 218 346 118 927 001 953 125 000 000 820 908 285 442 916 352 × 2 = 0 + 0.677 163 934 014 970 436 692 237 854 003 906 250 000 001 641 816 570 885 832 704;
  • 56) 0.677 163 934 014 970 436 692 237 854 003 906 250 000 001 641 816 570 885 832 704 × 2 = 1 + 0.354 327 868 029 940 873 384 475 708 007 812 500 000 003 283 633 141 771 665 408;
  • 57) 0.354 327 868 029 940 873 384 475 708 007 812 500 000 003 283 633 141 771 665 408 × 2 = 0 + 0.708 655 736 059 881 746 768 951 416 015 625 000 000 006 567 266 283 543 330 816;
  • 58) 0.708 655 736 059 881 746 768 951 416 015 625 000 000 006 567 266 283 543 330 816 × 2 = 1 + 0.417 311 472 119 763 493 537 902 832 031 250 000 000 013 134 532 567 086 661 632;
  • 59) 0.417 311 472 119 763 493 537 902 832 031 250 000 000 013 134 532 567 086 661 632 × 2 = 0 + 0.834 622 944 239 526 987 075 805 664 062 500 000 000 026 269 065 134 173 323 264;
  • 60) 0.834 622 944 239 526 987 075 805 664 062 500 000 000 026 269 065 134 173 323 264 × 2 = 1 + 0.669 245 888 479 053 974 151 611 328 125 000 000 000 052 538 130 268 346 646 528;
  • 61) 0.669 245 888 479 053 974 151 611 328 125 000 000 000 052 538 130 268 346 646 528 × 2 = 1 + 0.338 491 776 958 107 948 303 222 656 250 000 000 000 105 076 260 536 693 293 056;
  • 62) 0.338 491 776 958 107 948 303 222 656 250 000 000 000 105 076 260 536 693 293 056 × 2 = 0 + 0.676 983 553 916 215 896 606 445 312 500 000 000 000 210 152 521 073 386 586 112;
  • 63) 0.676 983 553 916 215 896 606 445 312 500 000 000 000 210 152 521 073 386 586 112 × 2 = 1 + 0.353 967 107 832 431 793 212 890 625 000 000 000 000 420 305 042 146 773 172 224;
  • 64) 0.353 967 107 832 431 793 212 890 625 000 000 000 000 420 305 042 146 773 172 224 × 2 = 0 + 0.707 934 215 664 863 586 425 781 250 000 000 000 000 840 610 084 293 546 344 448;
  • 65) 0.707 934 215 664 863 586 425 781 250 000 000 000 000 840 610 084 293 546 344 448 × 2 = 1 + 0.415 868 431 329 727 172 851 562 500 000 000 000 001 681 220 168 587 092 688 896;
  • 66) 0.415 868 431 329 727 172 851 562 500 000 000 000 001 681 220 168 587 092 688 896 × 2 = 0 + 0.831 736 862 659 454 345 703 125 000 000 000 000 003 362 440 337 174 185 377 792;
  • 67) 0.831 736 862 659 454 345 703 125 000 000 000 000 003 362 440 337 174 185 377 792 × 2 = 1 + 0.663 473 725 318 908 691 406 250 000 000 000 000 006 724 880 674 348 370 755 584;
  • 68) 0.663 473 725 318 908 691 406 250 000 000 000 000 006 724 880 674 348 370 755 584 × 2 = 1 + 0.326 947 450 637 817 382 812 500 000 000 000 000 013 449 761 348 696 741 511 168;
  • 69) 0.326 947 450 637 817 382 812 500 000 000 000 000 013 449 761 348 696 741 511 168 × 2 = 0 + 0.653 894 901 275 634 765 625 000 000 000 000 000 026 899 522 697 393 483 022 336;
  • 70) 0.653 894 901 275 634 765 625 000 000 000 000 000 026 899 522 697 393 483 022 336 × 2 = 1 + 0.307 789 802 551 269 531 250 000 000 000 000 000 053 799 045 394 786 966 044 672;
  • 71) 0.307 789 802 551 269 531 250 000 000 000 000 000 053 799 045 394 786 966 044 672 × 2 = 0 + 0.615 579 605 102 539 062 500 000 000 000 000 000 107 598 090 789 573 932 089 344;
  • 72) 0.615 579 605 102 539 062 500 000 000 000 000 000 107 598 090 789 573 932 089 344 × 2 = 1 + 0.231 159 210 205 078 125 000 000 000 000 000 000 215 196 181 579 147 864 178 688;
  • 73) 0.231 159 210 205 078 125 000 000 000 000 000 000 215 196 181 579 147 864 178 688 × 2 = 0 + 0.462 318 420 410 156 250 000 000 000 000 000 000 430 392 363 158 295 728 357 376;
  • 74) 0.462 318 420 410 156 250 000 000 000 000 000 000 430 392 363 158 295 728 357 376 × 2 = 0 + 0.924 636 840 820 312 500 000 000 000 000 000 000 860 784 726 316 591 456 714 752;
  • 75) 0.924 636 840 820 312 500 000 000 000 000 000 000 860 784 726 316 591 456 714 752 × 2 = 1 + 0.849 273 681 640 625 000 000 000 000 000 000 001 721 569 452 633 182 913 429 504;
  • 76) 0.849 273 681 640 625 000 000 000 000 000 000 001 721 569 452 633 182 913 429 504 × 2 = 1 + 0.698 547 363 281 250 000 000 000 000 000 000 003 443 138 905 266 365 826 859 008;
  • 77) 0.698 547 363 281 250 000 000 000 000 000 000 003 443 138 905 266 365 826 859 008 × 2 = 1 + 0.397 094 726 562 500 000 000 000 000 000 000 006 886 277 810 532 731 653 718 016;
  • 78) 0.397 094 726 562 500 000 000 000 000 000 000 006 886 277 810 532 731 653 718 016 × 2 = 0 + 0.794 189 453 125 000 000 000 000 000 000 000 013 772 555 621 065 463 307 436 032;
  • 79) 0.794 189 453 125 000 000 000 000 000 000 000 013 772 555 621 065 463 307 436 032 × 2 = 1 + 0.588 378 906 250 000 000 000 000 000 000 000 027 545 111 242 130 926 614 872 064;
  • 80) 0.588 378 906 250 000 000 000 000 000 000 000 027 545 111 242 130 926 614 872 064 × 2 = 1 + 0.176 757 812 500 000 000 000 000 000 000 000 055 090 222 484 261 853 229 744 128;
  • 81) 0.176 757 812 500 000 000 000 000 000 000 000 055 090 222 484 261 853 229 744 128 × 2 = 0 + 0.353 515 625 000 000 000 000 000 000 000 000 110 180 444 968 523 706 459 488 256;
  • 82) 0.353 515 625 000 000 000 000 000 000 000 000 110 180 444 968 523 706 459 488 256 × 2 = 0 + 0.707 031 250 000 000 000 000 000 000 000 000 220 360 889 937 047 412 918 976 512;
  • 83) 0.707 031 250 000 000 000 000 000 000 000 000 220 360 889 937 047 412 918 976 512 × 2 = 1 + 0.414 062 500 000 000 000 000 000 000 000 000 440 721 779 874 094 825 837 953 024;
  • 84) 0.414 062 500 000 000 000 000 000 000 000 000 440 721 779 874 094 825 837 953 024 × 2 = 0 + 0.828 125 000 000 000 000 000 000 000 000 000 881 443 559 748 189 651 675 906 048;
  • 85) 0.828 125 000 000 000 000 000 000 000 000 000 881 443 559 748 189 651 675 906 048 × 2 = 1 + 0.656 250 000 000 000 000 000 000 000 000 001 762 887 119 496 379 303 351 812 096;
  • 86) 0.656 250 000 000 000 000 000 000 000 000 001 762 887 119 496 379 303 351 812 096 × 2 = 1 + 0.312 500 000 000 000 000 000 000 000 000 003 525 774 238 992 758 606 703 624 192;
  • 87) 0.312 500 000 000 000 000 000 000 000 000 003 525 774 238 992 758 606 703 624 192 × 2 = 0 + 0.625 000 000 000 000 000 000 000 000 000 007 051 548 477 985 517 213 407 248 384;
  • 88) 0.625 000 000 000 000 000 000 000 000 000 007 051 548 477 985 517 213 407 248 384 × 2 = 1 + 0.250 000 000 000 000 000 000 000 000 000 014 103 096 955 971 034 426 814 496 768;
  • 89) 0.250 000 000 000 000 000 000 000 000 000 014 103 096 955 971 034 426 814 496 768 × 2 = 0 + 0.500 000 000 000 000 000 000 000 000 000 028 206 193 911 942 068 853 628 993 536;
  • 90) 0.500 000 000 000 000 000 000 000 000 000 028 206 193 911 942 068 853 628 993 536 × 2 = 1 + 0.000 000 000 000 000 000 000 000 000 000 056 412 387 823 884 137 707 257 987 072;
  • 91) 0.000 000 000 000 000 000 000 000 000 000 056 412 387 823 884 137 707 257 987 072 × 2 = 0 + 0.000 000 000 000 000 000 000 000 000 000 112 824 775 647 768 275 414 515 974 144;

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:


0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 029 978(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:

0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 029 978(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:


0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 029 978(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 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 029 978 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 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 029 944 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 Art of Strategic Ambiguity. NotebookLM YouTube Video of 6.59 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