0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 163 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 163(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
0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 163(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: 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 163.

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) 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 163 × 2 = 0 + 0.000 000 000 007 269 999 999 999 999 516 522 114 064 601 356 670 305 976 326;
  • 2) 0.000 000 000 007 269 999 999 999 999 516 522 114 064 601 356 670 305 976 326 × 2 = 0 + 0.000 000 000 014 539 999 999 999 999 033 044 228 129 202 713 340 611 952 652;
  • 3) 0.000 000 000 014 539 999 999 999 999 033 044 228 129 202 713 340 611 952 652 × 2 = 0 + 0.000 000 000 029 079 999 999 999 998 066 088 456 258 405 426 681 223 905 304;
  • 4) 0.000 000 000 029 079 999 999 999 998 066 088 456 258 405 426 681 223 905 304 × 2 = 0 + 0.000 000 000 058 159 999 999 999 996 132 176 912 516 810 853 362 447 810 608;
  • 5) 0.000 000 000 058 159 999 999 999 996 132 176 912 516 810 853 362 447 810 608 × 2 = 0 + 0.000 000 000 116 319 999 999 999 992 264 353 825 033 621 706 724 895 621 216;
  • 6) 0.000 000 000 116 319 999 999 999 992 264 353 825 033 621 706 724 895 621 216 × 2 = 0 + 0.000 000 000 232 639 999 999 999 984 528 707 650 067 243 413 449 791 242 432;
  • 7) 0.000 000 000 232 639 999 999 999 984 528 707 650 067 243 413 449 791 242 432 × 2 = 0 + 0.000 000 000 465 279 999 999 999 969 057 415 300 134 486 826 899 582 484 864;
  • 8) 0.000 000 000 465 279 999 999 999 969 057 415 300 134 486 826 899 582 484 864 × 2 = 0 + 0.000 000 000 930 559 999 999 999 938 114 830 600 268 973 653 799 164 969 728;
  • 9) 0.000 000 000 930 559 999 999 999 938 114 830 600 268 973 653 799 164 969 728 × 2 = 0 + 0.000 000 001 861 119 999 999 999 876 229 661 200 537 947 307 598 329 939 456;
  • 10) 0.000 000 001 861 119 999 999 999 876 229 661 200 537 947 307 598 329 939 456 × 2 = 0 + 0.000 000 003 722 239 999 999 999 752 459 322 401 075 894 615 196 659 878 912;
  • 11) 0.000 000 003 722 239 999 999 999 752 459 322 401 075 894 615 196 659 878 912 × 2 = 0 + 0.000 000 007 444 479 999 999 999 504 918 644 802 151 789 230 393 319 757 824;
  • 12) 0.000 000 007 444 479 999 999 999 504 918 644 802 151 789 230 393 319 757 824 × 2 = 0 + 0.000 000 014 888 959 999 999 999 009 837 289 604 303 578 460 786 639 515 648;
  • 13) 0.000 000 014 888 959 999 999 999 009 837 289 604 303 578 460 786 639 515 648 × 2 = 0 + 0.000 000 029 777 919 999 999 998 019 674 579 208 607 156 921 573 279 031 296;
  • 14) 0.000 000 029 777 919 999 999 998 019 674 579 208 607 156 921 573 279 031 296 × 2 = 0 + 0.000 000 059 555 839 999 999 996 039 349 158 417 214 313 843 146 558 062 592;
  • 15) 0.000 000 059 555 839 999 999 996 039 349 158 417 214 313 843 146 558 062 592 × 2 = 0 + 0.000 000 119 111 679 999 999 992 078 698 316 834 428 627 686 293 116 125 184;
  • 16) 0.000 000 119 111 679 999 999 992 078 698 316 834 428 627 686 293 116 125 184 × 2 = 0 + 0.000 000 238 223 359 999 999 984 157 396 633 668 857 255 372 586 232 250 368;
  • 17) 0.000 000 238 223 359 999 999 984 157 396 633 668 857 255 372 586 232 250 368 × 2 = 0 + 0.000 000 476 446 719 999 999 968 314 793 267 337 714 510 745 172 464 500 736;
  • 18) 0.000 000 476 446 719 999 999 968 314 793 267 337 714 510 745 172 464 500 736 × 2 = 0 + 0.000 000 952 893 439 999 999 936 629 586 534 675 429 021 490 344 929 001 472;
  • 19) 0.000 000 952 893 439 999 999 936 629 586 534 675 429 021 490 344 929 001 472 × 2 = 0 + 0.000 001 905 786 879 999 999 873 259 173 069 350 858 042 980 689 858 002 944;
  • 20) 0.000 001 905 786 879 999 999 873 259 173 069 350 858 042 980 689 858 002 944 × 2 = 0 + 0.000 003 811 573 759 999 999 746 518 346 138 701 716 085 961 379 716 005 888;
  • 21) 0.000 003 811 573 759 999 999 746 518 346 138 701 716 085 961 379 716 005 888 × 2 = 0 + 0.000 007 623 147 519 999 999 493 036 692 277 403 432 171 922 759 432 011 776;
  • 22) 0.000 007 623 147 519 999 999 493 036 692 277 403 432 171 922 759 432 011 776 × 2 = 0 + 0.000 015 246 295 039 999 998 986 073 384 554 806 864 343 845 518 864 023 552;
  • 23) 0.000 015 246 295 039 999 998 986 073 384 554 806 864 343 845 518 864 023 552 × 2 = 0 + 0.000 030 492 590 079 999 997 972 146 769 109 613 728 687 691 037 728 047 104;
  • 24) 0.000 030 492 590 079 999 997 972 146 769 109 613 728 687 691 037 728 047 104 × 2 = 0 + 0.000 060 985 180 159 999 995 944 293 538 219 227 457 375 382 075 456 094 208;
  • 25) 0.000 060 985 180 159 999 995 944 293 538 219 227 457 375 382 075 456 094 208 × 2 = 0 + 0.000 121 970 360 319 999 991 888 587 076 438 454 914 750 764 150 912 188 416;
  • 26) 0.000 121 970 360 319 999 991 888 587 076 438 454 914 750 764 150 912 188 416 × 2 = 0 + 0.000 243 940 720 639 999 983 777 174 152 876 909 829 501 528 301 824 376 832;
  • 27) 0.000 243 940 720 639 999 983 777 174 152 876 909 829 501 528 301 824 376 832 × 2 = 0 + 0.000 487 881 441 279 999 967 554 348 305 753 819 659 003 056 603 648 753 664;
  • 28) 0.000 487 881 441 279 999 967 554 348 305 753 819 659 003 056 603 648 753 664 × 2 = 0 + 0.000 975 762 882 559 999 935 108 696 611 507 639 318 006 113 207 297 507 328;
  • 29) 0.000 975 762 882 559 999 935 108 696 611 507 639 318 006 113 207 297 507 328 × 2 = 0 + 0.001 951 525 765 119 999 870 217 393 223 015 278 636 012 226 414 595 014 656;
  • 30) 0.001 951 525 765 119 999 870 217 393 223 015 278 636 012 226 414 595 014 656 × 2 = 0 + 0.003 903 051 530 239 999 740 434 786 446 030 557 272 024 452 829 190 029 312;
  • 31) 0.003 903 051 530 239 999 740 434 786 446 030 557 272 024 452 829 190 029 312 × 2 = 0 + 0.007 806 103 060 479 999 480 869 572 892 061 114 544 048 905 658 380 058 624;
  • 32) 0.007 806 103 060 479 999 480 869 572 892 061 114 544 048 905 658 380 058 624 × 2 = 0 + 0.015 612 206 120 959 998 961 739 145 784 122 229 088 097 811 316 760 117 248;
  • 33) 0.015 612 206 120 959 998 961 739 145 784 122 229 088 097 811 316 760 117 248 × 2 = 0 + 0.031 224 412 241 919 997 923 478 291 568 244 458 176 195 622 633 520 234 496;
  • 34) 0.031 224 412 241 919 997 923 478 291 568 244 458 176 195 622 633 520 234 496 × 2 = 0 + 0.062 448 824 483 839 995 846 956 583 136 488 916 352 391 245 267 040 468 992;
  • 35) 0.062 448 824 483 839 995 846 956 583 136 488 916 352 391 245 267 040 468 992 × 2 = 0 + 0.124 897 648 967 679 991 693 913 166 272 977 832 704 782 490 534 080 937 984;
  • 36) 0.124 897 648 967 679 991 693 913 166 272 977 832 704 782 490 534 080 937 984 × 2 = 0 + 0.249 795 297 935 359 983 387 826 332 545 955 665 409 564 981 068 161 875 968;
  • 37) 0.249 795 297 935 359 983 387 826 332 545 955 665 409 564 981 068 161 875 968 × 2 = 0 + 0.499 590 595 870 719 966 775 652 665 091 911 330 819 129 962 136 323 751 936;
  • 38) 0.499 590 595 870 719 966 775 652 665 091 911 330 819 129 962 136 323 751 936 × 2 = 0 + 0.999 181 191 741 439 933 551 305 330 183 822 661 638 259 924 272 647 503 872;
  • 39) 0.999 181 191 741 439 933 551 305 330 183 822 661 638 259 924 272 647 503 872 × 2 = 1 + 0.998 362 383 482 879 867 102 610 660 367 645 323 276 519 848 545 295 007 744;
  • 40) 0.998 362 383 482 879 867 102 610 660 367 645 323 276 519 848 545 295 007 744 × 2 = 1 + 0.996 724 766 965 759 734 205 221 320 735 290 646 553 039 697 090 590 015 488;
  • 41) 0.996 724 766 965 759 734 205 221 320 735 290 646 553 039 697 090 590 015 488 × 2 = 1 + 0.993 449 533 931 519 468 410 442 641 470 581 293 106 079 394 181 180 030 976;
  • 42) 0.993 449 533 931 519 468 410 442 641 470 581 293 106 079 394 181 180 030 976 × 2 = 1 + 0.986 899 067 863 038 936 820 885 282 941 162 586 212 158 788 362 360 061 952;
  • 43) 0.986 899 067 863 038 936 820 885 282 941 162 586 212 158 788 362 360 061 952 × 2 = 1 + 0.973 798 135 726 077 873 641 770 565 882 325 172 424 317 576 724 720 123 904;
  • 44) 0.973 798 135 726 077 873 641 770 565 882 325 172 424 317 576 724 720 123 904 × 2 = 1 + 0.947 596 271 452 155 747 283 541 131 764 650 344 848 635 153 449 440 247 808;
  • 45) 0.947 596 271 452 155 747 283 541 131 764 650 344 848 635 153 449 440 247 808 × 2 = 1 + 0.895 192 542 904 311 494 567 082 263 529 300 689 697 270 306 898 880 495 616;
  • 46) 0.895 192 542 904 311 494 567 082 263 529 300 689 697 270 306 898 880 495 616 × 2 = 1 + 0.790 385 085 808 622 989 134 164 527 058 601 379 394 540 613 797 760 991 232;
  • 47) 0.790 385 085 808 622 989 134 164 527 058 601 379 394 540 613 797 760 991 232 × 2 = 1 + 0.580 770 171 617 245 978 268 329 054 117 202 758 789 081 227 595 521 982 464;
  • 48) 0.580 770 171 617 245 978 268 329 054 117 202 758 789 081 227 595 521 982 464 × 2 = 1 + 0.161 540 343 234 491 956 536 658 108 234 405 517 578 162 455 191 043 964 928;
  • 49) 0.161 540 343 234 491 956 536 658 108 234 405 517 578 162 455 191 043 964 928 × 2 = 0 + 0.323 080 686 468 983 913 073 316 216 468 811 035 156 324 910 382 087 929 856;
  • 50) 0.323 080 686 468 983 913 073 316 216 468 811 035 156 324 910 382 087 929 856 × 2 = 0 + 0.646 161 372 937 967 826 146 632 432 937 622 070 312 649 820 764 175 859 712;
  • 51) 0.646 161 372 937 967 826 146 632 432 937 622 070 312 649 820 764 175 859 712 × 2 = 1 + 0.292 322 745 875 935 652 293 264 865 875 244 140 625 299 641 528 351 719 424;
  • 52) 0.292 322 745 875 935 652 293 264 865 875 244 140 625 299 641 528 351 719 424 × 2 = 0 + 0.584 645 491 751 871 304 586 529 731 750 488 281 250 599 283 056 703 438 848;
  • 53) 0.584 645 491 751 871 304 586 529 731 750 488 281 250 599 283 056 703 438 848 × 2 = 1 + 0.169 290 983 503 742 609 173 059 463 500 976 562 501 198 566 113 406 877 696;
  • 54) 0.169 290 983 503 742 609 173 059 463 500 976 562 501 198 566 113 406 877 696 × 2 = 0 + 0.338 581 967 007 485 218 346 118 927 001 953 125 002 397 132 226 813 755 392;
  • 55) 0.338 581 967 007 485 218 346 118 927 001 953 125 002 397 132 226 813 755 392 × 2 = 0 + 0.677 163 934 014 970 436 692 237 854 003 906 250 004 794 264 453 627 510 784;
  • 56) 0.677 163 934 014 970 436 692 237 854 003 906 250 004 794 264 453 627 510 784 × 2 = 1 + 0.354 327 868 029 940 873 384 475 708 007 812 500 009 588 528 907 255 021 568;
  • 57) 0.354 327 868 029 940 873 384 475 708 007 812 500 009 588 528 907 255 021 568 × 2 = 0 + 0.708 655 736 059 881 746 768 951 416 015 625 000 019 177 057 814 510 043 136;
  • 58) 0.708 655 736 059 881 746 768 951 416 015 625 000 019 177 057 814 510 043 136 × 2 = 1 + 0.417 311 472 119 763 493 537 902 832 031 250 000 038 354 115 629 020 086 272;
  • 59) 0.417 311 472 119 763 493 537 902 832 031 250 000 038 354 115 629 020 086 272 × 2 = 0 + 0.834 622 944 239 526 987 075 805 664 062 500 000 076 708 231 258 040 172 544;
  • 60) 0.834 622 944 239 526 987 075 805 664 062 500 000 076 708 231 258 040 172 544 × 2 = 1 + 0.669 245 888 479 053 974 151 611 328 125 000 000 153 416 462 516 080 345 088;
  • 61) 0.669 245 888 479 053 974 151 611 328 125 000 000 153 416 462 516 080 345 088 × 2 = 1 + 0.338 491 776 958 107 948 303 222 656 250 000 000 306 832 925 032 160 690 176;
  • 62) 0.338 491 776 958 107 948 303 222 656 250 000 000 306 832 925 032 160 690 176 × 2 = 0 + 0.676 983 553 916 215 896 606 445 312 500 000 000 613 665 850 064 321 380 352;
  • 63) 0.676 983 553 916 215 896 606 445 312 500 000 000 613 665 850 064 321 380 352 × 2 = 1 + 0.353 967 107 832 431 793 212 890 625 000 000 001 227 331 700 128 642 760 704;
  • 64) 0.353 967 107 832 431 793 212 890 625 000 000 001 227 331 700 128 642 760 704 × 2 = 0 + 0.707 934 215 664 863 586 425 781 250 000 000 002 454 663 400 257 285 521 408;
  • 65) 0.707 934 215 664 863 586 425 781 250 000 000 002 454 663 400 257 285 521 408 × 2 = 1 + 0.415 868 431 329 727 172 851 562 500 000 000 004 909 326 800 514 571 042 816;
  • 66) 0.415 868 431 329 727 172 851 562 500 000 000 004 909 326 800 514 571 042 816 × 2 = 0 + 0.831 736 862 659 454 345 703 125 000 000 000 009 818 653 601 029 142 085 632;
  • 67) 0.831 736 862 659 454 345 703 125 000 000 000 009 818 653 601 029 142 085 632 × 2 = 1 + 0.663 473 725 318 908 691 406 250 000 000 000 019 637 307 202 058 284 171 264;
  • 68) 0.663 473 725 318 908 691 406 250 000 000 000 019 637 307 202 058 284 171 264 × 2 = 1 + 0.326 947 450 637 817 382 812 500 000 000 000 039 274 614 404 116 568 342 528;
  • 69) 0.326 947 450 637 817 382 812 500 000 000 000 039 274 614 404 116 568 342 528 × 2 = 0 + 0.653 894 901 275 634 765 625 000 000 000 000 078 549 228 808 233 136 685 056;
  • 70) 0.653 894 901 275 634 765 625 000 000 000 000 078 549 228 808 233 136 685 056 × 2 = 1 + 0.307 789 802 551 269 531 250 000 000 000 000 157 098 457 616 466 273 370 112;
  • 71) 0.307 789 802 551 269 531 250 000 000 000 000 157 098 457 616 466 273 370 112 × 2 = 0 + 0.615 579 605 102 539 062 500 000 000 000 000 314 196 915 232 932 546 740 224;
  • 72) 0.615 579 605 102 539 062 500 000 000 000 000 314 196 915 232 932 546 740 224 × 2 = 1 + 0.231 159 210 205 078 125 000 000 000 000 000 628 393 830 465 865 093 480 448;
  • 73) 0.231 159 210 205 078 125 000 000 000 000 000 628 393 830 465 865 093 480 448 × 2 = 0 + 0.462 318 420 410 156 250 000 000 000 000 001 256 787 660 931 730 186 960 896;
  • 74) 0.462 318 420 410 156 250 000 000 000 000 001 256 787 660 931 730 186 960 896 × 2 = 0 + 0.924 636 840 820 312 500 000 000 000 000 002 513 575 321 863 460 373 921 792;
  • 75) 0.924 636 840 820 312 500 000 000 000 000 002 513 575 321 863 460 373 921 792 × 2 = 1 + 0.849 273 681 640 625 000 000 000 000 000 005 027 150 643 726 920 747 843 584;
  • 76) 0.849 273 681 640 625 000 000 000 000 000 005 027 150 643 726 920 747 843 584 × 2 = 1 + 0.698 547 363 281 250 000 000 000 000 000 010 054 301 287 453 841 495 687 168;
  • 77) 0.698 547 363 281 250 000 000 000 000 000 010 054 301 287 453 841 495 687 168 × 2 = 1 + 0.397 094 726 562 500 000 000 000 000 000 020 108 602 574 907 682 991 374 336;
  • 78) 0.397 094 726 562 500 000 000 000 000 000 020 108 602 574 907 682 991 374 336 × 2 = 0 + 0.794 189 453 125 000 000 000 000 000 000 040 217 205 149 815 365 982 748 672;
  • 79) 0.794 189 453 125 000 000 000 000 000 000 040 217 205 149 815 365 982 748 672 × 2 = 1 + 0.588 378 906 250 000 000 000 000 000 000 080 434 410 299 630 731 965 497 344;
  • 80) 0.588 378 906 250 000 000 000 000 000 000 080 434 410 299 630 731 965 497 344 × 2 = 1 + 0.176 757 812 500 000 000 000 000 000 000 160 868 820 599 261 463 930 994 688;
  • 81) 0.176 757 812 500 000 000 000 000 000 000 160 868 820 599 261 463 930 994 688 × 2 = 0 + 0.353 515 625 000 000 000 000 000 000 000 321 737 641 198 522 927 861 989 376;
  • 82) 0.353 515 625 000 000 000 000 000 000 000 321 737 641 198 522 927 861 989 376 × 2 = 0 + 0.707 031 250 000 000 000 000 000 000 000 643 475 282 397 045 855 723 978 752;
  • 83) 0.707 031 250 000 000 000 000 000 000 000 643 475 282 397 045 855 723 978 752 × 2 = 1 + 0.414 062 500 000 000 000 000 000 000 001 286 950 564 794 091 711 447 957 504;
  • 84) 0.414 062 500 000 000 000 000 000 000 001 286 950 564 794 091 711 447 957 504 × 2 = 0 + 0.828 125 000 000 000 000 000 000 000 002 573 901 129 588 183 422 895 915 008;
  • 85) 0.828 125 000 000 000 000 000 000 000 002 573 901 129 588 183 422 895 915 008 × 2 = 1 + 0.656 250 000 000 000 000 000 000 000 005 147 802 259 176 366 845 791 830 016;
  • 86) 0.656 250 000 000 000 000 000 000 000 005 147 802 259 176 366 845 791 830 016 × 2 = 1 + 0.312 500 000 000 000 000 000 000 000 010 295 604 518 352 733 691 583 660 032;
  • 87) 0.312 500 000 000 000 000 000 000 000 010 295 604 518 352 733 691 583 660 032 × 2 = 0 + 0.625 000 000 000 000 000 000 000 000 020 591 209 036 705 467 383 167 320 064;
  • 88) 0.625 000 000 000 000 000 000 000 000 020 591 209 036 705 467 383 167 320 064 × 2 = 1 + 0.250 000 000 000 000 000 000 000 000 041 182 418 073 410 934 766 334 640 128;
  • 89) 0.250 000 000 000 000 000 000 000 000 041 182 418 073 410 934 766 334 640 128 × 2 = 0 + 0.500 000 000 000 000 000 000 000 000 082 364 836 146 821 869 532 669 280 256;
  • 90) 0.500 000 000 000 000 000 000 000 000 082 364 836 146 821 869 532 669 280 256 × 2 = 1 + 0.000 000 000 000 000 000 000 000 000 164 729 672 293 643 739 065 338 560 512;
  • 91) 0.000 000 000 000 000 000 000 000 000 164 729 672 293 643 739 065 338 560 512 × 2 = 0 + 0.000 000 000 000 000 000 000 000 000 329 459 344 587 287 478 130 677 121 024;

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:


0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 163(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:

0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 163(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:


0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 163(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 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 163 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 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 064 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: Your Greatest Competitor, Your Most Powerful Ally. NotebookLM YouTube Video of 7.54 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