0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 153 039 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 153 039(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
0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 153 039(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: 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 153 039.

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) 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 153 039 × 2 = 0 + 0.000 000 000 007 269 999 999 999 999 516 522 114 064 601 356 670 306 078;
  • 2) 0.000 000 000 007 269 999 999 999 999 516 522 114 064 601 356 670 306 078 × 2 = 0 + 0.000 000 000 014 539 999 999 999 999 033 044 228 129 202 713 340 612 156;
  • 3) 0.000 000 000 014 539 999 999 999 999 033 044 228 129 202 713 340 612 156 × 2 = 0 + 0.000 000 000 029 079 999 999 999 998 066 088 456 258 405 426 681 224 312;
  • 4) 0.000 000 000 029 079 999 999 999 998 066 088 456 258 405 426 681 224 312 × 2 = 0 + 0.000 000 000 058 159 999 999 999 996 132 176 912 516 810 853 362 448 624;
  • 5) 0.000 000 000 058 159 999 999 999 996 132 176 912 516 810 853 362 448 624 × 2 = 0 + 0.000 000 000 116 319 999 999 999 992 264 353 825 033 621 706 724 897 248;
  • 6) 0.000 000 000 116 319 999 999 999 992 264 353 825 033 621 706 724 897 248 × 2 = 0 + 0.000 000 000 232 639 999 999 999 984 528 707 650 067 243 413 449 794 496;
  • 7) 0.000 000 000 232 639 999 999 999 984 528 707 650 067 243 413 449 794 496 × 2 = 0 + 0.000 000 000 465 279 999 999 999 969 057 415 300 134 486 826 899 588 992;
  • 8) 0.000 000 000 465 279 999 999 999 969 057 415 300 134 486 826 899 588 992 × 2 = 0 + 0.000 000 000 930 559 999 999 999 938 114 830 600 268 973 653 799 177 984;
  • 9) 0.000 000 000 930 559 999 999 999 938 114 830 600 268 973 653 799 177 984 × 2 = 0 + 0.000 000 001 861 119 999 999 999 876 229 661 200 537 947 307 598 355 968;
  • 10) 0.000 000 001 861 119 999 999 999 876 229 661 200 537 947 307 598 355 968 × 2 = 0 + 0.000 000 003 722 239 999 999 999 752 459 322 401 075 894 615 196 711 936;
  • 11) 0.000 000 003 722 239 999 999 999 752 459 322 401 075 894 615 196 711 936 × 2 = 0 + 0.000 000 007 444 479 999 999 999 504 918 644 802 151 789 230 393 423 872;
  • 12) 0.000 000 007 444 479 999 999 999 504 918 644 802 151 789 230 393 423 872 × 2 = 0 + 0.000 000 014 888 959 999 999 999 009 837 289 604 303 578 460 786 847 744;
  • 13) 0.000 000 014 888 959 999 999 999 009 837 289 604 303 578 460 786 847 744 × 2 = 0 + 0.000 000 029 777 919 999 999 998 019 674 579 208 607 156 921 573 695 488;
  • 14) 0.000 000 029 777 919 999 999 998 019 674 579 208 607 156 921 573 695 488 × 2 = 0 + 0.000 000 059 555 839 999 999 996 039 349 158 417 214 313 843 147 390 976;
  • 15) 0.000 000 059 555 839 999 999 996 039 349 158 417 214 313 843 147 390 976 × 2 = 0 + 0.000 000 119 111 679 999 999 992 078 698 316 834 428 627 686 294 781 952;
  • 16) 0.000 000 119 111 679 999 999 992 078 698 316 834 428 627 686 294 781 952 × 2 = 0 + 0.000 000 238 223 359 999 999 984 157 396 633 668 857 255 372 589 563 904;
  • 17) 0.000 000 238 223 359 999 999 984 157 396 633 668 857 255 372 589 563 904 × 2 = 0 + 0.000 000 476 446 719 999 999 968 314 793 267 337 714 510 745 179 127 808;
  • 18) 0.000 000 476 446 719 999 999 968 314 793 267 337 714 510 745 179 127 808 × 2 = 0 + 0.000 000 952 893 439 999 999 936 629 586 534 675 429 021 490 358 255 616;
  • 19) 0.000 000 952 893 439 999 999 936 629 586 534 675 429 021 490 358 255 616 × 2 = 0 + 0.000 001 905 786 879 999 999 873 259 173 069 350 858 042 980 716 511 232;
  • 20) 0.000 001 905 786 879 999 999 873 259 173 069 350 858 042 980 716 511 232 × 2 = 0 + 0.000 003 811 573 759 999 999 746 518 346 138 701 716 085 961 433 022 464;
  • 21) 0.000 003 811 573 759 999 999 746 518 346 138 701 716 085 961 433 022 464 × 2 = 0 + 0.000 007 623 147 519 999 999 493 036 692 277 403 432 171 922 866 044 928;
  • 22) 0.000 007 623 147 519 999 999 493 036 692 277 403 432 171 922 866 044 928 × 2 = 0 + 0.000 015 246 295 039 999 998 986 073 384 554 806 864 343 845 732 089 856;
  • 23) 0.000 015 246 295 039 999 998 986 073 384 554 806 864 343 845 732 089 856 × 2 = 0 + 0.000 030 492 590 079 999 997 972 146 769 109 613 728 687 691 464 179 712;
  • 24) 0.000 030 492 590 079 999 997 972 146 769 109 613 728 687 691 464 179 712 × 2 = 0 + 0.000 060 985 180 159 999 995 944 293 538 219 227 457 375 382 928 359 424;
  • 25) 0.000 060 985 180 159 999 995 944 293 538 219 227 457 375 382 928 359 424 × 2 = 0 + 0.000 121 970 360 319 999 991 888 587 076 438 454 914 750 765 856 718 848;
  • 26) 0.000 121 970 360 319 999 991 888 587 076 438 454 914 750 765 856 718 848 × 2 = 0 + 0.000 243 940 720 639 999 983 777 174 152 876 909 829 501 531 713 437 696;
  • 27) 0.000 243 940 720 639 999 983 777 174 152 876 909 829 501 531 713 437 696 × 2 = 0 + 0.000 487 881 441 279 999 967 554 348 305 753 819 659 003 063 426 875 392;
  • 28) 0.000 487 881 441 279 999 967 554 348 305 753 819 659 003 063 426 875 392 × 2 = 0 + 0.000 975 762 882 559 999 935 108 696 611 507 639 318 006 126 853 750 784;
  • 29) 0.000 975 762 882 559 999 935 108 696 611 507 639 318 006 126 853 750 784 × 2 = 0 + 0.001 951 525 765 119 999 870 217 393 223 015 278 636 012 253 707 501 568;
  • 30) 0.001 951 525 765 119 999 870 217 393 223 015 278 636 012 253 707 501 568 × 2 = 0 + 0.003 903 051 530 239 999 740 434 786 446 030 557 272 024 507 415 003 136;
  • 31) 0.003 903 051 530 239 999 740 434 786 446 030 557 272 024 507 415 003 136 × 2 = 0 + 0.007 806 103 060 479 999 480 869 572 892 061 114 544 049 014 830 006 272;
  • 32) 0.007 806 103 060 479 999 480 869 572 892 061 114 544 049 014 830 006 272 × 2 = 0 + 0.015 612 206 120 959 998 961 739 145 784 122 229 088 098 029 660 012 544;
  • 33) 0.015 612 206 120 959 998 961 739 145 784 122 229 088 098 029 660 012 544 × 2 = 0 + 0.031 224 412 241 919 997 923 478 291 568 244 458 176 196 059 320 025 088;
  • 34) 0.031 224 412 241 919 997 923 478 291 568 244 458 176 196 059 320 025 088 × 2 = 0 + 0.062 448 824 483 839 995 846 956 583 136 488 916 352 392 118 640 050 176;
  • 35) 0.062 448 824 483 839 995 846 956 583 136 488 916 352 392 118 640 050 176 × 2 = 0 + 0.124 897 648 967 679 991 693 913 166 272 977 832 704 784 237 280 100 352;
  • 36) 0.124 897 648 967 679 991 693 913 166 272 977 832 704 784 237 280 100 352 × 2 = 0 + 0.249 795 297 935 359 983 387 826 332 545 955 665 409 568 474 560 200 704;
  • 37) 0.249 795 297 935 359 983 387 826 332 545 955 665 409 568 474 560 200 704 × 2 = 0 + 0.499 590 595 870 719 966 775 652 665 091 911 330 819 136 949 120 401 408;
  • 38) 0.499 590 595 870 719 966 775 652 665 091 911 330 819 136 949 120 401 408 × 2 = 0 + 0.999 181 191 741 439 933 551 305 330 183 822 661 638 273 898 240 802 816;
  • 39) 0.999 181 191 741 439 933 551 305 330 183 822 661 638 273 898 240 802 816 × 2 = 1 + 0.998 362 383 482 879 867 102 610 660 367 645 323 276 547 796 481 605 632;
  • 40) 0.998 362 383 482 879 867 102 610 660 367 645 323 276 547 796 481 605 632 × 2 = 1 + 0.996 724 766 965 759 734 205 221 320 735 290 646 553 095 592 963 211 264;
  • 41) 0.996 724 766 965 759 734 205 221 320 735 290 646 553 095 592 963 211 264 × 2 = 1 + 0.993 449 533 931 519 468 410 442 641 470 581 293 106 191 185 926 422 528;
  • 42) 0.993 449 533 931 519 468 410 442 641 470 581 293 106 191 185 926 422 528 × 2 = 1 + 0.986 899 067 863 038 936 820 885 282 941 162 586 212 382 371 852 845 056;
  • 43) 0.986 899 067 863 038 936 820 885 282 941 162 586 212 382 371 852 845 056 × 2 = 1 + 0.973 798 135 726 077 873 641 770 565 882 325 172 424 764 743 705 690 112;
  • 44) 0.973 798 135 726 077 873 641 770 565 882 325 172 424 764 743 705 690 112 × 2 = 1 + 0.947 596 271 452 155 747 283 541 131 764 650 344 849 529 487 411 380 224;
  • 45) 0.947 596 271 452 155 747 283 541 131 764 650 344 849 529 487 411 380 224 × 2 = 1 + 0.895 192 542 904 311 494 567 082 263 529 300 689 699 058 974 822 760 448;
  • 46) 0.895 192 542 904 311 494 567 082 263 529 300 689 699 058 974 822 760 448 × 2 = 1 + 0.790 385 085 808 622 989 134 164 527 058 601 379 398 117 949 645 520 896;
  • 47) 0.790 385 085 808 622 989 134 164 527 058 601 379 398 117 949 645 520 896 × 2 = 1 + 0.580 770 171 617 245 978 268 329 054 117 202 758 796 235 899 291 041 792;
  • 48) 0.580 770 171 617 245 978 268 329 054 117 202 758 796 235 899 291 041 792 × 2 = 1 + 0.161 540 343 234 491 956 536 658 108 234 405 517 592 471 798 582 083 584;
  • 49) 0.161 540 343 234 491 956 536 658 108 234 405 517 592 471 798 582 083 584 × 2 = 0 + 0.323 080 686 468 983 913 073 316 216 468 811 035 184 943 597 164 167 168;
  • 50) 0.323 080 686 468 983 913 073 316 216 468 811 035 184 943 597 164 167 168 × 2 = 0 + 0.646 161 372 937 967 826 146 632 432 937 622 070 369 887 194 328 334 336;
  • 51) 0.646 161 372 937 967 826 146 632 432 937 622 070 369 887 194 328 334 336 × 2 = 1 + 0.292 322 745 875 935 652 293 264 865 875 244 140 739 774 388 656 668 672;
  • 52) 0.292 322 745 875 935 652 293 264 865 875 244 140 739 774 388 656 668 672 × 2 = 0 + 0.584 645 491 751 871 304 586 529 731 750 488 281 479 548 777 313 337 344;
  • 53) 0.584 645 491 751 871 304 586 529 731 750 488 281 479 548 777 313 337 344 × 2 = 1 + 0.169 290 983 503 742 609 173 059 463 500 976 562 959 097 554 626 674 688;
  • 54) 0.169 290 983 503 742 609 173 059 463 500 976 562 959 097 554 626 674 688 × 2 = 0 + 0.338 581 967 007 485 218 346 118 927 001 953 125 918 195 109 253 349 376;
  • 55) 0.338 581 967 007 485 218 346 118 927 001 953 125 918 195 109 253 349 376 × 2 = 0 + 0.677 163 934 014 970 436 692 237 854 003 906 251 836 390 218 506 698 752;
  • 56) 0.677 163 934 014 970 436 692 237 854 003 906 251 836 390 218 506 698 752 × 2 = 1 + 0.354 327 868 029 940 873 384 475 708 007 812 503 672 780 437 013 397 504;
  • 57) 0.354 327 868 029 940 873 384 475 708 007 812 503 672 780 437 013 397 504 × 2 = 0 + 0.708 655 736 059 881 746 768 951 416 015 625 007 345 560 874 026 795 008;
  • 58) 0.708 655 736 059 881 746 768 951 416 015 625 007 345 560 874 026 795 008 × 2 = 1 + 0.417 311 472 119 763 493 537 902 832 031 250 014 691 121 748 053 590 016;
  • 59) 0.417 311 472 119 763 493 537 902 832 031 250 014 691 121 748 053 590 016 × 2 = 0 + 0.834 622 944 239 526 987 075 805 664 062 500 029 382 243 496 107 180 032;
  • 60) 0.834 622 944 239 526 987 075 805 664 062 500 029 382 243 496 107 180 032 × 2 = 1 + 0.669 245 888 479 053 974 151 611 328 125 000 058 764 486 992 214 360 064;
  • 61) 0.669 245 888 479 053 974 151 611 328 125 000 058 764 486 992 214 360 064 × 2 = 1 + 0.338 491 776 958 107 948 303 222 656 250 000 117 528 973 984 428 720 128;
  • 62) 0.338 491 776 958 107 948 303 222 656 250 000 117 528 973 984 428 720 128 × 2 = 0 + 0.676 983 553 916 215 896 606 445 312 500 000 235 057 947 968 857 440 256;
  • 63) 0.676 983 553 916 215 896 606 445 312 500 000 235 057 947 968 857 440 256 × 2 = 1 + 0.353 967 107 832 431 793 212 890 625 000 000 470 115 895 937 714 880 512;
  • 64) 0.353 967 107 832 431 793 212 890 625 000 000 470 115 895 937 714 880 512 × 2 = 0 + 0.707 934 215 664 863 586 425 781 250 000 000 940 231 791 875 429 761 024;
  • 65) 0.707 934 215 664 863 586 425 781 250 000 000 940 231 791 875 429 761 024 × 2 = 1 + 0.415 868 431 329 727 172 851 562 500 000 001 880 463 583 750 859 522 048;
  • 66) 0.415 868 431 329 727 172 851 562 500 000 001 880 463 583 750 859 522 048 × 2 = 0 + 0.831 736 862 659 454 345 703 125 000 000 003 760 927 167 501 719 044 096;
  • 67) 0.831 736 862 659 454 345 703 125 000 000 003 760 927 167 501 719 044 096 × 2 = 1 + 0.663 473 725 318 908 691 406 250 000 000 007 521 854 335 003 438 088 192;
  • 68) 0.663 473 725 318 908 691 406 250 000 000 007 521 854 335 003 438 088 192 × 2 = 1 + 0.326 947 450 637 817 382 812 500 000 000 015 043 708 670 006 876 176 384;
  • 69) 0.326 947 450 637 817 382 812 500 000 000 015 043 708 670 006 876 176 384 × 2 = 0 + 0.653 894 901 275 634 765 625 000 000 000 030 087 417 340 013 752 352 768;
  • 70) 0.653 894 901 275 634 765 625 000 000 000 030 087 417 340 013 752 352 768 × 2 = 1 + 0.307 789 802 551 269 531 250 000 000 000 060 174 834 680 027 504 705 536;
  • 71) 0.307 789 802 551 269 531 250 000 000 000 060 174 834 680 027 504 705 536 × 2 = 0 + 0.615 579 605 102 539 062 500 000 000 000 120 349 669 360 055 009 411 072;
  • 72) 0.615 579 605 102 539 062 500 000 000 000 120 349 669 360 055 009 411 072 × 2 = 1 + 0.231 159 210 205 078 125 000 000 000 000 240 699 338 720 110 018 822 144;
  • 73) 0.231 159 210 205 078 125 000 000 000 000 240 699 338 720 110 018 822 144 × 2 = 0 + 0.462 318 420 410 156 250 000 000 000 000 481 398 677 440 220 037 644 288;
  • 74) 0.462 318 420 410 156 250 000 000 000 000 481 398 677 440 220 037 644 288 × 2 = 0 + 0.924 636 840 820 312 500 000 000 000 000 962 797 354 880 440 075 288 576;
  • 75) 0.924 636 840 820 312 500 000 000 000 000 962 797 354 880 440 075 288 576 × 2 = 1 + 0.849 273 681 640 625 000 000 000 000 001 925 594 709 760 880 150 577 152;
  • 76) 0.849 273 681 640 625 000 000 000 000 001 925 594 709 760 880 150 577 152 × 2 = 1 + 0.698 547 363 281 250 000 000 000 000 003 851 189 419 521 760 301 154 304;
  • 77) 0.698 547 363 281 250 000 000 000 000 003 851 189 419 521 760 301 154 304 × 2 = 1 + 0.397 094 726 562 500 000 000 000 000 007 702 378 839 043 520 602 308 608;
  • 78) 0.397 094 726 562 500 000 000 000 000 007 702 378 839 043 520 602 308 608 × 2 = 0 + 0.794 189 453 125 000 000 000 000 000 015 404 757 678 087 041 204 617 216;
  • 79) 0.794 189 453 125 000 000 000 000 000 015 404 757 678 087 041 204 617 216 × 2 = 1 + 0.588 378 906 250 000 000 000 000 000 030 809 515 356 174 082 409 234 432;
  • 80) 0.588 378 906 250 000 000 000 000 000 030 809 515 356 174 082 409 234 432 × 2 = 1 + 0.176 757 812 500 000 000 000 000 000 061 619 030 712 348 164 818 468 864;
  • 81) 0.176 757 812 500 000 000 000 000 000 061 619 030 712 348 164 818 468 864 × 2 = 0 + 0.353 515 625 000 000 000 000 000 000 123 238 061 424 696 329 636 937 728;
  • 82) 0.353 515 625 000 000 000 000 000 000 123 238 061 424 696 329 636 937 728 × 2 = 0 + 0.707 031 250 000 000 000 000 000 000 246 476 122 849 392 659 273 875 456;
  • 83) 0.707 031 250 000 000 000 000 000 000 246 476 122 849 392 659 273 875 456 × 2 = 1 + 0.414 062 500 000 000 000 000 000 000 492 952 245 698 785 318 547 750 912;
  • 84) 0.414 062 500 000 000 000 000 000 000 492 952 245 698 785 318 547 750 912 × 2 = 0 + 0.828 125 000 000 000 000 000 000 000 985 904 491 397 570 637 095 501 824;
  • 85) 0.828 125 000 000 000 000 000 000 000 985 904 491 397 570 637 095 501 824 × 2 = 1 + 0.656 250 000 000 000 000 000 000 001 971 808 982 795 141 274 191 003 648;
  • 86) 0.656 250 000 000 000 000 000 000 001 971 808 982 795 141 274 191 003 648 × 2 = 1 + 0.312 500 000 000 000 000 000 000 003 943 617 965 590 282 548 382 007 296;
  • 87) 0.312 500 000 000 000 000 000 000 003 943 617 965 590 282 548 382 007 296 × 2 = 0 + 0.625 000 000 000 000 000 000 000 007 887 235 931 180 565 096 764 014 592;
  • 88) 0.625 000 000 000 000 000 000 000 007 887 235 931 180 565 096 764 014 592 × 2 = 1 + 0.250 000 000 000 000 000 000 000 015 774 471 862 361 130 193 528 029 184;
  • 89) 0.250 000 000 000 000 000 000 000 015 774 471 862 361 130 193 528 029 184 × 2 = 0 + 0.500 000 000 000 000 000 000 000 031 548 943 724 722 260 387 056 058 368;
  • 90) 0.500 000 000 000 000 000 000 000 031 548 943 724 722 260 387 056 058 368 × 2 = 1 + 0.000 000 000 000 000 000 000 000 063 097 887 449 444 520 774 112 116 736;
  • 91) 0.000 000 000 000 000 000 000 000 063 097 887 449 444 520 774 112 116 736 × 2 = 0 + 0.000 000 000 000 000 000 000 000 126 195 774 898 889 041 548 224 233 472;

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:


0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 153 039(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:

0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 153 039(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:


0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 153 039(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 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 153 039 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 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 989 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: Reputation: Bridge Between Company's Actions & Market Value. NotebookLM YouTube Video of 6.55 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