0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 904 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 904(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
0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 904(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: 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 904.

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) 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 904 × 2 = 0 + 0.000 000 000 007 269 999 999 999 999 516 522 114 064 601 356 670 305 808;
  • 2) 0.000 000 000 007 269 999 999 999 999 516 522 114 064 601 356 670 305 808 × 2 = 0 + 0.000 000 000 014 539 999 999 999 999 033 044 228 129 202 713 340 611 616;
  • 3) 0.000 000 000 014 539 999 999 999 999 033 044 228 129 202 713 340 611 616 × 2 = 0 + 0.000 000 000 029 079 999 999 999 998 066 088 456 258 405 426 681 223 232;
  • 4) 0.000 000 000 029 079 999 999 999 998 066 088 456 258 405 426 681 223 232 × 2 = 0 + 0.000 000 000 058 159 999 999 999 996 132 176 912 516 810 853 362 446 464;
  • 5) 0.000 000 000 058 159 999 999 999 996 132 176 912 516 810 853 362 446 464 × 2 = 0 + 0.000 000 000 116 319 999 999 999 992 264 353 825 033 621 706 724 892 928;
  • 6) 0.000 000 000 116 319 999 999 999 992 264 353 825 033 621 706 724 892 928 × 2 = 0 + 0.000 000 000 232 639 999 999 999 984 528 707 650 067 243 413 449 785 856;
  • 7) 0.000 000 000 232 639 999 999 999 984 528 707 650 067 243 413 449 785 856 × 2 = 0 + 0.000 000 000 465 279 999 999 999 969 057 415 300 134 486 826 899 571 712;
  • 8) 0.000 000 000 465 279 999 999 999 969 057 415 300 134 486 826 899 571 712 × 2 = 0 + 0.000 000 000 930 559 999 999 999 938 114 830 600 268 973 653 799 143 424;
  • 9) 0.000 000 000 930 559 999 999 999 938 114 830 600 268 973 653 799 143 424 × 2 = 0 + 0.000 000 001 861 119 999 999 999 876 229 661 200 537 947 307 598 286 848;
  • 10) 0.000 000 001 861 119 999 999 999 876 229 661 200 537 947 307 598 286 848 × 2 = 0 + 0.000 000 003 722 239 999 999 999 752 459 322 401 075 894 615 196 573 696;
  • 11) 0.000 000 003 722 239 999 999 999 752 459 322 401 075 894 615 196 573 696 × 2 = 0 + 0.000 000 007 444 479 999 999 999 504 918 644 802 151 789 230 393 147 392;
  • 12) 0.000 000 007 444 479 999 999 999 504 918 644 802 151 789 230 393 147 392 × 2 = 0 + 0.000 000 014 888 959 999 999 999 009 837 289 604 303 578 460 786 294 784;
  • 13) 0.000 000 014 888 959 999 999 999 009 837 289 604 303 578 460 786 294 784 × 2 = 0 + 0.000 000 029 777 919 999 999 998 019 674 579 208 607 156 921 572 589 568;
  • 14) 0.000 000 029 777 919 999 999 998 019 674 579 208 607 156 921 572 589 568 × 2 = 0 + 0.000 000 059 555 839 999 999 996 039 349 158 417 214 313 843 145 179 136;
  • 15) 0.000 000 059 555 839 999 999 996 039 349 158 417 214 313 843 145 179 136 × 2 = 0 + 0.000 000 119 111 679 999 999 992 078 698 316 834 428 627 686 290 358 272;
  • 16) 0.000 000 119 111 679 999 999 992 078 698 316 834 428 627 686 290 358 272 × 2 = 0 + 0.000 000 238 223 359 999 999 984 157 396 633 668 857 255 372 580 716 544;
  • 17) 0.000 000 238 223 359 999 999 984 157 396 633 668 857 255 372 580 716 544 × 2 = 0 + 0.000 000 476 446 719 999 999 968 314 793 267 337 714 510 745 161 433 088;
  • 18) 0.000 000 476 446 719 999 999 968 314 793 267 337 714 510 745 161 433 088 × 2 = 0 + 0.000 000 952 893 439 999 999 936 629 586 534 675 429 021 490 322 866 176;
  • 19) 0.000 000 952 893 439 999 999 936 629 586 534 675 429 021 490 322 866 176 × 2 = 0 + 0.000 001 905 786 879 999 999 873 259 173 069 350 858 042 980 645 732 352;
  • 20) 0.000 001 905 786 879 999 999 873 259 173 069 350 858 042 980 645 732 352 × 2 = 0 + 0.000 003 811 573 759 999 999 746 518 346 138 701 716 085 961 291 464 704;
  • 21) 0.000 003 811 573 759 999 999 746 518 346 138 701 716 085 961 291 464 704 × 2 = 0 + 0.000 007 623 147 519 999 999 493 036 692 277 403 432 171 922 582 929 408;
  • 22) 0.000 007 623 147 519 999 999 493 036 692 277 403 432 171 922 582 929 408 × 2 = 0 + 0.000 015 246 295 039 999 998 986 073 384 554 806 864 343 845 165 858 816;
  • 23) 0.000 015 246 295 039 999 998 986 073 384 554 806 864 343 845 165 858 816 × 2 = 0 + 0.000 030 492 590 079 999 997 972 146 769 109 613 728 687 690 331 717 632;
  • 24) 0.000 030 492 590 079 999 997 972 146 769 109 613 728 687 690 331 717 632 × 2 = 0 + 0.000 060 985 180 159 999 995 944 293 538 219 227 457 375 380 663 435 264;
  • 25) 0.000 060 985 180 159 999 995 944 293 538 219 227 457 375 380 663 435 264 × 2 = 0 + 0.000 121 970 360 319 999 991 888 587 076 438 454 914 750 761 326 870 528;
  • 26) 0.000 121 970 360 319 999 991 888 587 076 438 454 914 750 761 326 870 528 × 2 = 0 + 0.000 243 940 720 639 999 983 777 174 152 876 909 829 501 522 653 741 056;
  • 27) 0.000 243 940 720 639 999 983 777 174 152 876 909 829 501 522 653 741 056 × 2 = 0 + 0.000 487 881 441 279 999 967 554 348 305 753 819 659 003 045 307 482 112;
  • 28) 0.000 487 881 441 279 999 967 554 348 305 753 819 659 003 045 307 482 112 × 2 = 0 + 0.000 975 762 882 559 999 935 108 696 611 507 639 318 006 090 614 964 224;
  • 29) 0.000 975 762 882 559 999 935 108 696 611 507 639 318 006 090 614 964 224 × 2 = 0 + 0.001 951 525 765 119 999 870 217 393 223 015 278 636 012 181 229 928 448;
  • 30) 0.001 951 525 765 119 999 870 217 393 223 015 278 636 012 181 229 928 448 × 2 = 0 + 0.003 903 051 530 239 999 740 434 786 446 030 557 272 024 362 459 856 896;
  • 31) 0.003 903 051 530 239 999 740 434 786 446 030 557 272 024 362 459 856 896 × 2 = 0 + 0.007 806 103 060 479 999 480 869 572 892 061 114 544 048 724 919 713 792;
  • 32) 0.007 806 103 060 479 999 480 869 572 892 061 114 544 048 724 919 713 792 × 2 = 0 + 0.015 612 206 120 959 998 961 739 145 784 122 229 088 097 449 839 427 584;
  • 33) 0.015 612 206 120 959 998 961 739 145 784 122 229 088 097 449 839 427 584 × 2 = 0 + 0.031 224 412 241 919 997 923 478 291 568 244 458 176 194 899 678 855 168;
  • 34) 0.031 224 412 241 919 997 923 478 291 568 244 458 176 194 899 678 855 168 × 2 = 0 + 0.062 448 824 483 839 995 846 956 583 136 488 916 352 389 799 357 710 336;
  • 35) 0.062 448 824 483 839 995 846 956 583 136 488 916 352 389 799 357 710 336 × 2 = 0 + 0.124 897 648 967 679 991 693 913 166 272 977 832 704 779 598 715 420 672;
  • 36) 0.124 897 648 967 679 991 693 913 166 272 977 832 704 779 598 715 420 672 × 2 = 0 + 0.249 795 297 935 359 983 387 826 332 545 955 665 409 559 197 430 841 344;
  • 37) 0.249 795 297 935 359 983 387 826 332 545 955 665 409 559 197 430 841 344 × 2 = 0 + 0.499 590 595 870 719 966 775 652 665 091 911 330 819 118 394 861 682 688;
  • 38) 0.499 590 595 870 719 966 775 652 665 091 911 330 819 118 394 861 682 688 × 2 = 0 + 0.999 181 191 741 439 933 551 305 330 183 822 661 638 236 789 723 365 376;
  • 39) 0.999 181 191 741 439 933 551 305 330 183 822 661 638 236 789 723 365 376 × 2 = 1 + 0.998 362 383 482 879 867 102 610 660 367 645 323 276 473 579 446 730 752;
  • 40) 0.998 362 383 482 879 867 102 610 660 367 645 323 276 473 579 446 730 752 × 2 = 1 + 0.996 724 766 965 759 734 205 221 320 735 290 646 552 947 158 893 461 504;
  • 41) 0.996 724 766 965 759 734 205 221 320 735 290 646 552 947 158 893 461 504 × 2 = 1 + 0.993 449 533 931 519 468 410 442 641 470 581 293 105 894 317 786 923 008;
  • 42) 0.993 449 533 931 519 468 410 442 641 470 581 293 105 894 317 786 923 008 × 2 = 1 + 0.986 899 067 863 038 936 820 885 282 941 162 586 211 788 635 573 846 016;
  • 43) 0.986 899 067 863 038 936 820 885 282 941 162 586 211 788 635 573 846 016 × 2 = 1 + 0.973 798 135 726 077 873 641 770 565 882 325 172 423 577 271 147 692 032;
  • 44) 0.973 798 135 726 077 873 641 770 565 882 325 172 423 577 271 147 692 032 × 2 = 1 + 0.947 596 271 452 155 747 283 541 131 764 650 344 847 154 542 295 384 064;
  • 45) 0.947 596 271 452 155 747 283 541 131 764 650 344 847 154 542 295 384 064 × 2 = 1 + 0.895 192 542 904 311 494 567 082 263 529 300 689 694 309 084 590 768 128;
  • 46) 0.895 192 542 904 311 494 567 082 263 529 300 689 694 309 084 590 768 128 × 2 = 1 + 0.790 385 085 808 622 989 134 164 527 058 601 379 388 618 169 181 536 256;
  • 47) 0.790 385 085 808 622 989 134 164 527 058 601 379 388 618 169 181 536 256 × 2 = 1 + 0.580 770 171 617 245 978 268 329 054 117 202 758 777 236 338 363 072 512;
  • 48) 0.580 770 171 617 245 978 268 329 054 117 202 758 777 236 338 363 072 512 × 2 = 1 + 0.161 540 343 234 491 956 536 658 108 234 405 517 554 472 676 726 145 024;
  • 49) 0.161 540 343 234 491 956 536 658 108 234 405 517 554 472 676 726 145 024 × 2 = 0 + 0.323 080 686 468 983 913 073 316 216 468 811 035 108 945 353 452 290 048;
  • 50) 0.323 080 686 468 983 913 073 316 216 468 811 035 108 945 353 452 290 048 × 2 = 0 + 0.646 161 372 937 967 826 146 632 432 937 622 070 217 890 706 904 580 096;
  • 51) 0.646 161 372 937 967 826 146 632 432 937 622 070 217 890 706 904 580 096 × 2 = 1 + 0.292 322 745 875 935 652 293 264 865 875 244 140 435 781 413 809 160 192;
  • 52) 0.292 322 745 875 935 652 293 264 865 875 244 140 435 781 413 809 160 192 × 2 = 0 + 0.584 645 491 751 871 304 586 529 731 750 488 280 871 562 827 618 320 384;
  • 53) 0.584 645 491 751 871 304 586 529 731 750 488 280 871 562 827 618 320 384 × 2 = 1 + 0.169 290 983 503 742 609 173 059 463 500 976 561 743 125 655 236 640 768;
  • 54) 0.169 290 983 503 742 609 173 059 463 500 976 561 743 125 655 236 640 768 × 2 = 0 + 0.338 581 967 007 485 218 346 118 927 001 953 123 486 251 310 473 281 536;
  • 55) 0.338 581 967 007 485 218 346 118 927 001 953 123 486 251 310 473 281 536 × 2 = 0 + 0.677 163 934 014 970 436 692 237 854 003 906 246 972 502 620 946 563 072;
  • 56) 0.677 163 934 014 970 436 692 237 854 003 906 246 972 502 620 946 563 072 × 2 = 1 + 0.354 327 868 029 940 873 384 475 708 007 812 493 945 005 241 893 126 144;
  • 57) 0.354 327 868 029 940 873 384 475 708 007 812 493 945 005 241 893 126 144 × 2 = 0 + 0.708 655 736 059 881 746 768 951 416 015 624 987 890 010 483 786 252 288;
  • 58) 0.708 655 736 059 881 746 768 951 416 015 624 987 890 010 483 786 252 288 × 2 = 1 + 0.417 311 472 119 763 493 537 902 832 031 249 975 780 020 967 572 504 576;
  • 59) 0.417 311 472 119 763 493 537 902 832 031 249 975 780 020 967 572 504 576 × 2 = 0 + 0.834 622 944 239 526 987 075 805 664 062 499 951 560 041 935 145 009 152;
  • 60) 0.834 622 944 239 526 987 075 805 664 062 499 951 560 041 935 145 009 152 × 2 = 1 + 0.669 245 888 479 053 974 151 611 328 124 999 903 120 083 870 290 018 304;
  • 61) 0.669 245 888 479 053 974 151 611 328 124 999 903 120 083 870 290 018 304 × 2 = 1 + 0.338 491 776 958 107 948 303 222 656 249 999 806 240 167 740 580 036 608;
  • 62) 0.338 491 776 958 107 948 303 222 656 249 999 806 240 167 740 580 036 608 × 2 = 0 + 0.676 983 553 916 215 896 606 445 312 499 999 612 480 335 481 160 073 216;
  • 63) 0.676 983 553 916 215 896 606 445 312 499 999 612 480 335 481 160 073 216 × 2 = 1 + 0.353 967 107 832 431 793 212 890 624 999 999 224 960 670 962 320 146 432;
  • 64) 0.353 967 107 832 431 793 212 890 624 999 999 224 960 670 962 320 146 432 × 2 = 0 + 0.707 934 215 664 863 586 425 781 249 999 998 449 921 341 924 640 292 864;
  • 65) 0.707 934 215 664 863 586 425 781 249 999 998 449 921 341 924 640 292 864 × 2 = 1 + 0.415 868 431 329 727 172 851 562 499 999 996 899 842 683 849 280 585 728;
  • 66) 0.415 868 431 329 727 172 851 562 499 999 996 899 842 683 849 280 585 728 × 2 = 0 + 0.831 736 862 659 454 345 703 124 999 999 993 799 685 367 698 561 171 456;
  • 67) 0.831 736 862 659 454 345 703 124 999 999 993 799 685 367 698 561 171 456 × 2 = 1 + 0.663 473 725 318 908 691 406 249 999 999 987 599 370 735 397 122 342 912;
  • 68) 0.663 473 725 318 908 691 406 249 999 999 987 599 370 735 397 122 342 912 × 2 = 1 + 0.326 947 450 637 817 382 812 499 999 999 975 198 741 470 794 244 685 824;
  • 69) 0.326 947 450 637 817 382 812 499 999 999 975 198 741 470 794 244 685 824 × 2 = 0 + 0.653 894 901 275 634 765 624 999 999 999 950 397 482 941 588 489 371 648;
  • 70) 0.653 894 901 275 634 765 624 999 999 999 950 397 482 941 588 489 371 648 × 2 = 1 + 0.307 789 802 551 269 531 249 999 999 999 900 794 965 883 176 978 743 296;
  • 71) 0.307 789 802 551 269 531 249 999 999 999 900 794 965 883 176 978 743 296 × 2 = 0 + 0.615 579 605 102 539 062 499 999 999 999 801 589 931 766 353 957 486 592;
  • 72) 0.615 579 605 102 539 062 499 999 999 999 801 589 931 766 353 957 486 592 × 2 = 1 + 0.231 159 210 205 078 124 999 999 999 999 603 179 863 532 707 914 973 184;
  • 73) 0.231 159 210 205 078 124 999 999 999 999 603 179 863 532 707 914 973 184 × 2 = 0 + 0.462 318 420 410 156 249 999 999 999 999 206 359 727 065 415 829 946 368;
  • 74) 0.462 318 420 410 156 249 999 999 999 999 206 359 727 065 415 829 946 368 × 2 = 0 + 0.924 636 840 820 312 499 999 999 999 998 412 719 454 130 831 659 892 736;
  • 75) 0.924 636 840 820 312 499 999 999 999 998 412 719 454 130 831 659 892 736 × 2 = 1 + 0.849 273 681 640 624 999 999 999 999 996 825 438 908 261 663 319 785 472;
  • 76) 0.849 273 681 640 624 999 999 999 999 996 825 438 908 261 663 319 785 472 × 2 = 1 + 0.698 547 363 281 249 999 999 999 999 993 650 877 816 523 326 639 570 944;
  • 77) 0.698 547 363 281 249 999 999 999 999 993 650 877 816 523 326 639 570 944 × 2 = 1 + 0.397 094 726 562 499 999 999 999 999 987 301 755 633 046 653 279 141 888;
  • 78) 0.397 094 726 562 499 999 999 999 999 987 301 755 633 046 653 279 141 888 × 2 = 0 + 0.794 189 453 124 999 999 999 999 999 974 603 511 266 093 306 558 283 776;
  • 79) 0.794 189 453 124 999 999 999 999 999 974 603 511 266 093 306 558 283 776 × 2 = 1 + 0.588 378 906 249 999 999 999 999 999 949 207 022 532 186 613 116 567 552;
  • 80) 0.588 378 906 249 999 999 999 999 999 949 207 022 532 186 613 116 567 552 × 2 = 1 + 0.176 757 812 499 999 999 999 999 999 898 414 045 064 373 226 233 135 104;
  • 81) 0.176 757 812 499 999 999 999 999 999 898 414 045 064 373 226 233 135 104 × 2 = 0 + 0.353 515 624 999 999 999 999 999 999 796 828 090 128 746 452 466 270 208;
  • 82) 0.353 515 624 999 999 999 999 999 999 796 828 090 128 746 452 466 270 208 × 2 = 0 + 0.707 031 249 999 999 999 999 999 999 593 656 180 257 492 904 932 540 416;
  • 83) 0.707 031 249 999 999 999 999 999 999 593 656 180 257 492 904 932 540 416 × 2 = 1 + 0.414 062 499 999 999 999 999 999 999 187 312 360 514 985 809 865 080 832;
  • 84) 0.414 062 499 999 999 999 999 999 999 187 312 360 514 985 809 865 080 832 × 2 = 0 + 0.828 124 999 999 999 999 999 999 998 374 624 721 029 971 619 730 161 664;
  • 85) 0.828 124 999 999 999 999 999 999 998 374 624 721 029 971 619 730 161 664 × 2 = 1 + 0.656 249 999 999 999 999 999 999 996 749 249 442 059 943 239 460 323 328;
  • 86) 0.656 249 999 999 999 999 999 999 996 749 249 442 059 943 239 460 323 328 × 2 = 1 + 0.312 499 999 999 999 999 999 999 993 498 498 884 119 886 478 920 646 656;
  • 87) 0.312 499 999 999 999 999 999 999 993 498 498 884 119 886 478 920 646 656 × 2 = 0 + 0.624 999 999 999 999 999 999 999 986 996 997 768 239 772 957 841 293 312;
  • 88) 0.624 999 999 999 999 999 999 999 986 996 997 768 239 772 957 841 293 312 × 2 = 1 + 0.249 999 999 999 999 999 999 999 973 993 995 536 479 545 915 682 586 624;
  • 89) 0.249 999 999 999 999 999 999 999 973 993 995 536 479 545 915 682 586 624 × 2 = 0 + 0.499 999 999 999 999 999 999 999 947 987 991 072 959 091 831 365 173 248;
  • 90) 0.499 999 999 999 999 999 999 999 947 987 991 072 959 091 831 365 173 248 × 2 = 0 + 0.999 999 999 999 999 999 999 999 895 975 982 145 918 183 662 730 346 496;
  • 91) 0.999 999 999 999 999 999 999 999 895 975 982 145 918 183 662 730 346 496 × 2 = 1 + 0.999 999 999 999 999 999 999 999 791 951 964 291 836 367 325 460 692 992;

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:


0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 904(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0011 1111 1111 0010 1001 0101 1010 1011 0101 0011 1011 0010 1101 001(2)

5. Positive number before normalization:

0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 904(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0011 1111 1111 0010 1001 0101 1010 1011 0101 0011 1011 0010 1101 001(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:


0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 904(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0011 1111 1111 0010 1001 0101 1010 1011 0101 0011 1011 0010 1101 001(2) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0011 1111 1111 0010 1001 0101 1010 1011 0101 0011 1011 0010 1101 001(2) × 20 =

1.1111 1111 1001 0100 1010 1101 0101 1010 1001 1101 1001 0110 1001(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 1001


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 1001 =

1111 1111 1001 0100 1010 1101 0101 1010 1001 1101 1001 0110 1001


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 1001


Decimal number 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 904 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 1001

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How to convert numbers from the decimal system (base ten) to 64 bit double precision IEEE 754 binary floating point standard

Follow the steps below to convert a base 10 decimal number to 64 bit double precision IEEE 754 binary floating point:

  • 1. If the number to be converted is negative, start with its the positive version.
  • 2. First convert the integer part. Divide repeatedly by 2 the positive representation of the integer number that is to be converted to binary, until we get a quotient that is equal to zero, keeping track of each remainder.
  • 3. Construct the base 2 representation of the positive integer part of the number, by taking all the remainders from the previous operations, starting from the bottom of the list constructed above. Thus, the last remainder of the divisions becomes the first symbol (the leftmost) of the base two number, while the first remainder becomes the last symbol (the rightmost).
  • 4. Then convert the fractional part. Multiply the number repeatedly by 2, until we get a fractional part that is equal to zero, keeping track of each integer part of the results.
  • 5. Construct the base 2 representation of the fractional part of the number, by taking all the integer parts of the multiplying operations, starting from the top of the list constructed above (they should appear in the binary representation, from left to right, in the order they have been calculated).
  • 6. Normalize the binary representation of the number, shifting the decimal mark (the decimal point) "n" positions either to the left, or to the right, so that only one non zero digit remains to the left of the decimal mark.
  • 7. Adjust the exponent in 11 bit excess/bias notation and then convert it from decimal (base 10) to 11 bit binary, by using the same technique of repeatedly dividing by 2, as shown above:
    Exponent (adjusted) = Exponent (unadjusted) + 2(11-1) - 1
  • 8. Normalize mantissa, remove the leading (leftmost) bit, since it's allways '1' (and the decimal mark, if the case) and adjust its length to 52 bits, either by removing the excess bits from the right (losing precision...) or by adding extra bits set on '0' to the right.
  • 9. Sign (it takes 1 bit) is either 1 for a negative or 0 for a positive number.

Example: convert the negative number -31.640 215 from the decimal system (base ten) to 64 bit double precision IEEE 754 binary floating point:

  • 1. Start with the positive version of the number:

    |-31.640 215| = 31.640 215

  • 2. First convert the integer part, 31. Divide it repeatedly by 2, keeping track of each remainder, until we get a quotient that is equal to zero:
    • division = quotient + remainder;
    • 31 ÷ 2 = 15 + 1;
    • 15 ÷ 2 = 7 + 1;
    • 7 ÷ 2 = 3 + 1;
    • 3 ÷ 2 = 1 + 1;
    • 1 ÷ 2 = 0 + 1;
    • We have encountered a quotient that is ZERO => FULL STOP
  • 3. Construct the base 2 representation of the integer part of the number by taking all the remainders of the previous dividing operations, starting from the bottom of the list constructed above:

    31(10) = 1 1111(2)

  • 4. Then, convert the fractional part, 0.640 215. Multiply repeatedly by 2, keeping track of each integer part of the results, until we get a fractional part that is equal to zero:
    • #) multiplying = integer + fractional part;
    • 1) 0.640 215 × 2 = 1 + 0.280 43;
    • 2) 0.280 43 × 2 = 0 + 0.560 86;
    • 3) 0.560 86 × 2 = 1 + 0.121 72;
    • 4) 0.121 72 × 2 = 0 + 0.243 44;
    • 5) 0.243 44 × 2 = 0 + 0.486 88;
    • 6) 0.486 88 × 2 = 0 + 0.973 76;
    • 7) 0.973 76 × 2 = 1 + 0.947 52;
    • 8) 0.947 52 × 2 = 1 + 0.895 04;
    • 9) 0.895 04 × 2 = 1 + 0.790 08;
    • 10) 0.790 08 × 2 = 1 + 0.580 16;
    • 11) 0.580 16 × 2 = 1 + 0.160 32;
    • 12) 0.160 32 × 2 = 0 + 0.320 64;
    • 13) 0.320 64 × 2 = 0 + 0.641 28;
    • 14) 0.641 28 × 2 = 1 + 0.282 56;
    • 15) 0.282 56 × 2 = 0 + 0.565 12;
    • 16) 0.565 12 × 2 = 1 + 0.130 24;
    • 17) 0.130 24 × 2 = 0 + 0.260 48;
    • 18) 0.260 48 × 2 = 0 + 0.520 96;
    • 19) 0.520 96 × 2 = 1 + 0.041 92;
    • 20) 0.041 92 × 2 = 0 + 0.083 84;
    • 21) 0.083 84 × 2 = 0 + 0.167 68;
    • 22) 0.167 68 × 2 = 0 + 0.335 36;
    • 23) 0.335 36 × 2 = 0 + 0.670 72;
    • 24) 0.670 72 × 2 = 1 + 0.341 44;
    • 25) 0.341 44 × 2 = 0 + 0.682 88;
    • 26) 0.682 88 × 2 = 1 + 0.365 76;
    • 27) 0.365 76 × 2 = 0 + 0.731 52;
    • 28) 0.731 52 × 2 = 1 + 0.463 04;
    • 29) 0.463 04 × 2 = 0 + 0.926 08;
    • 30) 0.926 08 × 2 = 1 + 0.852 16;
    • 31) 0.852 16 × 2 = 1 + 0.704 32;
    • 32) 0.704 32 × 2 = 1 + 0.408 64;
    • 33) 0.408 64 × 2 = 0 + 0.817 28;
    • 34) 0.817 28 × 2 = 1 + 0.634 56;
    • 35) 0.634 56 × 2 = 1 + 0.269 12;
    • 36) 0.269 12 × 2 = 0 + 0.538 24;
    • 37) 0.538 24 × 2 = 1 + 0.076 48;
    • 38) 0.076 48 × 2 = 0 + 0.152 96;
    • 39) 0.152 96 × 2 = 0 + 0.305 92;
    • 40) 0.305 92 × 2 = 0 + 0.611 84;
    • 41) 0.611 84 × 2 = 1 + 0.223 68;
    • 42) 0.223 68 × 2 = 0 + 0.447 36;
    • 43) 0.447 36 × 2 = 0 + 0.894 72;
    • 44) 0.894 72 × 2 = 1 + 0.789 44;
    • 45) 0.789 44 × 2 = 1 + 0.578 88;
    • 46) 0.578 88 × 2 = 1 + 0.157 76;
    • 47) 0.157 76 × 2 = 0 + 0.315 52;
    • 48) 0.315 52 × 2 = 0 + 0.631 04;
    • 49) 0.631 04 × 2 = 1 + 0.262 08;
    • 50) 0.262 08 × 2 = 0 + 0.524 16;
    • 51) 0.524 16 × 2 = 1 + 0.048 32;
    • 52) 0.048 32 × 2 = 0 + 0.096 64;
    • 53) 0.096 64 × 2 = 0 + 0.193 28;
    • We didn't get any fractional part that was equal to zero. But we had enough iterations (over Mantissa limit = 52) and at least one integer part that was different from zero => FULL STOP (losing precision...).
  • 5. Construct the base 2 representation of the fractional part of the number, by taking all the integer parts of the previous multiplying operations, starting from the top of the constructed list above:

    0.640 215(10) = 0.1010 0011 1110 0101 0010 0001 0101 0111 0110 1000 1001 1100 1010 0(2)

  • 6. Summarizing - the positive number before normalization:

    31.640 215(10) = 1 1111.1010 0011 1110 0101 0010 0001 0101 0111 0110 1000 1001 1100 1010 0(2)

  • 7. Normalize the binary representation of the number, shifting the decimal mark 4 positions to the left so that only one non-zero digit stays to the left of the decimal mark:

    31.640 215(10) =
    1 1111.1010 0011 1110 0101 0010 0001 0101 0111 0110 1000 1001 1100 1010 0(2) =
    1 1111.1010 0011 1110 0101 0010 0001 0101 0111 0110 1000 1001 1100 1010 0(2) × 20 =
    1.1111 1010 0011 1110 0101 0010 0001 0101 0111 0110 1000 1001 1100 1010 0(2) × 24

  • 8. Up to this moment, there are the following elements that would feed into the 64 bit double precision IEEE 754 binary floating point representation:

    Sign: 1 (a negative number)

    Exponent (unadjusted): 4

    Mantissa (not-normalized): 1.1111 1010 0011 1110 0101 0010 0001 0101 0111 0110 1000 1001 1100 1010 0

  • 9. Adjust the exponent in 11 bit excess/bias notation and then convert it from decimal (base 10) to 11 bit binary (base 2), by using the same technique of repeatedly dividing it by 2, as shown above:

    Exponent (adjusted) = Exponent (unadjusted) + 2(11-1) - 1 = (4 + 1023)(10) = 1027(10) =
    100 0000 0011(2)

  • 10. Normalize mantissa, remove the leading (leftmost) bit, since it's allways '1' (and the decimal sign) and adjust its length to 52 bits, by removing the excess bits, from the right (losing precision...):

    Mantissa (not-normalized): 1.1111 1010 0011 1110 0101 0010 0001 0101 0111 0110 1000 1001 1100 1010 0

    Mantissa (normalized): 1111 1010 0011 1110 0101 0010 0001 0101 0111 0110 1000 1001 1100

  • Conclusion:

    Sign (1 bit) = 1 (a negative number)

    Exponent (8 bits) = 100 0000 0011

    Mantissa (52 bits) = 1111 1010 0011 1110 0101 0010 0001 0101 0111 0110 1000 1001 1100

  • Number -31.640 215, converted from decimal system (base 10) to 64 bit double precision IEEE 754 binary floating point =
    1 - 100 0000 0011 - 1111 1010 0011 1110 0101 0010 0001 0101 0111 0110 1000 1001 1100