0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 991 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 991(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
0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 991(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: 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 991.

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) 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 991 × 2 = 0 + 0.000 000 000 007 269 999 999 999 999 516 522 114 064 601 356 670 305 982;
  • 2) 0.000 000 000 007 269 999 999 999 999 516 522 114 064 601 356 670 305 982 × 2 = 0 + 0.000 000 000 014 539 999 999 999 999 033 044 228 129 202 713 340 611 964;
  • 3) 0.000 000 000 014 539 999 999 999 999 033 044 228 129 202 713 340 611 964 × 2 = 0 + 0.000 000 000 029 079 999 999 999 998 066 088 456 258 405 426 681 223 928;
  • 4) 0.000 000 000 029 079 999 999 999 998 066 088 456 258 405 426 681 223 928 × 2 = 0 + 0.000 000 000 058 159 999 999 999 996 132 176 912 516 810 853 362 447 856;
  • 5) 0.000 000 000 058 159 999 999 999 996 132 176 912 516 810 853 362 447 856 × 2 = 0 + 0.000 000 000 116 319 999 999 999 992 264 353 825 033 621 706 724 895 712;
  • 6) 0.000 000 000 116 319 999 999 999 992 264 353 825 033 621 706 724 895 712 × 2 = 0 + 0.000 000 000 232 639 999 999 999 984 528 707 650 067 243 413 449 791 424;
  • 7) 0.000 000 000 232 639 999 999 999 984 528 707 650 067 243 413 449 791 424 × 2 = 0 + 0.000 000 000 465 279 999 999 999 969 057 415 300 134 486 826 899 582 848;
  • 8) 0.000 000 000 465 279 999 999 999 969 057 415 300 134 486 826 899 582 848 × 2 = 0 + 0.000 000 000 930 559 999 999 999 938 114 830 600 268 973 653 799 165 696;
  • 9) 0.000 000 000 930 559 999 999 999 938 114 830 600 268 973 653 799 165 696 × 2 = 0 + 0.000 000 001 861 119 999 999 999 876 229 661 200 537 947 307 598 331 392;
  • 10) 0.000 000 001 861 119 999 999 999 876 229 661 200 537 947 307 598 331 392 × 2 = 0 + 0.000 000 003 722 239 999 999 999 752 459 322 401 075 894 615 196 662 784;
  • 11) 0.000 000 003 722 239 999 999 999 752 459 322 401 075 894 615 196 662 784 × 2 = 0 + 0.000 000 007 444 479 999 999 999 504 918 644 802 151 789 230 393 325 568;
  • 12) 0.000 000 007 444 479 999 999 999 504 918 644 802 151 789 230 393 325 568 × 2 = 0 + 0.000 000 014 888 959 999 999 999 009 837 289 604 303 578 460 786 651 136;
  • 13) 0.000 000 014 888 959 999 999 999 009 837 289 604 303 578 460 786 651 136 × 2 = 0 + 0.000 000 029 777 919 999 999 998 019 674 579 208 607 156 921 573 302 272;
  • 14) 0.000 000 029 777 919 999 999 998 019 674 579 208 607 156 921 573 302 272 × 2 = 0 + 0.000 000 059 555 839 999 999 996 039 349 158 417 214 313 843 146 604 544;
  • 15) 0.000 000 059 555 839 999 999 996 039 349 158 417 214 313 843 146 604 544 × 2 = 0 + 0.000 000 119 111 679 999 999 992 078 698 316 834 428 627 686 293 209 088;
  • 16) 0.000 000 119 111 679 999 999 992 078 698 316 834 428 627 686 293 209 088 × 2 = 0 + 0.000 000 238 223 359 999 999 984 157 396 633 668 857 255 372 586 418 176;
  • 17) 0.000 000 238 223 359 999 999 984 157 396 633 668 857 255 372 586 418 176 × 2 = 0 + 0.000 000 476 446 719 999 999 968 314 793 267 337 714 510 745 172 836 352;
  • 18) 0.000 000 476 446 719 999 999 968 314 793 267 337 714 510 745 172 836 352 × 2 = 0 + 0.000 000 952 893 439 999 999 936 629 586 534 675 429 021 490 345 672 704;
  • 19) 0.000 000 952 893 439 999 999 936 629 586 534 675 429 021 490 345 672 704 × 2 = 0 + 0.000 001 905 786 879 999 999 873 259 173 069 350 858 042 980 691 345 408;
  • 20) 0.000 001 905 786 879 999 999 873 259 173 069 350 858 042 980 691 345 408 × 2 = 0 + 0.000 003 811 573 759 999 999 746 518 346 138 701 716 085 961 382 690 816;
  • 21) 0.000 003 811 573 759 999 999 746 518 346 138 701 716 085 961 382 690 816 × 2 = 0 + 0.000 007 623 147 519 999 999 493 036 692 277 403 432 171 922 765 381 632;
  • 22) 0.000 007 623 147 519 999 999 493 036 692 277 403 432 171 922 765 381 632 × 2 = 0 + 0.000 015 246 295 039 999 998 986 073 384 554 806 864 343 845 530 763 264;
  • 23) 0.000 015 246 295 039 999 998 986 073 384 554 806 864 343 845 530 763 264 × 2 = 0 + 0.000 030 492 590 079 999 997 972 146 769 109 613 728 687 691 061 526 528;
  • 24) 0.000 030 492 590 079 999 997 972 146 769 109 613 728 687 691 061 526 528 × 2 = 0 + 0.000 060 985 180 159 999 995 944 293 538 219 227 457 375 382 123 053 056;
  • 25) 0.000 060 985 180 159 999 995 944 293 538 219 227 457 375 382 123 053 056 × 2 = 0 + 0.000 121 970 360 319 999 991 888 587 076 438 454 914 750 764 246 106 112;
  • 26) 0.000 121 970 360 319 999 991 888 587 076 438 454 914 750 764 246 106 112 × 2 = 0 + 0.000 243 940 720 639 999 983 777 174 152 876 909 829 501 528 492 212 224;
  • 27) 0.000 243 940 720 639 999 983 777 174 152 876 909 829 501 528 492 212 224 × 2 = 0 + 0.000 487 881 441 279 999 967 554 348 305 753 819 659 003 056 984 424 448;
  • 28) 0.000 487 881 441 279 999 967 554 348 305 753 819 659 003 056 984 424 448 × 2 = 0 + 0.000 975 762 882 559 999 935 108 696 611 507 639 318 006 113 968 848 896;
  • 29) 0.000 975 762 882 559 999 935 108 696 611 507 639 318 006 113 968 848 896 × 2 = 0 + 0.001 951 525 765 119 999 870 217 393 223 015 278 636 012 227 937 697 792;
  • 30) 0.001 951 525 765 119 999 870 217 393 223 015 278 636 012 227 937 697 792 × 2 = 0 + 0.003 903 051 530 239 999 740 434 786 446 030 557 272 024 455 875 395 584;
  • 31) 0.003 903 051 530 239 999 740 434 786 446 030 557 272 024 455 875 395 584 × 2 = 0 + 0.007 806 103 060 479 999 480 869 572 892 061 114 544 048 911 750 791 168;
  • 32) 0.007 806 103 060 479 999 480 869 572 892 061 114 544 048 911 750 791 168 × 2 = 0 + 0.015 612 206 120 959 998 961 739 145 784 122 229 088 097 823 501 582 336;
  • 33) 0.015 612 206 120 959 998 961 739 145 784 122 229 088 097 823 501 582 336 × 2 = 0 + 0.031 224 412 241 919 997 923 478 291 568 244 458 176 195 647 003 164 672;
  • 34) 0.031 224 412 241 919 997 923 478 291 568 244 458 176 195 647 003 164 672 × 2 = 0 + 0.062 448 824 483 839 995 846 956 583 136 488 916 352 391 294 006 329 344;
  • 35) 0.062 448 824 483 839 995 846 956 583 136 488 916 352 391 294 006 329 344 × 2 = 0 + 0.124 897 648 967 679 991 693 913 166 272 977 832 704 782 588 012 658 688;
  • 36) 0.124 897 648 967 679 991 693 913 166 272 977 832 704 782 588 012 658 688 × 2 = 0 + 0.249 795 297 935 359 983 387 826 332 545 955 665 409 565 176 025 317 376;
  • 37) 0.249 795 297 935 359 983 387 826 332 545 955 665 409 565 176 025 317 376 × 2 = 0 + 0.499 590 595 870 719 966 775 652 665 091 911 330 819 130 352 050 634 752;
  • 38) 0.499 590 595 870 719 966 775 652 665 091 911 330 819 130 352 050 634 752 × 2 = 0 + 0.999 181 191 741 439 933 551 305 330 183 822 661 638 260 704 101 269 504;
  • 39) 0.999 181 191 741 439 933 551 305 330 183 822 661 638 260 704 101 269 504 × 2 = 1 + 0.998 362 383 482 879 867 102 610 660 367 645 323 276 521 408 202 539 008;
  • 40) 0.998 362 383 482 879 867 102 610 660 367 645 323 276 521 408 202 539 008 × 2 = 1 + 0.996 724 766 965 759 734 205 221 320 735 290 646 553 042 816 405 078 016;
  • 41) 0.996 724 766 965 759 734 205 221 320 735 290 646 553 042 816 405 078 016 × 2 = 1 + 0.993 449 533 931 519 468 410 442 641 470 581 293 106 085 632 810 156 032;
  • 42) 0.993 449 533 931 519 468 410 442 641 470 581 293 106 085 632 810 156 032 × 2 = 1 + 0.986 899 067 863 038 936 820 885 282 941 162 586 212 171 265 620 312 064;
  • 43) 0.986 899 067 863 038 936 820 885 282 941 162 586 212 171 265 620 312 064 × 2 = 1 + 0.973 798 135 726 077 873 641 770 565 882 325 172 424 342 531 240 624 128;
  • 44) 0.973 798 135 726 077 873 641 770 565 882 325 172 424 342 531 240 624 128 × 2 = 1 + 0.947 596 271 452 155 747 283 541 131 764 650 344 848 685 062 481 248 256;
  • 45) 0.947 596 271 452 155 747 283 541 131 764 650 344 848 685 062 481 248 256 × 2 = 1 + 0.895 192 542 904 311 494 567 082 263 529 300 689 697 370 124 962 496 512;
  • 46) 0.895 192 542 904 311 494 567 082 263 529 300 689 697 370 124 962 496 512 × 2 = 1 + 0.790 385 085 808 622 989 134 164 527 058 601 379 394 740 249 924 993 024;
  • 47) 0.790 385 085 808 622 989 134 164 527 058 601 379 394 740 249 924 993 024 × 2 = 1 + 0.580 770 171 617 245 978 268 329 054 117 202 758 789 480 499 849 986 048;
  • 48) 0.580 770 171 617 245 978 268 329 054 117 202 758 789 480 499 849 986 048 × 2 = 1 + 0.161 540 343 234 491 956 536 658 108 234 405 517 578 960 999 699 972 096;
  • 49) 0.161 540 343 234 491 956 536 658 108 234 405 517 578 960 999 699 972 096 × 2 = 0 + 0.323 080 686 468 983 913 073 316 216 468 811 035 157 921 999 399 944 192;
  • 50) 0.323 080 686 468 983 913 073 316 216 468 811 035 157 921 999 399 944 192 × 2 = 0 + 0.646 161 372 937 967 826 146 632 432 937 622 070 315 843 998 799 888 384;
  • 51) 0.646 161 372 937 967 826 146 632 432 937 622 070 315 843 998 799 888 384 × 2 = 1 + 0.292 322 745 875 935 652 293 264 865 875 244 140 631 687 997 599 776 768;
  • 52) 0.292 322 745 875 935 652 293 264 865 875 244 140 631 687 997 599 776 768 × 2 = 0 + 0.584 645 491 751 871 304 586 529 731 750 488 281 263 375 995 199 553 536;
  • 53) 0.584 645 491 751 871 304 586 529 731 750 488 281 263 375 995 199 553 536 × 2 = 1 + 0.169 290 983 503 742 609 173 059 463 500 976 562 526 751 990 399 107 072;
  • 54) 0.169 290 983 503 742 609 173 059 463 500 976 562 526 751 990 399 107 072 × 2 = 0 + 0.338 581 967 007 485 218 346 118 927 001 953 125 053 503 980 798 214 144;
  • 55) 0.338 581 967 007 485 218 346 118 927 001 953 125 053 503 980 798 214 144 × 2 = 0 + 0.677 163 934 014 970 436 692 237 854 003 906 250 107 007 961 596 428 288;
  • 56) 0.677 163 934 014 970 436 692 237 854 003 906 250 107 007 961 596 428 288 × 2 = 1 + 0.354 327 868 029 940 873 384 475 708 007 812 500 214 015 923 192 856 576;
  • 57) 0.354 327 868 029 940 873 384 475 708 007 812 500 214 015 923 192 856 576 × 2 = 0 + 0.708 655 736 059 881 746 768 951 416 015 625 000 428 031 846 385 713 152;
  • 58) 0.708 655 736 059 881 746 768 951 416 015 625 000 428 031 846 385 713 152 × 2 = 1 + 0.417 311 472 119 763 493 537 902 832 031 250 000 856 063 692 771 426 304;
  • 59) 0.417 311 472 119 763 493 537 902 832 031 250 000 856 063 692 771 426 304 × 2 = 0 + 0.834 622 944 239 526 987 075 805 664 062 500 001 712 127 385 542 852 608;
  • 60) 0.834 622 944 239 526 987 075 805 664 062 500 001 712 127 385 542 852 608 × 2 = 1 + 0.669 245 888 479 053 974 151 611 328 125 000 003 424 254 771 085 705 216;
  • 61) 0.669 245 888 479 053 974 151 611 328 125 000 003 424 254 771 085 705 216 × 2 = 1 + 0.338 491 776 958 107 948 303 222 656 250 000 006 848 509 542 171 410 432;
  • 62) 0.338 491 776 958 107 948 303 222 656 250 000 006 848 509 542 171 410 432 × 2 = 0 + 0.676 983 553 916 215 896 606 445 312 500 000 013 697 019 084 342 820 864;
  • 63) 0.676 983 553 916 215 896 606 445 312 500 000 013 697 019 084 342 820 864 × 2 = 1 + 0.353 967 107 832 431 793 212 890 625 000 000 027 394 038 168 685 641 728;
  • 64) 0.353 967 107 832 431 793 212 890 625 000 000 027 394 038 168 685 641 728 × 2 = 0 + 0.707 934 215 664 863 586 425 781 250 000 000 054 788 076 337 371 283 456;
  • 65) 0.707 934 215 664 863 586 425 781 250 000 000 054 788 076 337 371 283 456 × 2 = 1 + 0.415 868 431 329 727 172 851 562 500 000 000 109 576 152 674 742 566 912;
  • 66) 0.415 868 431 329 727 172 851 562 500 000 000 109 576 152 674 742 566 912 × 2 = 0 + 0.831 736 862 659 454 345 703 125 000 000 000 219 152 305 349 485 133 824;
  • 67) 0.831 736 862 659 454 345 703 125 000 000 000 219 152 305 349 485 133 824 × 2 = 1 + 0.663 473 725 318 908 691 406 250 000 000 000 438 304 610 698 970 267 648;
  • 68) 0.663 473 725 318 908 691 406 250 000 000 000 438 304 610 698 970 267 648 × 2 = 1 + 0.326 947 450 637 817 382 812 500 000 000 000 876 609 221 397 940 535 296;
  • 69) 0.326 947 450 637 817 382 812 500 000 000 000 876 609 221 397 940 535 296 × 2 = 0 + 0.653 894 901 275 634 765 625 000 000 000 001 753 218 442 795 881 070 592;
  • 70) 0.653 894 901 275 634 765 625 000 000 000 001 753 218 442 795 881 070 592 × 2 = 1 + 0.307 789 802 551 269 531 250 000 000 000 003 506 436 885 591 762 141 184;
  • 71) 0.307 789 802 551 269 531 250 000 000 000 003 506 436 885 591 762 141 184 × 2 = 0 + 0.615 579 605 102 539 062 500 000 000 000 007 012 873 771 183 524 282 368;
  • 72) 0.615 579 605 102 539 062 500 000 000 000 007 012 873 771 183 524 282 368 × 2 = 1 + 0.231 159 210 205 078 125 000 000 000 000 014 025 747 542 367 048 564 736;
  • 73) 0.231 159 210 205 078 125 000 000 000 000 014 025 747 542 367 048 564 736 × 2 = 0 + 0.462 318 420 410 156 250 000 000 000 000 028 051 495 084 734 097 129 472;
  • 74) 0.462 318 420 410 156 250 000 000 000 000 028 051 495 084 734 097 129 472 × 2 = 0 + 0.924 636 840 820 312 500 000 000 000 000 056 102 990 169 468 194 258 944;
  • 75) 0.924 636 840 820 312 500 000 000 000 000 056 102 990 169 468 194 258 944 × 2 = 1 + 0.849 273 681 640 625 000 000 000 000 000 112 205 980 338 936 388 517 888;
  • 76) 0.849 273 681 640 625 000 000 000 000 000 112 205 980 338 936 388 517 888 × 2 = 1 + 0.698 547 363 281 250 000 000 000 000 000 224 411 960 677 872 777 035 776;
  • 77) 0.698 547 363 281 250 000 000 000 000 000 224 411 960 677 872 777 035 776 × 2 = 1 + 0.397 094 726 562 500 000 000 000 000 000 448 823 921 355 745 554 071 552;
  • 78) 0.397 094 726 562 500 000 000 000 000 000 448 823 921 355 745 554 071 552 × 2 = 0 + 0.794 189 453 125 000 000 000 000 000 000 897 647 842 711 491 108 143 104;
  • 79) 0.794 189 453 125 000 000 000 000 000 000 897 647 842 711 491 108 143 104 × 2 = 1 + 0.588 378 906 250 000 000 000 000 000 001 795 295 685 422 982 216 286 208;
  • 80) 0.588 378 906 250 000 000 000 000 000 001 795 295 685 422 982 216 286 208 × 2 = 1 + 0.176 757 812 500 000 000 000 000 000 003 590 591 370 845 964 432 572 416;
  • 81) 0.176 757 812 500 000 000 000 000 000 003 590 591 370 845 964 432 572 416 × 2 = 0 + 0.353 515 625 000 000 000 000 000 000 007 181 182 741 691 928 865 144 832;
  • 82) 0.353 515 625 000 000 000 000 000 000 007 181 182 741 691 928 865 144 832 × 2 = 0 + 0.707 031 250 000 000 000 000 000 000 014 362 365 483 383 857 730 289 664;
  • 83) 0.707 031 250 000 000 000 000 000 000 014 362 365 483 383 857 730 289 664 × 2 = 1 + 0.414 062 500 000 000 000 000 000 000 028 724 730 966 767 715 460 579 328;
  • 84) 0.414 062 500 000 000 000 000 000 000 028 724 730 966 767 715 460 579 328 × 2 = 0 + 0.828 125 000 000 000 000 000 000 000 057 449 461 933 535 430 921 158 656;
  • 85) 0.828 125 000 000 000 000 000 000 000 057 449 461 933 535 430 921 158 656 × 2 = 1 + 0.656 250 000 000 000 000 000 000 000 114 898 923 867 070 861 842 317 312;
  • 86) 0.656 250 000 000 000 000 000 000 000 114 898 923 867 070 861 842 317 312 × 2 = 1 + 0.312 500 000 000 000 000 000 000 000 229 797 847 734 141 723 684 634 624;
  • 87) 0.312 500 000 000 000 000 000 000 000 229 797 847 734 141 723 684 634 624 × 2 = 0 + 0.625 000 000 000 000 000 000 000 000 459 595 695 468 283 447 369 269 248;
  • 88) 0.625 000 000 000 000 000 000 000 000 459 595 695 468 283 447 369 269 248 × 2 = 1 + 0.250 000 000 000 000 000 000 000 000 919 191 390 936 566 894 738 538 496;
  • 89) 0.250 000 000 000 000 000 000 000 000 919 191 390 936 566 894 738 538 496 × 2 = 0 + 0.500 000 000 000 000 000 000 000 001 838 382 781 873 133 789 477 076 992;
  • 90) 0.500 000 000 000 000 000 000 000 001 838 382 781 873 133 789 477 076 992 × 2 = 1 + 0.000 000 000 000 000 000 000 000 003 676 765 563 746 267 578 954 153 984;
  • 91) 0.000 000 000 000 000 000 000 000 003 676 765 563 746 267 578 954 153 984 × 2 = 0 + 0.000 000 000 000 000 000 000 000 007 353 531 127 492 535 157 908 307 968;

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:


0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 991(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:

0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 991(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:


0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 991(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 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 991 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

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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