0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 987 903 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 987 903(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
0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 987 903(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: 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 987 903.

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) 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 987 903 × 2 = 0 + 0.000 000 000 007 269 999 999 999 999 516 522 114 064 601 356 670 305 975 806;
  • 2) 0.000 000 000 007 269 999 999 999 999 516 522 114 064 601 356 670 305 975 806 × 2 = 0 + 0.000 000 000 014 539 999 999 999 999 033 044 228 129 202 713 340 611 951 612;
  • 3) 0.000 000 000 014 539 999 999 999 999 033 044 228 129 202 713 340 611 951 612 × 2 = 0 + 0.000 000 000 029 079 999 999 999 998 066 088 456 258 405 426 681 223 903 224;
  • 4) 0.000 000 000 029 079 999 999 999 998 066 088 456 258 405 426 681 223 903 224 × 2 = 0 + 0.000 000 000 058 159 999 999 999 996 132 176 912 516 810 853 362 447 806 448;
  • 5) 0.000 000 000 058 159 999 999 999 996 132 176 912 516 810 853 362 447 806 448 × 2 = 0 + 0.000 000 000 116 319 999 999 999 992 264 353 825 033 621 706 724 895 612 896;
  • 6) 0.000 000 000 116 319 999 999 999 992 264 353 825 033 621 706 724 895 612 896 × 2 = 0 + 0.000 000 000 232 639 999 999 999 984 528 707 650 067 243 413 449 791 225 792;
  • 7) 0.000 000 000 232 639 999 999 999 984 528 707 650 067 243 413 449 791 225 792 × 2 = 0 + 0.000 000 000 465 279 999 999 999 969 057 415 300 134 486 826 899 582 451 584;
  • 8) 0.000 000 000 465 279 999 999 999 969 057 415 300 134 486 826 899 582 451 584 × 2 = 0 + 0.000 000 000 930 559 999 999 999 938 114 830 600 268 973 653 799 164 903 168;
  • 9) 0.000 000 000 930 559 999 999 999 938 114 830 600 268 973 653 799 164 903 168 × 2 = 0 + 0.000 000 001 861 119 999 999 999 876 229 661 200 537 947 307 598 329 806 336;
  • 10) 0.000 000 001 861 119 999 999 999 876 229 661 200 537 947 307 598 329 806 336 × 2 = 0 + 0.000 000 003 722 239 999 999 999 752 459 322 401 075 894 615 196 659 612 672;
  • 11) 0.000 000 003 722 239 999 999 999 752 459 322 401 075 894 615 196 659 612 672 × 2 = 0 + 0.000 000 007 444 479 999 999 999 504 918 644 802 151 789 230 393 319 225 344;
  • 12) 0.000 000 007 444 479 999 999 999 504 918 644 802 151 789 230 393 319 225 344 × 2 = 0 + 0.000 000 014 888 959 999 999 999 009 837 289 604 303 578 460 786 638 450 688;
  • 13) 0.000 000 014 888 959 999 999 999 009 837 289 604 303 578 460 786 638 450 688 × 2 = 0 + 0.000 000 029 777 919 999 999 998 019 674 579 208 607 156 921 573 276 901 376;
  • 14) 0.000 000 029 777 919 999 999 998 019 674 579 208 607 156 921 573 276 901 376 × 2 = 0 + 0.000 000 059 555 839 999 999 996 039 349 158 417 214 313 843 146 553 802 752;
  • 15) 0.000 000 059 555 839 999 999 996 039 349 158 417 214 313 843 146 553 802 752 × 2 = 0 + 0.000 000 119 111 679 999 999 992 078 698 316 834 428 627 686 293 107 605 504;
  • 16) 0.000 000 119 111 679 999 999 992 078 698 316 834 428 627 686 293 107 605 504 × 2 = 0 + 0.000 000 238 223 359 999 999 984 157 396 633 668 857 255 372 586 215 211 008;
  • 17) 0.000 000 238 223 359 999 999 984 157 396 633 668 857 255 372 586 215 211 008 × 2 = 0 + 0.000 000 476 446 719 999 999 968 314 793 267 337 714 510 745 172 430 422 016;
  • 18) 0.000 000 476 446 719 999 999 968 314 793 267 337 714 510 745 172 430 422 016 × 2 = 0 + 0.000 000 952 893 439 999 999 936 629 586 534 675 429 021 490 344 860 844 032;
  • 19) 0.000 000 952 893 439 999 999 936 629 586 534 675 429 021 490 344 860 844 032 × 2 = 0 + 0.000 001 905 786 879 999 999 873 259 173 069 350 858 042 980 689 721 688 064;
  • 20) 0.000 001 905 786 879 999 999 873 259 173 069 350 858 042 980 689 721 688 064 × 2 = 0 + 0.000 003 811 573 759 999 999 746 518 346 138 701 716 085 961 379 443 376 128;
  • 21) 0.000 003 811 573 759 999 999 746 518 346 138 701 716 085 961 379 443 376 128 × 2 = 0 + 0.000 007 623 147 519 999 999 493 036 692 277 403 432 171 922 758 886 752 256;
  • 22) 0.000 007 623 147 519 999 999 493 036 692 277 403 432 171 922 758 886 752 256 × 2 = 0 + 0.000 015 246 295 039 999 998 986 073 384 554 806 864 343 845 517 773 504 512;
  • 23) 0.000 015 246 295 039 999 998 986 073 384 554 806 864 343 845 517 773 504 512 × 2 = 0 + 0.000 030 492 590 079 999 997 972 146 769 109 613 728 687 691 035 547 009 024;
  • 24) 0.000 030 492 590 079 999 997 972 146 769 109 613 728 687 691 035 547 009 024 × 2 = 0 + 0.000 060 985 180 159 999 995 944 293 538 219 227 457 375 382 071 094 018 048;
  • 25) 0.000 060 985 180 159 999 995 944 293 538 219 227 457 375 382 071 094 018 048 × 2 = 0 + 0.000 121 970 360 319 999 991 888 587 076 438 454 914 750 764 142 188 036 096;
  • 26) 0.000 121 970 360 319 999 991 888 587 076 438 454 914 750 764 142 188 036 096 × 2 = 0 + 0.000 243 940 720 639 999 983 777 174 152 876 909 829 501 528 284 376 072 192;
  • 27) 0.000 243 940 720 639 999 983 777 174 152 876 909 829 501 528 284 376 072 192 × 2 = 0 + 0.000 487 881 441 279 999 967 554 348 305 753 819 659 003 056 568 752 144 384;
  • 28) 0.000 487 881 441 279 999 967 554 348 305 753 819 659 003 056 568 752 144 384 × 2 = 0 + 0.000 975 762 882 559 999 935 108 696 611 507 639 318 006 113 137 504 288 768;
  • 29) 0.000 975 762 882 559 999 935 108 696 611 507 639 318 006 113 137 504 288 768 × 2 = 0 + 0.001 951 525 765 119 999 870 217 393 223 015 278 636 012 226 275 008 577 536;
  • 30) 0.001 951 525 765 119 999 870 217 393 223 015 278 636 012 226 275 008 577 536 × 2 = 0 + 0.003 903 051 530 239 999 740 434 786 446 030 557 272 024 452 550 017 155 072;
  • 31) 0.003 903 051 530 239 999 740 434 786 446 030 557 272 024 452 550 017 155 072 × 2 = 0 + 0.007 806 103 060 479 999 480 869 572 892 061 114 544 048 905 100 034 310 144;
  • 32) 0.007 806 103 060 479 999 480 869 572 892 061 114 544 048 905 100 034 310 144 × 2 = 0 + 0.015 612 206 120 959 998 961 739 145 784 122 229 088 097 810 200 068 620 288;
  • 33) 0.015 612 206 120 959 998 961 739 145 784 122 229 088 097 810 200 068 620 288 × 2 = 0 + 0.031 224 412 241 919 997 923 478 291 568 244 458 176 195 620 400 137 240 576;
  • 34) 0.031 224 412 241 919 997 923 478 291 568 244 458 176 195 620 400 137 240 576 × 2 = 0 + 0.062 448 824 483 839 995 846 956 583 136 488 916 352 391 240 800 274 481 152;
  • 35) 0.062 448 824 483 839 995 846 956 583 136 488 916 352 391 240 800 274 481 152 × 2 = 0 + 0.124 897 648 967 679 991 693 913 166 272 977 832 704 782 481 600 548 962 304;
  • 36) 0.124 897 648 967 679 991 693 913 166 272 977 832 704 782 481 600 548 962 304 × 2 = 0 + 0.249 795 297 935 359 983 387 826 332 545 955 665 409 564 963 201 097 924 608;
  • 37) 0.249 795 297 935 359 983 387 826 332 545 955 665 409 564 963 201 097 924 608 × 2 = 0 + 0.499 590 595 870 719 966 775 652 665 091 911 330 819 129 926 402 195 849 216;
  • 38) 0.499 590 595 870 719 966 775 652 665 091 911 330 819 129 926 402 195 849 216 × 2 = 0 + 0.999 181 191 741 439 933 551 305 330 183 822 661 638 259 852 804 391 698 432;
  • 39) 0.999 181 191 741 439 933 551 305 330 183 822 661 638 259 852 804 391 698 432 × 2 = 1 + 0.998 362 383 482 879 867 102 610 660 367 645 323 276 519 705 608 783 396 864;
  • 40) 0.998 362 383 482 879 867 102 610 660 367 645 323 276 519 705 608 783 396 864 × 2 = 1 + 0.996 724 766 965 759 734 205 221 320 735 290 646 553 039 411 217 566 793 728;
  • 41) 0.996 724 766 965 759 734 205 221 320 735 290 646 553 039 411 217 566 793 728 × 2 = 1 + 0.993 449 533 931 519 468 410 442 641 470 581 293 106 078 822 435 133 587 456;
  • 42) 0.993 449 533 931 519 468 410 442 641 470 581 293 106 078 822 435 133 587 456 × 2 = 1 + 0.986 899 067 863 038 936 820 885 282 941 162 586 212 157 644 870 267 174 912;
  • 43) 0.986 899 067 863 038 936 820 885 282 941 162 586 212 157 644 870 267 174 912 × 2 = 1 + 0.973 798 135 726 077 873 641 770 565 882 325 172 424 315 289 740 534 349 824;
  • 44) 0.973 798 135 726 077 873 641 770 565 882 325 172 424 315 289 740 534 349 824 × 2 = 1 + 0.947 596 271 452 155 747 283 541 131 764 650 344 848 630 579 481 068 699 648;
  • 45) 0.947 596 271 452 155 747 283 541 131 764 650 344 848 630 579 481 068 699 648 × 2 = 1 + 0.895 192 542 904 311 494 567 082 263 529 300 689 697 261 158 962 137 399 296;
  • 46) 0.895 192 542 904 311 494 567 082 263 529 300 689 697 261 158 962 137 399 296 × 2 = 1 + 0.790 385 085 808 622 989 134 164 527 058 601 379 394 522 317 924 274 798 592;
  • 47) 0.790 385 085 808 622 989 134 164 527 058 601 379 394 522 317 924 274 798 592 × 2 = 1 + 0.580 770 171 617 245 978 268 329 054 117 202 758 789 044 635 848 549 597 184;
  • 48) 0.580 770 171 617 245 978 268 329 054 117 202 758 789 044 635 848 549 597 184 × 2 = 1 + 0.161 540 343 234 491 956 536 658 108 234 405 517 578 089 271 697 099 194 368;
  • 49) 0.161 540 343 234 491 956 536 658 108 234 405 517 578 089 271 697 099 194 368 × 2 = 0 + 0.323 080 686 468 983 913 073 316 216 468 811 035 156 178 543 394 198 388 736;
  • 50) 0.323 080 686 468 983 913 073 316 216 468 811 035 156 178 543 394 198 388 736 × 2 = 0 + 0.646 161 372 937 967 826 146 632 432 937 622 070 312 357 086 788 396 777 472;
  • 51) 0.646 161 372 937 967 826 146 632 432 937 622 070 312 357 086 788 396 777 472 × 2 = 1 + 0.292 322 745 875 935 652 293 264 865 875 244 140 624 714 173 576 793 554 944;
  • 52) 0.292 322 745 875 935 652 293 264 865 875 244 140 624 714 173 576 793 554 944 × 2 = 0 + 0.584 645 491 751 871 304 586 529 731 750 488 281 249 428 347 153 587 109 888;
  • 53) 0.584 645 491 751 871 304 586 529 731 750 488 281 249 428 347 153 587 109 888 × 2 = 1 + 0.169 290 983 503 742 609 173 059 463 500 976 562 498 856 694 307 174 219 776;
  • 54) 0.169 290 983 503 742 609 173 059 463 500 976 562 498 856 694 307 174 219 776 × 2 = 0 + 0.338 581 967 007 485 218 346 118 927 001 953 124 997 713 388 614 348 439 552;
  • 55) 0.338 581 967 007 485 218 346 118 927 001 953 124 997 713 388 614 348 439 552 × 2 = 0 + 0.677 163 934 014 970 436 692 237 854 003 906 249 995 426 777 228 696 879 104;
  • 56) 0.677 163 934 014 970 436 692 237 854 003 906 249 995 426 777 228 696 879 104 × 2 = 1 + 0.354 327 868 029 940 873 384 475 708 007 812 499 990 853 554 457 393 758 208;
  • 57) 0.354 327 868 029 940 873 384 475 708 007 812 499 990 853 554 457 393 758 208 × 2 = 0 + 0.708 655 736 059 881 746 768 951 416 015 624 999 981 707 108 914 787 516 416;
  • 58) 0.708 655 736 059 881 746 768 951 416 015 624 999 981 707 108 914 787 516 416 × 2 = 1 + 0.417 311 472 119 763 493 537 902 832 031 249 999 963 414 217 829 575 032 832;
  • 59) 0.417 311 472 119 763 493 537 902 832 031 249 999 963 414 217 829 575 032 832 × 2 = 0 + 0.834 622 944 239 526 987 075 805 664 062 499 999 926 828 435 659 150 065 664;
  • 60) 0.834 622 944 239 526 987 075 805 664 062 499 999 926 828 435 659 150 065 664 × 2 = 1 + 0.669 245 888 479 053 974 151 611 328 124 999 999 853 656 871 318 300 131 328;
  • 61) 0.669 245 888 479 053 974 151 611 328 124 999 999 853 656 871 318 300 131 328 × 2 = 1 + 0.338 491 776 958 107 948 303 222 656 249 999 999 707 313 742 636 600 262 656;
  • 62) 0.338 491 776 958 107 948 303 222 656 249 999 999 707 313 742 636 600 262 656 × 2 = 0 + 0.676 983 553 916 215 896 606 445 312 499 999 999 414 627 485 273 200 525 312;
  • 63) 0.676 983 553 916 215 896 606 445 312 499 999 999 414 627 485 273 200 525 312 × 2 = 1 + 0.353 967 107 832 431 793 212 890 624 999 999 998 829 254 970 546 401 050 624;
  • 64) 0.353 967 107 832 431 793 212 890 624 999 999 998 829 254 970 546 401 050 624 × 2 = 0 + 0.707 934 215 664 863 586 425 781 249 999 999 997 658 509 941 092 802 101 248;
  • 65) 0.707 934 215 664 863 586 425 781 249 999 999 997 658 509 941 092 802 101 248 × 2 = 1 + 0.415 868 431 329 727 172 851 562 499 999 999 995 317 019 882 185 604 202 496;
  • 66) 0.415 868 431 329 727 172 851 562 499 999 999 995 317 019 882 185 604 202 496 × 2 = 0 + 0.831 736 862 659 454 345 703 124 999 999 999 990 634 039 764 371 208 404 992;
  • 67) 0.831 736 862 659 454 345 703 124 999 999 999 990 634 039 764 371 208 404 992 × 2 = 1 + 0.663 473 725 318 908 691 406 249 999 999 999 981 268 079 528 742 416 809 984;
  • 68) 0.663 473 725 318 908 691 406 249 999 999 999 981 268 079 528 742 416 809 984 × 2 = 1 + 0.326 947 450 637 817 382 812 499 999 999 999 962 536 159 057 484 833 619 968;
  • 69) 0.326 947 450 637 817 382 812 499 999 999 999 962 536 159 057 484 833 619 968 × 2 = 0 + 0.653 894 901 275 634 765 624 999 999 999 999 925 072 318 114 969 667 239 936;
  • 70) 0.653 894 901 275 634 765 624 999 999 999 999 925 072 318 114 969 667 239 936 × 2 = 1 + 0.307 789 802 551 269 531 249 999 999 999 999 850 144 636 229 939 334 479 872;
  • 71) 0.307 789 802 551 269 531 249 999 999 999 999 850 144 636 229 939 334 479 872 × 2 = 0 + 0.615 579 605 102 539 062 499 999 999 999 999 700 289 272 459 878 668 959 744;
  • 72) 0.615 579 605 102 539 062 499 999 999 999 999 700 289 272 459 878 668 959 744 × 2 = 1 + 0.231 159 210 205 078 124 999 999 999 999 999 400 578 544 919 757 337 919 488;
  • 73) 0.231 159 210 205 078 124 999 999 999 999 999 400 578 544 919 757 337 919 488 × 2 = 0 + 0.462 318 420 410 156 249 999 999 999 999 998 801 157 089 839 514 675 838 976;
  • 74) 0.462 318 420 410 156 249 999 999 999 999 998 801 157 089 839 514 675 838 976 × 2 = 0 + 0.924 636 840 820 312 499 999 999 999 999 997 602 314 179 679 029 351 677 952;
  • 75) 0.924 636 840 820 312 499 999 999 999 999 997 602 314 179 679 029 351 677 952 × 2 = 1 + 0.849 273 681 640 624 999 999 999 999 999 995 204 628 359 358 058 703 355 904;
  • 76) 0.849 273 681 640 624 999 999 999 999 999 995 204 628 359 358 058 703 355 904 × 2 = 1 + 0.698 547 363 281 249 999 999 999 999 999 990 409 256 718 716 117 406 711 808;
  • 77) 0.698 547 363 281 249 999 999 999 999 999 990 409 256 718 716 117 406 711 808 × 2 = 1 + 0.397 094 726 562 499 999 999 999 999 999 980 818 513 437 432 234 813 423 616;
  • 78) 0.397 094 726 562 499 999 999 999 999 999 980 818 513 437 432 234 813 423 616 × 2 = 0 + 0.794 189 453 124 999 999 999 999 999 999 961 637 026 874 864 469 626 847 232;
  • 79) 0.794 189 453 124 999 999 999 999 999 999 961 637 026 874 864 469 626 847 232 × 2 = 1 + 0.588 378 906 249 999 999 999 999 999 999 923 274 053 749 728 939 253 694 464;
  • 80) 0.588 378 906 249 999 999 999 999 999 999 923 274 053 749 728 939 253 694 464 × 2 = 1 + 0.176 757 812 499 999 999 999 999 999 999 846 548 107 499 457 878 507 388 928;
  • 81) 0.176 757 812 499 999 999 999 999 999 999 846 548 107 499 457 878 507 388 928 × 2 = 0 + 0.353 515 624 999 999 999 999 999 999 999 693 096 214 998 915 757 014 777 856;
  • 82) 0.353 515 624 999 999 999 999 999 999 999 693 096 214 998 915 757 014 777 856 × 2 = 0 + 0.707 031 249 999 999 999 999 999 999 999 386 192 429 997 831 514 029 555 712;
  • 83) 0.707 031 249 999 999 999 999 999 999 999 386 192 429 997 831 514 029 555 712 × 2 = 1 + 0.414 062 499 999 999 999 999 999 999 998 772 384 859 995 663 028 059 111 424;
  • 84) 0.414 062 499 999 999 999 999 999 999 998 772 384 859 995 663 028 059 111 424 × 2 = 0 + 0.828 124 999 999 999 999 999 999 999 997 544 769 719 991 326 056 118 222 848;
  • 85) 0.828 124 999 999 999 999 999 999 999 997 544 769 719 991 326 056 118 222 848 × 2 = 1 + 0.656 249 999 999 999 999 999 999 999 995 089 539 439 982 652 112 236 445 696;
  • 86) 0.656 249 999 999 999 999 999 999 999 995 089 539 439 982 652 112 236 445 696 × 2 = 1 + 0.312 499 999 999 999 999 999 999 999 990 179 078 879 965 304 224 472 891 392;
  • 87) 0.312 499 999 999 999 999 999 999 999 990 179 078 879 965 304 224 472 891 392 × 2 = 0 + 0.624 999 999 999 999 999 999 999 999 980 358 157 759 930 608 448 945 782 784;
  • 88) 0.624 999 999 999 999 999 999 999 999 980 358 157 759 930 608 448 945 782 784 × 2 = 1 + 0.249 999 999 999 999 999 999 999 999 960 716 315 519 861 216 897 891 565 568;
  • 89) 0.249 999 999 999 999 999 999 999 999 960 716 315 519 861 216 897 891 565 568 × 2 = 0 + 0.499 999 999 999 999 999 999 999 999 921 432 631 039 722 433 795 783 131 136;
  • 90) 0.499 999 999 999 999 999 999 999 999 921 432 631 039 722 433 795 783 131 136 × 2 = 0 + 0.999 999 999 999 999 999 999 999 999 842 865 262 079 444 867 591 566 262 272;
  • 91) 0.999 999 999 999 999 999 999 999 999 842 865 262 079 444 867 591 566 262 272 × 2 = 1 + 0.999 999 999 999 999 999 999 999 999 685 730 524 158 889 735 183 132 524 544;

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:


0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 987 903(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:

0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 987 903(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:


0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 987 903(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 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 987 903 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