0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 979 5 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 979 5(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
0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 979 5(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: 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 979 5.

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) 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 979 5 × 2 = 0 + 0.000 000 000 007 269 999 999 999 999 516 522 114 064 601 356 670 305 959;
  • 2) 0.000 000 000 007 269 999 999 999 999 516 522 114 064 601 356 670 305 959 × 2 = 0 + 0.000 000 000 014 539 999 999 999 999 033 044 228 129 202 713 340 611 918;
  • 3) 0.000 000 000 014 539 999 999 999 999 033 044 228 129 202 713 340 611 918 × 2 = 0 + 0.000 000 000 029 079 999 999 999 998 066 088 456 258 405 426 681 223 836;
  • 4) 0.000 000 000 029 079 999 999 999 998 066 088 456 258 405 426 681 223 836 × 2 = 0 + 0.000 000 000 058 159 999 999 999 996 132 176 912 516 810 853 362 447 672;
  • 5) 0.000 000 000 058 159 999 999 999 996 132 176 912 516 810 853 362 447 672 × 2 = 0 + 0.000 000 000 116 319 999 999 999 992 264 353 825 033 621 706 724 895 344;
  • 6) 0.000 000 000 116 319 999 999 999 992 264 353 825 033 621 706 724 895 344 × 2 = 0 + 0.000 000 000 232 639 999 999 999 984 528 707 650 067 243 413 449 790 688;
  • 7) 0.000 000 000 232 639 999 999 999 984 528 707 650 067 243 413 449 790 688 × 2 = 0 + 0.000 000 000 465 279 999 999 999 969 057 415 300 134 486 826 899 581 376;
  • 8) 0.000 000 000 465 279 999 999 999 969 057 415 300 134 486 826 899 581 376 × 2 = 0 + 0.000 000 000 930 559 999 999 999 938 114 830 600 268 973 653 799 162 752;
  • 9) 0.000 000 000 930 559 999 999 999 938 114 830 600 268 973 653 799 162 752 × 2 = 0 + 0.000 000 001 861 119 999 999 999 876 229 661 200 537 947 307 598 325 504;
  • 10) 0.000 000 001 861 119 999 999 999 876 229 661 200 537 947 307 598 325 504 × 2 = 0 + 0.000 000 003 722 239 999 999 999 752 459 322 401 075 894 615 196 651 008;
  • 11) 0.000 000 003 722 239 999 999 999 752 459 322 401 075 894 615 196 651 008 × 2 = 0 + 0.000 000 007 444 479 999 999 999 504 918 644 802 151 789 230 393 302 016;
  • 12) 0.000 000 007 444 479 999 999 999 504 918 644 802 151 789 230 393 302 016 × 2 = 0 + 0.000 000 014 888 959 999 999 999 009 837 289 604 303 578 460 786 604 032;
  • 13) 0.000 000 014 888 959 999 999 999 009 837 289 604 303 578 460 786 604 032 × 2 = 0 + 0.000 000 029 777 919 999 999 998 019 674 579 208 607 156 921 573 208 064;
  • 14) 0.000 000 029 777 919 999 999 998 019 674 579 208 607 156 921 573 208 064 × 2 = 0 + 0.000 000 059 555 839 999 999 996 039 349 158 417 214 313 843 146 416 128;
  • 15) 0.000 000 059 555 839 999 999 996 039 349 158 417 214 313 843 146 416 128 × 2 = 0 + 0.000 000 119 111 679 999 999 992 078 698 316 834 428 627 686 292 832 256;
  • 16) 0.000 000 119 111 679 999 999 992 078 698 316 834 428 627 686 292 832 256 × 2 = 0 + 0.000 000 238 223 359 999 999 984 157 396 633 668 857 255 372 585 664 512;
  • 17) 0.000 000 238 223 359 999 999 984 157 396 633 668 857 255 372 585 664 512 × 2 = 0 + 0.000 000 476 446 719 999 999 968 314 793 267 337 714 510 745 171 329 024;
  • 18) 0.000 000 476 446 719 999 999 968 314 793 267 337 714 510 745 171 329 024 × 2 = 0 + 0.000 000 952 893 439 999 999 936 629 586 534 675 429 021 490 342 658 048;
  • 19) 0.000 000 952 893 439 999 999 936 629 586 534 675 429 021 490 342 658 048 × 2 = 0 + 0.000 001 905 786 879 999 999 873 259 173 069 350 858 042 980 685 316 096;
  • 20) 0.000 001 905 786 879 999 999 873 259 173 069 350 858 042 980 685 316 096 × 2 = 0 + 0.000 003 811 573 759 999 999 746 518 346 138 701 716 085 961 370 632 192;
  • 21) 0.000 003 811 573 759 999 999 746 518 346 138 701 716 085 961 370 632 192 × 2 = 0 + 0.000 007 623 147 519 999 999 493 036 692 277 403 432 171 922 741 264 384;
  • 22) 0.000 007 623 147 519 999 999 493 036 692 277 403 432 171 922 741 264 384 × 2 = 0 + 0.000 015 246 295 039 999 998 986 073 384 554 806 864 343 845 482 528 768;
  • 23) 0.000 015 246 295 039 999 998 986 073 384 554 806 864 343 845 482 528 768 × 2 = 0 + 0.000 030 492 590 079 999 997 972 146 769 109 613 728 687 690 965 057 536;
  • 24) 0.000 030 492 590 079 999 997 972 146 769 109 613 728 687 690 965 057 536 × 2 = 0 + 0.000 060 985 180 159 999 995 944 293 538 219 227 457 375 381 930 115 072;
  • 25) 0.000 060 985 180 159 999 995 944 293 538 219 227 457 375 381 930 115 072 × 2 = 0 + 0.000 121 970 360 319 999 991 888 587 076 438 454 914 750 763 860 230 144;
  • 26) 0.000 121 970 360 319 999 991 888 587 076 438 454 914 750 763 860 230 144 × 2 = 0 + 0.000 243 940 720 639 999 983 777 174 152 876 909 829 501 527 720 460 288;
  • 27) 0.000 243 940 720 639 999 983 777 174 152 876 909 829 501 527 720 460 288 × 2 = 0 + 0.000 487 881 441 279 999 967 554 348 305 753 819 659 003 055 440 920 576;
  • 28) 0.000 487 881 441 279 999 967 554 348 305 753 819 659 003 055 440 920 576 × 2 = 0 + 0.000 975 762 882 559 999 935 108 696 611 507 639 318 006 110 881 841 152;
  • 29) 0.000 975 762 882 559 999 935 108 696 611 507 639 318 006 110 881 841 152 × 2 = 0 + 0.001 951 525 765 119 999 870 217 393 223 015 278 636 012 221 763 682 304;
  • 30) 0.001 951 525 765 119 999 870 217 393 223 015 278 636 012 221 763 682 304 × 2 = 0 + 0.003 903 051 530 239 999 740 434 786 446 030 557 272 024 443 527 364 608;
  • 31) 0.003 903 051 530 239 999 740 434 786 446 030 557 272 024 443 527 364 608 × 2 = 0 + 0.007 806 103 060 479 999 480 869 572 892 061 114 544 048 887 054 729 216;
  • 32) 0.007 806 103 060 479 999 480 869 572 892 061 114 544 048 887 054 729 216 × 2 = 0 + 0.015 612 206 120 959 998 961 739 145 784 122 229 088 097 774 109 458 432;
  • 33) 0.015 612 206 120 959 998 961 739 145 784 122 229 088 097 774 109 458 432 × 2 = 0 + 0.031 224 412 241 919 997 923 478 291 568 244 458 176 195 548 218 916 864;
  • 34) 0.031 224 412 241 919 997 923 478 291 568 244 458 176 195 548 218 916 864 × 2 = 0 + 0.062 448 824 483 839 995 846 956 583 136 488 916 352 391 096 437 833 728;
  • 35) 0.062 448 824 483 839 995 846 956 583 136 488 916 352 391 096 437 833 728 × 2 = 0 + 0.124 897 648 967 679 991 693 913 166 272 977 832 704 782 192 875 667 456;
  • 36) 0.124 897 648 967 679 991 693 913 166 272 977 832 704 782 192 875 667 456 × 2 = 0 + 0.249 795 297 935 359 983 387 826 332 545 955 665 409 564 385 751 334 912;
  • 37) 0.249 795 297 935 359 983 387 826 332 545 955 665 409 564 385 751 334 912 × 2 = 0 + 0.499 590 595 870 719 966 775 652 665 091 911 330 819 128 771 502 669 824;
  • 38) 0.499 590 595 870 719 966 775 652 665 091 911 330 819 128 771 502 669 824 × 2 = 0 + 0.999 181 191 741 439 933 551 305 330 183 822 661 638 257 543 005 339 648;
  • 39) 0.999 181 191 741 439 933 551 305 330 183 822 661 638 257 543 005 339 648 × 2 = 1 + 0.998 362 383 482 879 867 102 610 660 367 645 323 276 515 086 010 679 296;
  • 40) 0.998 362 383 482 879 867 102 610 660 367 645 323 276 515 086 010 679 296 × 2 = 1 + 0.996 724 766 965 759 734 205 221 320 735 290 646 553 030 172 021 358 592;
  • 41) 0.996 724 766 965 759 734 205 221 320 735 290 646 553 030 172 021 358 592 × 2 = 1 + 0.993 449 533 931 519 468 410 442 641 470 581 293 106 060 344 042 717 184;
  • 42) 0.993 449 533 931 519 468 410 442 641 470 581 293 106 060 344 042 717 184 × 2 = 1 + 0.986 899 067 863 038 936 820 885 282 941 162 586 212 120 688 085 434 368;
  • 43) 0.986 899 067 863 038 936 820 885 282 941 162 586 212 120 688 085 434 368 × 2 = 1 + 0.973 798 135 726 077 873 641 770 565 882 325 172 424 241 376 170 868 736;
  • 44) 0.973 798 135 726 077 873 641 770 565 882 325 172 424 241 376 170 868 736 × 2 = 1 + 0.947 596 271 452 155 747 283 541 131 764 650 344 848 482 752 341 737 472;
  • 45) 0.947 596 271 452 155 747 283 541 131 764 650 344 848 482 752 341 737 472 × 2 = 1 + 0.895 192 542 904 311 494 567 082 263 529 300 689 696 965 504 683 474 944;
  • 46) 0.895 192 542 904 311 494 567 082 263 529 300 689 696 965 504 683 474 944 × 2 = 1 + 0.790 385 085 808 622 989 134 164 527 058 601 379 393 931 009 366 949 888;
  • 47) 0.790 385 085 808 622 989 134 164 527 058 601 379 393 931 009 366 949 888 × 2 = 1 + 0.580 770 171 617 245 978 268 329 054 117 202 758 787 862 018 733 899 776;
  • 48) 0.580 770 171 617 245 978 268 329 054 117 202 758 787 862 018 733 899 776 × 2 = 1 + 0.161 540 343 234 491 956 536 658 108 234 405 517 575 724 037 467 799 552;
  • 49) 0.161 540 343 234 491 956 536 658 108 234 405 517 575 724 037 467 799 552 × 2 = 0 + 0.323 080 686 468 983 913 073 316 216 468 811 035 151 448 074 935 599 104;
  • 50) 0.323 080 686 468 983 913 073 316 216 468 811 035 151 448 074 935 599 104 × 2 = 0 + 0.646 161 372 937 967 826 146 632 432 937 622 070 302 896 149 871 198 208;
  • 51) 0.646 161 372 937 967 826 146 632 432 937 622 070 302 896 149 871 198 208 × 2 = 1 + 0.292 322 745 875 935 652 293 264 865 875 244 140 605 792 299 742 396 416;
  • 52) 0.292 322 745 875 935 652 293 264 865 875 244 140 605 792 299 742 396 416 × 2 = 0 + 0.584 645 491 751 871 304 586 529 731 750 488 281 211 584 599 484 792 832;
  • 53) 0.584 645 491 751 871 304 586 529 731 750 488 281 211 584 599 484 792 832 × 2 = 1 + 0.169 290 983 503 742 609 173 059 463 500 976 562 423 169 198 969 585 664;
  • 54) 0.169 290 983 503 742 609 173 059 463 500 976 562 423 169 198 969 585 664 × 2 = 0 + 0.338 581 967 007 485 218 346 118 927 001 953 124 846 338 397 939 171 328;
  • 55) 0.338 581 967 007 485 218 346 118 927 001 953 124 846 338 397 939 171 328 × 2 = 0 + 0.677 163 934 014 970 436 692 237 854 003 906 249 692 676 795 878 342 656;
  • 56) 0.677 163 934 014 970 436 692 237 854 003 906 249 692 676 795 878 342 656 × 2 = 1 + 0.354 327 868 029 940 873 384 475 708 007 812 499 385 353 591 756 685 312;
  • 57) 0.354 327 868 029 940 873 384 475 708 007 812 499 385 353 591 756 685 312 × 2 = 0 + 0.708 655 736 059 881 746 768 951 416 015 624 998 770 707 183 513 370 624;
  • 58) 0.708 655 736 059 881 746 768 951 416 015 624 998 770 707 183 513 370 624 × 2 = 1 + 0.417 311 472 119 763 493 537 902 832 031 249 997 541 414 367 026 741 248;
  • 59) 0.417 311 472 119 763 493 537 902 832 031 249 997 541 414 367 026 741 248 × 2 = 0 + 0.834 622 944 239 526 987 075 805 664 062 499 995 082 828 734 053 482 496;
  • 60) 0.834 622 944 239 526 987 075 805 664 062 499 995 082 828 734 053 482 496 × 2 = 1 + 0.669 245 888 479 053 974 151 611 328 124 999 990 165 657 468 106 964 992;
  • 61) 0.669 245 888 479 053 974 151 611 328 124 999 990 165 657 468 106 964 992 × 2 = 1 + 0.338 491 776 958 107 948 303 222 656 249 999 980 331 314 936 213 929 984;
  • 62) 0.338 491 776 958 107 948 303 222 656 249 999 980 331 314 936 213 929 984 × 2 = 0 + 0.676 983 553 916 215 896 606 445 312 499 999 960 662 629 872 427 859 968;
  • 63) 0.676 983 553 916 215 896 606 445 312 499 999 960 662 629 872 427 859 968 × 2 = 1 + 0.353 967 107 832 431 793 212 890 624 999 999 921 325 259 744 855 719 936;
  • 64) 0.353 967 107 832 431 793 212 890 624 999 999 921 325 259 744 855 719 936 × 2 = 0 + 0.707 934 215 664 863 586 425 781 249 999 999 842 650 519 489 711 439 872;
  • 65) 0.707 934 215 664 863 586 425 781 249 999 999 842 650 519 489 711 439 872 × 2 = 1 + 0.415 868 431 329 727 172 851 562 499 999 999 685 301 038 979 422 879 744;
  • 66) 0.415 868 431 329 727 172 851 562 499 999 999 685 301 038 979 422 879 744 × 2 = 0 + 0.831 736 862 659 454 345 703 124 999 999 999 370 602 077 958 845 759 488;
  • 67) 0.831 736 862 659 454 345 703 124 999 999 999 370 602 077 958 845 759 488 × 2 = 1 + 0.663 473 725 318 908 691 406 249 999 999 998 741 204 155 917 691 518 976;
  • 68) 0.663 473 725 318 908 691 406 249 999 999 998 741 204 155 917 691 518 976 × 2 = 1 + 0.326 947 450 637 817 382 812 499 999 999 997 482 408 311 835 383 037 952;
  • 69) 0.326 947 450 637 817 382 812 499 999 999 997 482 408 311 835 383 037 952 × 2 = 0 + 0.653 894 901 275 634 765 624 999 999 999 994 964 816 623 670 766 075 904;
  • 70) 0.653 894 901 275 634 765 624 999 999 999 994 964 816 623 670 766 075 904 × 2 = 1 + 0.307 789 802 551 269 531 249 999 999 999 989 929 633 247 341 532 151 808;
  • 71) 0.307 789 802 551 269 531 249 999 999 999 989 929 633 247 341 532 151 808 × 2 = 0 + 0.615 579 605 102 539 062 499 999 999 999 979 859 266 494 683 064 303 616;
  • 72) 0.615 579 605 102 539 062 499 999 999 999 979 859 266 494 683 064 303 616 × 2 = 1 + 0.231 159 210 205 078 124 999 999 999 999 959 718 532 989 366 128 607 232;
  • 73) 0.231 159 210 205 078 124 999 999 999 999 959 718 532 989 366 128 607 232 × 2 = 0 + 0.462 318 420 410 156 249 999 999 999 999 919 437 065 978 732 257 214 464;
  • 74) 0.462 318 420 410 156 249 999 999 999 999 919 437 065 978 732 257 214 464 × 2 = 0 + 0.924 636 840 820 312 499 999 999 999 999 838 874 131 957 464 514 428 928;
  • 75) 0.924 636 840 820 312 499 999 999 999 999 838 874 131 957 464 514 428 928 × 2 = 1 + 0.849 273 681 640 624 999 999 999 999 999 677 748 263 914 929 028 857 856;
  • 76) 0.849 273 681 640 624 999 999 999 999 999 677 748 263 914 929 028 857 856 × 2 = 1 + 0.698 547 363 281 249 999 999 999 999 999 355 496 527 829 858 057 715 712;
  • 77) 0.698 547 363 281 249 999 999 999 999 999 355 496 527 829 858 057 715 712 × 2 = 1 + 0.397 094 726 562 499 999 999 999 999 998 710 993 055 659 716 115 431 424;
  • 78) 0.397 094 726 562 499 999 999 999 999 998 710 993 055 659 716 115 431 424 × 2 = 0 + 0.794 189 453 124 999 999 999 999 999 997 421 986 111 319 432 230 862 848;
  • 79) 0.794 189 453 124 999 999 999 999 999 997 421 986 111 319 432 230 862 848 × 2 = 1 + 0.588 378 906 249 999 999 999 999 999 994 843 972 222 638 864 461 725 696;
  • 80) 0.588 378 906 249 999 999 999 999 999 994 843 972 222 638 864 461 725 696 × 2 = 1 + 0.176 757 812 499 999 999 999 999 999 989 687 944 445 277 728 923 451 392;
  • 81) 0.176 757 812 499 999 999 999 999 999 989 687 944 445 277 728 923 451 392 × 2 = 0 + 0.353 515 624 999 999 999 999 999 999 979 375 888 890 555 457 846 902 784;
  • 82) 0.353 515 624 999 999 999 999 999 999 979 375 888 890 555 457 846 902 784 × 2 = 0 + 0.707 031 249 999 999 999 999 999 999 958 751 777 781 110 915 693 805 568;
  • 83) 0.707 031 249 999 999 999 999 999 999 958 751 777 781 110 915 693 805 568 × 2 = 1 + 0.414 062 499 999 999 999 999 999 999 917 503 555 562 221 831 387 611 136;
  • 84) 0.414 062 499 999 999 999 999 999 999 917 503 555 562 221 831 387 611 136 × 2 = 0 + 0.828 124 999 999 999 999 999 999 999 835 007 111 124 443 662 775 222 272;
  • 85) 0.828 124 999 999 999 999 999 999 999 835 007 111 124 443 662 775 222 272 × 2 = 1 + 0.656 249 999 999 999 999 999 999 999 670 014 222 248 887 325 550 444 544;
  • 86) 0.656 249 999 999 999 999 999 999 999 670 014 222 248 887 325 550 444 544 × 2 = 1 + 0.312 499 999 999 999 999 999 999 999 340 028 444 497 774 651 100 889 088;
  • 87) 0.312 499 999 999 999 999 999 999 999 340 028 444 497 774 651 100 889 088 × 2 = 0 + 0.624 999 999 999 999 999 999 999 998 680 056 888 995 549 302 201 778 176;
  • 88) 0.624 999 999 999 999 999 999 999 998 680 056 888 995 549 302 201 778 176 × 2 = 1 + 0.249 999 999 999 999 999 999 999 997 360 113 777 991 098 604 403 556 352;
  • 89) 0.249 999 999 999 999 999 999 999 997 360 113 777 991 098 604 403 556 352 × 2 = 0 + 0.499 999 999 999 999 999 999 999 994 720 227 555 982 197 208 807 112 704;
  • 90) 0.499 999 999 999 999 999 999 999 994 720 227 555 982 197 208 807 112 704 × 2 = 0 + 0.999 999 999 999 999 999 999 999 989 440 455 111 964 394 417 614 225 408;
  • 91) 0.999 999 999 999 999 999 999 999 989 440 455 111 964 394 417 614 225 408 × 2 = 1 + 0.999 999 999 999 999 999 999 999 978 880 910 223 928 788 835 228 450 816;

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:


0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 979 5(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:

0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 979 5(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:


0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 979 5(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 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 979 5 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