0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 989 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 989(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
0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 989(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: 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 989.

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) 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 989 × 2 = 0 + 0.000 000 000 007 269 999 999 999 999 516 522 114 064 601 356 670 305 978;
  • 2) 0.000 000 000 007 269 999 999 999 999 516 522 114 064 601 356 670 305 978 × 2 = 0 + 0.000 000 000 014 539 999 999 999 999 033 044 228 129 202 713 340 611 956;
  • 3) 0.000 000 000 014 539 999 999 999 999 033 044 228 129 202 713 340 611 956 × 2 = 0 + 0.000 000 000 029 079 999 999 999 998 066 088 456 258 405 426 681 223 912;
  • 4) 0.000 000 000 029 079 999 999 999 998 066 088 456 258 405 426 681 223 912 × 2 = 0 + 0.000 000 000 058 159 999 999 999 996 132 176 912 516 810 853 362 447 824;
  • 5) 0.000 000 000 058 159 999 999 999 996 132 176 912 516 810 853 362 447 824 × 2 = 0 + 0.000 000 000 116 319 999 999 999 992 264 353 825 033 621 706 724 895 648;
  • 6) 0.000 000 000 116 319 999 999 999 992 264 353 825 033 621 706 724 895 648 × 2 = 0 + 0.000 000 000 232 639 999 999 999 984 528 707 650 067 243 413 449 791 296;
  • 7) 0.000 000 000 232 639 999 999 999 984 528 707 650 067 243 413 449 791 296 × 2 = 0 + 0.000 000 000 465 279 999 999 999 969 057 415 300 134 486 826 899 582 592;
  • 8) 0.000 000 000 465 279 999 999 999 969 057 415 300 134 486 826 899 582 592 × 2 = 0 + 0.000 000 000 930 559 999 999 999 938 114 830 600 268 973 653 799 165 184;
  • 9) 0.000 000 000 930 559 999 999 999 938 114 830 600 268 973 653 799 165 184 × 2 = 0 + 0.000 000 001 861 119 999 999 999 876 229 661 200 537 947 307 598 330 368;
  • 10) 0.000 000 001 861 119 999 999 999 876 229 661 200 537 947 307 598 330 368 × 2 = 0 + 0.000 000 003 722 239 999 999 999 752 459 322 401 075 894 615 196 660 736;
  • 11) 0.000 000 003 722 239 999 999 999 752 459 322 401 075 894 615 196 660 736 × 2 = 0 + 0.000 000 007 444 479 999 999 999 504 918 644 802 151 789 230 393 321 472;
  • 12) 0.000 000 007 444 479 999 999 999 504 918 644 802 151 789 230 393 321 472 × 2 = 0 + 0.000 000 014 888 959 999 999 999 009 837 289 604 303 578 460 786 642 944;
  • 13) 0.000 000 014 888 959 999 999 999 009 837 289 604 303 578 460 786 642 944 × 2 = 0 + 0.000 000 029 777 919 999 999 998 019 674 579 208 607 156 921 573 285 888;
  • 14) 0.000 000 029 777 919 999 999 998 019 674 579 208 607 156 921 573 285 888 × 2 = 0 + 0.000 000 059 555 839 999 999 996 039 349 158 417 214 313 843 146 571 776;
  • 15) 0.000 000 059 555 839 999 999 996 039 349 158 417 214 313 843 146 571 776 × 2 = 0 + 0.000 000 119 111 679 999 999 992 078 698 316 834 428 627 686 293 143 552;
  • 16) 0.000 000 119 111 679 999 999 992 078 698 316 834 428 627 686 293 143 552 × 2 = 0 + 0.000 000 238 223 359 999 999 984 157 396 633 668 857 255 372 586 287 104;
  • 17) 0.000 000 238 223 359 999 999 984 157 396 633 668 857 255 372 586 287 104 × 2 = 0 + 0.000 000 476 446 719 999 999 968 314 793 267 337 714 510 745 172 574 208;
  • 18) 0.000 000 476 446 719 999 999 968 314 793 267 337 714 510 745 172 574 208 × 2 = 0 + 0.000 000 952 893 439 999 999 936 629 586 534 675 429 021 490 345 148 416;
  • 19) 0.000 000 952 893 439 999 999 936 629 586 534 675 429 021 490 345 148 416 × 2 = 0 + 0.000 001 905 786 879 999 999 873 259 173 069 350 858 042 980 690 296 832;
  • 20) 0.000 001 905 786 879 999 999 873 259 173 069 350 858 042 980 690 296 832 × 2 = 0 + 0.000 003 811 573 759 999 999 746 518 346 138 701 716 085 961 380 593 664;
  • 21) 0.000 003 811 573 759 999 999 746 518 346 138 701 716 085 961 380 593 664 × 2 = 0 + 0.000 007 623 147 519 999 999 493 036 692 277 403 432 171 922 761 187 328;
  • 22) 0.000 007 623 147 519 999 999 493 036 692 277 403 432 171 922 761 187 328 × 2 = 0 + 0.000 015 246 295 039 999 998 986 073 384 554 806 864 343 845 522 374 656;
  • 23) 0.000 015 246 295 039 999 998 986 073 384 554 806 864 343 845 522 374 656 × 2 = 0 + 0.000 030 492 590 079 999 997 972 146 769 109 613 728 687 691 044 749 312;
  • 24) 0.000 030 492 590 079 999 997 972 146 769 109 613 728 687 691 044 749 312 × 2 = 0 + 0.000 060 985 180 159 999 995 944 293 538 219 227 457 375 382 089 498 624;
  • 25) 0.000 060 985 180 159 999 995 944 293 538 219 227 457 375 382 089 498 624 × 2 = 0 + 0.000 121 970 360 319 999 991 888 587 076 438 454 914 750 764 178 997 248;
  • 26) 0.000 121 970 360 319 999 991 888 587 076 438 454 914 750 764 178 997 248 × 2 = 0 + 0.000 243 940 720 639 999 983 777 174 152 876 909 829 501 528 357 994 496;
  • 27) 0.000 243 940 720 639 999 983 777 174 152 876 909 829 501 528 357 994 496 × 2 = 0 + 0.000 487 881 441 279 999 967 554 348 305 753 819 659 003 056 715 988 992;
  • 28) 0.000 487 881 441 279 999 967 554 348 305 753 819 659 003 056 715 988 992 × 2 = 0 + 0.000 975 762 882 559 999 935 108 696 611 507 639 318 006 113 431 977 984;
  • 29) 0.000 975 762 882 559 999 935 108 696 611 507 639 318 006 113 431 977 984 × 2 = 0 + 0.001 951 525 765 119 999 870 217 393 223 015 278 636 012 226 863 955 968;
  • 30) 0.001 951 525 765 119 999 870 217 393 223 015 278 636 012 226 863 955 968 × 2 = 0 + 0.003 903 051 530 239 999 740 434 786 446 030 557 272 024 453 727 911 936;
  • 31) 0.003 903 051 530 239 999 740 434 786 446 030 557 272 024 453 727 911 936 × 2 = 0 + 0.007 806 103 060 479 999 480 869 572 892 061 114 544 048 907 455 823 872;
  • 32) 0.007 806 103 060 479 999 480 869 572 892 061 114 544 048 907 455 823 872 × 2 = 0 + 0.015 612 206 120 959 998 961 739 145 784 122 229 088 097 814 911 647 744;
  • 33) 0.015 612 206 120 959 998 961 739 145 784 122 229 088 097 814 911 647 744 × 2 = 0 + 0.031 224 412 241 919 997 923 478 291 568 244 458 176 195 629 823 295 488;
  • 34) 0.031 224 412 241 919 997 923 478 291 568 244 458 176 195 629 823 295 488 × 2 = 0 + 0.062 448 824 483 839 995 846 956 583 136 488 916 352 391 259 646 590 976;
  • 35) 0.062 448 824 483 839 995 846 956 583 136 488 916 352 391 259 646 590 976 × 2 = 0 + 0.124 897 648 967 679 991 693 913 166 272 977 832 704 782 519 293 181 952;
  • 36) 0.124 897 648 967 679 991 693 913 166 272 977 832 704 782 519 293 181 952 × 2 = 0 + 0.249 795 297 935 359 983 387 826 332 545 955 665 409 565 038 586 363 904;
  • 37) 0.249 795 297 935 359 983 387 826 332 545 955 665 409 565 038 586 363 904 × 2 = 0 + 0.499 590 595 870 719 966 775 652 665 091 911 330 819 130 077 172 727 808;
  • 38) 0.499 590 595 870 719 966 775 652 665 091 911 330 819 130 077 172 727 808 × 2 = 0 + 0.999 181 191 741 439 933 551 305 330 183 822 661 638 260 154 345 455 616;
  • 39) 0.999 181 191 741 439 933 551 305 330 183 822 661 638 260 154 345 455 616 × 2 = 1 + 0.998 362 383 482 879 867 102 610 660 367 645 323 276 520 308 690 911 232;
  • 40) 0.998 362 383 482 879 867 102 610 660 367 645 323 276 520 308 690 911 232 × 2 = 1 + 0.996 724 766 965 759 734 205 221 320 735 290 646 553 040 617 381 822 464;
  • 41) 0.996 724 766 965 759 734 205 221 320 735 290 646 553 040 617 381 822 464 × 2 = 1 + 0.993 449 533 931 519 468 410 442 641 470 581 293 106 081 234 763 644 928;
  • 42) 0.993 449 533 931 519 468 410 442 641 470 581 293 106 081 234 763 644 928 × 2 = 1 + 0.986 899 067 863 038 936 820 885 282 941 162 586 212 162 469 527 289 856;
  • 43) 0.986 899 067 863 038 936 820 885 282 941 162 586 212 162 469 527 289 856 × 2 = 1 + 0.973 798 135 726 077 873 641 770 565 882 325 172 424 324 939 054 579 712;
  • 44) 0.973 798 135 726 077 873 641 770 565 882 325 172 424 324 939 054 579 712 × 2 = 1 + 0.947 596 271 452 155 747 283 541 131 764 650 344 848 649 878 109 159 424;
  • 45) 0.947 596 271 452 155 747 283 541 131 764 650 344 848 649 878 109 159 424 × 2 = 1 + 0.895 192 542 904 311 494 567 082 263 529 300 689 697 299 756 218 318 848;
  • 46) 0.895 192 542 904 311 494 567 082 263 529 300 689 697 299 756 218 318 848 × 2 = 1 + 0.790 385 085 808 622 989 134 164 527 058 601 379 394 599 512 436 637 696;
  • 47) 0.790 385 085 808 622 989 134 164 527 058 601 379 394 599 512 436 637 696 × 2 = 1 + 0.580 770 171 617 245 978 268 329 054 117 202 758 789 199 024 873 275 392;
  • 48) 0.580 770 171 617 245 978 268 329 054 117 202 758 789 199 024 873 275 392 × 2 = 1 + 0.161 540 343 234 491 956 536 658 108 234 405 517 578 398 049 746 550 784;
  • 49) 0.161 540 343 234 491 956 536 658 108 234 405 517 578 398 049 746 550 784 × 2 = 0 + 0.323 080 686 468 983 913 073 316 216 468 811 035 156 796 099 493 101 568;
  • 50) 0.323 080 686 468 983 913 073 316 216 468 811 035 156 796 099 493 101 568 × 2 = 0 + 0.646 161 372 937 967 826 146 632 432 937 622 070 313 592 198 986 203 136;
  • 51) 0.646 161 372 937 967 826 146 632 432 937 622 070 313 592 198 986 203 136 × 2 = 1 + 0.292 322 745 875 935 652 293 264 865 875 244 140 627 184 397 972 406 272;
  • 52) 0.292 322 745 875 935 652 293 264 865 875 244 140 627 184 397 972 406 272 × 2 = 0 + 0.584 645 491 751 871 304 586 529 731 750 488 281 254 368 795 944 812 544;
  • 53) 0.584 645 491 751 871 304 586 529 731 750 488 281 254 368 795 944 812 544 × 2 = 1 + 0.169 290 983 503 742 609 173 059 463 500 976 562 508 737 591 889 625 088;
  • 54) 0.169 290 983 503 742 609 173 059 463 500 976 562 508 737 591 889 625 088 × 2 = 0 + 0.338 581 967 007 485 218 346 118 927 001 953 125 017 475 183 779 250 176;
  • 55) 0.338 581 967 007 485 218 346 118 927 001 953 125 017 475 183 779 250 176 × 2 = 0 + 0.677 163 934 014 970 436 692 237 854 003 906 250 034 950 367 558 500 352;
  • 56) 0.677 163 934 014 970 436 692 237 854 003 906 250 034 950 367 558 500 352 × 2 = 1 + 0.354 327 868 029 940 873 384 475 708 007 812 500 069 900 735 117 000 704;
  • 57) 0.354 327 868 029 940 873 384 475 708 007 812 500 069 900 735 117 000 704 × 2 = 0 + 0.708 655 736 059 881 746 768 951 416 015 625 000 139 801 470 234 001 408;
  • 58) 0.708 655 736 059 881 746 768 951 416 015 625 000 139 801 470 234 001 408 × 2 = 1 + 0.417 311 472 119 763 493 537 902 832 031 250 000 279 602 940 468 002 816;
  • 59) 0.417 311 472 119 763 493 537 902 832 031 250 000 279 602 940 468 002 816 × 2 = 0 + 0.834 622 944 239 526 987 075 805 664 062 500 000 559 205 880 936 005 632;
  • 60) 0.834 622 944 239 526 987 075 805 664 062 500 000 559 205 880 936 005 632 × 2 = 1 + 0.669 245 888 479 053 974 151 611 328 125 000 001 118 411 761 872 011 264;
  • 61) 0.669 245 888 479 053 974 151 611 328 125 000 001 118 411 761 872 011 264 × 2 = 1 + 0.338 491 776 958 107 948 303 222 656 250 000 002 236 823 523 744 022 528;
  • 62) 0.338 491 776 958 107 948 303 222 656 250 000 002 236 823 523 744 022 528 × 2 = 0 + 0.676 983 553 916 215 896 606 445 312 500 000 004 473 647 047 488 045 056;
  • 63) 0.676 983 553 916 215 896 606 445 312 500 000 004 473 647 047 488 045 056 × 2 = 1 + 0.353 967 107 832 431 793 212 890 625 000 000 008 947 294 094 976 090 112;
  • 64) 0.353 967 107 832 431 793 212 890 625 000 000 008 947 294 094 976 090 112 × 2 = 0 + 0.707 934 215 664 863 586 425 781 250 000 000 017 894 588 189 952 180 224;
  • 65) 0.707 934 215 664 863 586 425 781 250 000 000 017 894 588 189 952 180 224 × 2 = 1 + 0.415 868 431 329 727 172 851 562 500 000 000 035 789 176 379 904 360 448;
  • 66) 0.415 868 431 329 727 172 851 562 500 000 000 035 789 176 379 904 360 448 × 2 = 0 + 0.831 736 862 659 454 345 703 125 000 000 000 071 578 352 759 808 720 896;
  • 67) 0.831 736 862 659 454 345 703 125 000 000 000 071 578 352 759 808 720 896 × 2 = 1 + 0.663 473 725 318 908 691 406 250 000 000 000 143 156 705 519 617 441 792;
  • 68) 0.663 473 725 318 908 691 406 250 000 000 000 143 156 705 519 617 441 792 × 2 = 1 + 0.326 947 450 637 817 382 812 500 000 000 000 286 313 411 039 234 883 584;
  • 69) 0.326 947 450 637 817 382 812 500 000 000 000 286 313 411 039 234 883 584 × 2 = 0 + 0.653 894 901 275 634 765 625 000 000 000 000 572 626 822 078 469 767 168;
  • 70) 0.653 894 901 275 634 765 625 000 000 000 000 572 626 822 078 469 767 168 × 2 = 1 + 0.307 789 802 551 269 531 250 000 000 000 001 145 253 644 156 939 534 336;
  • 71) 0.307 789 802 551 269 531 250 000 000 000 001 145 253 644 156 939 534 336 × 2 = 0 + 0.615 579 605 102 539 062 500 000 000 000 002 290 507 288 313 879 068 672;
  • 72) 0.615 579 605 102 539 062 500 000 000 000 002 290 507 288 313 879 068 672 × 2 = 1 + 0.231 159 210 205 078 125 000 000 000 000 004 581 014 576 627 758 137 344;
  • 73) 0.231 159 210 205 078 125 000 000 000 000 004 581 014 576 627 758 137 344 × 2 = 0 + 0.462 318 420 410 156 250 000 000 000 000 009 162 029 153 255 516 274 688;
  • 74) 0.462 318 420 410 156 250 000 000 000 000 009 162 029 153 255 516 274 688 × 2 = 0 + 0.924 636 840 820 312 500 000 000 000 000 018 324 058 306 511 032 549 376;
  • 75) 0.924 636 840 820 312 500 000 000 000 000 018 324 058 306 511 032 549 376 × 2 = 1 + 0.849 273 681 640 625 000 000 000 000 000 036 648 116 613 022 065 098 752;
  • 76) 0.849 273 681 640 625 000 000 000 000 000 036 648 116 613 022 065 098 752 × 2 = 1 + 0.698 547 363 281 250 000 000 000 000 000 073 296 233 226 044 130 197 504;
  • 77) 0.698 547 363 281 250 000 000 000 000 000 073 296 233 226 044 130 197 504 × 2 = 1 + 0.397 094 726 562 500 000 000 000 000 000 146 592 466 452 088 260 395 008;
  • 78) 0.397 094 726 562 500 000 000 000 000 000 146 592 466 452 088 260 395 008 × 2 = 0 + 0.794 189 453 125 000 000 000 000 000 000 293 184 932 904 176 520 790 016;
  • 79) 0.794 189 453 125 000 000 000 000 000 000 293 184 932 904 176 520 790 016 × 2 = 1 + 0.588 378 906 250 000 000 000 000 000 000 586 369 865 808 353 041 580 032;
  • 80) 0.588 378 906 250 000 000 000 000 000 000 586 369 865 808 353 041 580 032 × 2 = 1 + 0.176 757 812 500 000 000 000 000 000 001 172 739 731 616 706 083 160 064;
  • 81) 0.176 757 812 500 000 000 000 000 000 001 172 739 731 616 706 083 160 064 × 2 = 0 + 0.353 515 625 000 000 000 000 000 000 002 345 479 463 233 412 166 320 128;
  • 82) 0.353 515 625 000 000 000 000 000 000 002 345 479 463 233 412 166 320 128 × 2 = 0 + 0.707 031 250 000 000 000 000 000 000 004 690 958 926 466 824 332 640 256;
  • 83) 0.707 031 250 000 000 000 000 000 000 004 690 958 926 466 824 332 640 256 × 2 = 1 + 0.414 062 500 000 000 000 000 000 000 009 381 917 852 933 648 665 280 512;
  • 84) 0.414 062 500 000 000 000 000 000 000 009 381 917 852 933 648 665 280 512 × 2 = 0 + 0.828 125 000 000 000 000 000 000 000 018 763 835 705 867 297 330 561 024;
  • 85) 0.828 125 000 000 000 000 000 000 000 018 763 835 705 867 297 330 561 024 × 2 = 1 + 0.656 250 000 000 000 000 000 000 000 037 527 671 411 734 594 661 122 048;
  • 86) 0.656 250 000 000 000 000 000 000 000 037 527 671 411 734 594 661 122 048 × 2 = 1 + 0.312 500 000 000 000 000 000 000 000 075 055 342 823 469 189 322 244 096;
  • 87) 0.312 500 000 000 000 000 000 000 000 075 055 342 823 469 189 322 244 096 × 2 = 0 + 0.625 000 000 000 000 000 000 000 000 150 110 685 646 938 378 644 488 192;
  • 88) 0.625 000 000 000 000 000 000 000 000 150 110 685 646 938 378 644 488 192 × 2 = 1 + 0.250 000 000 000 000 000 000 000 000 300 221 371 293 876 757 288 976 384;
  • 89) 0.250 000 000 000 000 000 000 000 000 300 221 371 293 876 757 288 976 384 × 2 = 0 + 0.500 000 000 000 000 000 000 000 000 600 442 742 587 753 514 577 952 768;
  • 90) 0.500 000 000 000 000 000 000 000 000 600 442 742 587 753 514 577 952 768 × 2 = 1 + 0.000 000 000 000 000 000 000 000 001 200 885 485 175 507 029 155 905 536;
  • 91) 0.000 000 000 000 000 000 000 000 001 200 885 485 175 507 029 155 905 536 × 2 = 0 + 0.000 000 000 000 000 000 000 000 002 401 770 970 351 014 058 311 811 072;

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:


0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 989(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:

0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 989(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:


0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 989(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 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 989 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