0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 031 76 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 031 76(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
0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 031 76(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: 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 031 76.

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) 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 031 76 × 2 = 0 + 0.000 000 000 007 269 999 999 999 999 516 522 114 064 601 356 670 305 976 063 52;
  • 2) 0.000 000 000 007 269 999 999 999 999 516 522 114 064 601 356 670 305 976 063 52 × 2 = 0 + 0.000 000 000 014 539 999 999 999 999 033 044 228 129 202 713 340 611 952 127 04;
  • 3) 0.000 000 000 014 539 999 999 999 999 033 044 228 129 202 713 340 611 952 127 04 × 2 = 0 + 0.000 000 000 029 079 999 999 999 998 066 088 456 258 405 426 681 223 904 254 08;
  • 4) 0.000 000 000 029 079 999 999 999 998 066 088 456 258 405 426 681 223 904 254 08 × 2 = 0 + 0.000 000 000 058 159 999 999 999 996 132 176 912 516 810 853 362 447 808 508 16;
  • 5) 0.000 000 000 058 159 999 999 999 996 132 176 912 516 810 853 362 447 808 508 16 × 2 = 0 + 0.000 000 000 116 319 999 999 999 992 264 353 825 033 621 706 724 895 617 016 32;
  • 6) 0.000 000 000 116 319 999 999 999 992 264 353 825 033 621 706 724 895 617 016 32 × 2 = 0 + 0.000 000 000 232 639 999 999 999 984 528 707 650 067 243 413 449 791 234 032 64;
  • 7) 0.000 000 000 232 639 999 999 999 984 528 707 650 067 243 413 449 791 234 032 64 × 2 = 0 + 0.000 000 000 465 279 999 999 999 969 057 415 300 134 486 826 899 582 468 065 28;
  • 8) 0.000 000 000 465 279 999 999 999 969 057 415 300 134 486 826 899 582 468 065 28 × 2 = 0 + 0.000 000 000 930 559 999 999 999 938 114 830 600 268 973 653 799 164 936 130 56;
  • 9) 0.000 000 000 930 559 999 999 999 938 114 830 600 268 973 653 799 164 936 130 56 × 2 = 0 + 0.000 000 001 861 119 999 999 999 876 229 661 200 537 947 307 598 329 872 261 12;
  • 10) 0.000 000 001 861 119 999 999 999 876 229 661 200 537 947 307 598 329 872 261 12 × 2 = 0 + 0.000 000 003 722 239 999 999 999 752 459 322 401 075 894 615 196 659 744 522 24;
  • 11) 0.000 000 003 722 239 999 999 999 752 459 322 401 075 894 615 196 659 744 522 24 × 2 = 0 + 0.000 000 007 444 479 999 999 999 504 918 644 802 151 789 230 393 319 489 044 48;
  • 12) 0.000 000 007 444 479 999 999 999 504 918 644 802 151 789 230 393 319 489 044 48 × 2 = 0 + 0.000 000 014 888 959 999 999 999 009 837 289 604 303 578 460 786 638 978 088 96;
  • 13) 0.000 000 014 888 959 999 999 999 009 837 289 604 303 578 460 786 638 978 088 96 × 2 = 0 + 0.000 000 029 777 919 999 999 998 019 674 579 208 607 156 921 573 277 956 177 92;
  • 14) 0.000 000 029 777 919 999 999 998 019 674 579 208 607 156 921 573 277 956 177 92 × 2 = 0 + 0.000 000 059 555 839 999 999 996 039 349 158 417 214 313 843 146 555 912 355 84;
  • 15) 0.000 000 059 555 839 999 999 996 039 349 158 417 214 313 843 146 555 912 355 84 × 2 = 0 + 0.000 000 119 111 679 999 999 992 078 698 316 834 428 627 686 293 111 824 711 68;
  • 16) 0.000 000 119 111 679 999 999 992 078 698 316 834 428 627 686 293 111 824 711 68 × 2 = 0 + 0.000 000 238 223 359 999 999 984 157 396 633 668 857 255 372 586 223 649 423 36;
  • 17) 0.000 000 238 223 359 999 999 984 157 396 633 668 857 255 372 586 223 649 423 36 × 2 = 0 + 0.000 000 476 446 719 999 999 968 314 793 267 337 714 510 745 172 447 298 846 72;
  • 18) 0.000 000 476 446 719 999 999 968 314 793 267 337 714 510 745 172 447 298 846 72 × 2 = 0 + 0.000 000 952 893 439 999 999 936 629 586 534 675 429 021 490 344 894 597 693 44;
  • 19) 0.000 000 952 893 439 999 999 936 629 586 534 675 429 021 490 344 894 597 693 44 × 2 = 0 + 0.000 001 905 786 879 999 999 873 259 173 069 350 858 042 980 689 789 195 386 88;
  • 20) 0.000 001 905 786 879 999 999 873 259 173 069 350 858 042 980 689 789 195 386 88 × 2 = 0 + 0.000 003 811 573 759 999 999 746 518 346 138 701 716 085 961 379 578 390 773 76;
  • 21) 0.000 003 811 573 759 999 999 746 518 346 138 701 716 085 961 379 578 390 773 76 × 2 = 0 + 0.000 007 623 147 519 999 999 493 036 692 277 403 432 171 922 759 156 781 547 52;
  • 22) 0.000 007 623 147 519 999 999 493 036 692 277 403 432 171 922 759 156 781 547 52 × 2 = 0 + 0.000 015 246 295 039 999 998 986 073 384 554 806 864 343 845 518 313 563 095 04;
  • 23) 0.000 015 246 295 039 999 998 986 073 384 554 806 864 343 845 518 313 563 095 04 × 2 = 0 + 0.000 030 492 590 079 999 997 972 146 769 109 613 728 687 691 036 627 126 190 08;
  • 24) 0.000 030 492 590 079 999 997 972 146 769 109 613 728 687 691 036 627 126 190 08 × 2 = 0 + 0.000 060 985 180 159 999 995 944 293 538 219 227 457 375 382 073 254 252 380 16;
  • 25) 0.000 060 985 180 159 999 995 944 293 538 219 227 457 375 382 073 254 252 380 16 × 2 = 0 + 0.000 121 970 360 319 999 991 888 587 076 438 454 914 750 764 146 508 504 760 32;
  • 26) 0.000 121 970 360 319 999 991 888 587 076 438 454 914 750 764 146 508 504 760 32 × 2 = 0 + 0.000 243 940 720 639 999 983 777 174 152 876 909 829 501 528 293 017 009 520 64;
  • 27) 0.000 243 940 720 639 999 983 777 174 152 876 909 829 501 528 293 017 009 520 64 × 2 = 0 + 0.000 487 881 441 279 999 967 554 348 305 753 819 659 003 056 586 034 019 041 28;
  • 28) 0.000 487 881 441 279 999 967 554 348 305 753 819 659 003 056 586 034 019 041 28 × 2 = 0 + 0.000 975 762 882 559 999 935 108 696 611 507 639 318 006 113 172 068 038 082 56;
  • 29) 0.000 975 762 882 559 999 935 108 696 611 507 639 318 006 113 172 068 038 082 56 × 2 = 0 + 0.001 951 525 765 119 999 870 217 393 223 015 278 636 012 226 344 136 076 165 12;
  • 30) 0.001 951 525 765 119 999 870 217 393 223 015 278 636 012 226 344 136 076 165 12 × 2 = 0 + 0.003 903 051 530 239 999 740 434 786 446 030 557 272 024 452 688 272 152 330 24;
  • 31) 0.003 903 051 530 239 999 740 434 786 446 030 557 272 024 452 688 272 152 330 24 × 2 = 0 + 0.007 806 103 060 479 999 480 869 572 892 061 114 544 048 905 376 544 304 660 48;
  • 32) 0.007 806 103 060 479 999 480 869 572 892 061 114 544 048 905 376 544 304 660 48 × 2 = 0 + 0.015 612 206 120 959 998 961 739 145 784 122 229 088 097 810 753 088 609 320 96;
  • 33) 0.015 612 206 120 959 998 961 739 145 784 122 229 088 097 810 753 088 609 320 96 × 2 = 0 + 0.031 224 412 241 919 997 923 478 291 568 244 458 176 195 621 506 177 218 641 92;
  • 34) 0.031 224 412 241 919 997 923 478 291 568 244 458 176 195 621 506 177 218 641 92 × 2 = 0 + 0.062 448 824 483 839 995 846 956 583 136 488 916 352 391 243 012 354 437 283 84;
  • 35) 0.062 448 824 483 839 995 846 956 583 136 488 916 352 391 243 012 354 437 283 84 × 2 = 0 + 0.124 897 648 967 679 991 693 913 166 272 977 832 704 782 486 024 708 874 567 68;
  • 36) 0.124 897 648 967 679 991 693 913 166 272 977 832 704 782 486 024 708 874 567 68 × 2 = 0 + 0.249 795 297 935 359 983 387 826 332 545 955 665 409 564 972 049 417 749 135 36;
  • 37) 0.249 795 297 935 359 983 387 826 332 545 955 665 409 564 972 049 417 749 135 36 × 2 = 0 + 0.499 590 595 870 719 966 775 652 665 091 911 330 819 129 944 098 835 498 270 72;
  • 38) 0.499 590 595 870 719 966 775 652 665 091 911 330 819 129 944 098 835 498 270 72 × 2 = 0 + 0.999 181 191 741 439 933 551 305 330 183 822 661 638 259 888 197 670 996 541 44;
  • 39) 0.999 181 191 741 439 933 551 305 330 183 822 661 638 259 888 197 670 996 541 44 × 2 = 1 + 0.998 362 383 482 879 867 102 610 660 367 645 323 276 519 776 395 341 993 082 88;
  • 40) 0.998 362 383 482 879 867 102 610 660 367 645 323 276 519 776 395 341 993 082 88 × 2 = 1 + 0.996 724 766 965 759 734 205 221 320 735 290 646 553 039 552 790 683 986 165 76;
  • 41) 0.996 724 766 965 759 734 205 221 320 735 290 646 553 039 552 790 683 986 165 76 × 2 = 1 + 0.993 449 533 931 519 468 410 442 641 470 581 293 106 079 105 581 367 972 331 52;
  • 42) 0.993 449 533 931 519 468 410 442 641 470 581 293 106 079 105 581 367 972 331 52 × 2 = 1 + 0.986 899 067 863 038 936 820 885 282 941 162 586 212 158 211 162 735 944 663 04;
  • 43) 0.986 899 067 863 038 936 820 885 282 941 162 586 212 158 211 162 735 944 663 04 × 2 = 1 + 0.973 798 135 726 077 873 641 770 565 882 325 172 424 316 422 325 471 889 326 08;
  • 44) 0.973 798 135 726 077 873 641 770 565 882 325 172 424 316 422 325 471 889 326 08 × 2 = 1 + 0.947 596 271 452 155 747 283 541 131 764 650 344 848 632 844 650 943 778 652 16;
  • 45) 0.947 596 271 452 155 747 283 541 131 764 650 344 848 632 844 650 943 778 652 16 × 2 = 1 + 0.895 192 542 904 311 494 567 082 263 529 300 689 697 265 689 301 887 557 304 32;
  • 46) 0.895 192 542 904 311 494 567 082 263 529 300 689 697 265 689 301 887 557 304 32 × 2 = 1 + 0.790 385 085 808 622 989 134 164 527 058 601 379 394 531 378 603 775 114 608 64;
  • 47) 0.790 385 085 808 622 989 134 164 527 058 601 379 394 531 378 603 775 114 608 64 × 2 = 1 + 0.580 770 171 617 245 978 268 329 054 117 202 758 789 062 757 207 550 229 217 28;
  • 48) 0.580 770 171 617 245 978 268 329 054 117 202 758 789 062 757 207 550 229 217 28 × 2 = 1 + 0.161 540 343 234 491 956 536 658 108 234 405 517 578 125 514 415 100 458 434 56;
  • 49) 0.161 540 343 234 491 956 536 658 108 234 405 517 578 125 514 415 100 458 434 56 × 2 = 0 + 0.323 080 686 468 983 913 073 316 216 468 811 035 156 251 028 830 200 916 869 12;
  • 50) 0.323 080 686 468 983 913 073 316 216 468 811 035 156 251 028 830 200 916 869 12 × 2 = 0 + 0.646 161 372 937 967 826 146 632 432 937 622 070 312 502 057 660 401 833 738 24;
  • 51) 0.646 161 372 937 967 826 146 632 432 937 622 070 312 502 057 660 401 833 738 24 × 2 = 1 + 0.292 322 745 875 935 652 293 264 865 875 244 140 625 004 115 320 803 667 476 48;
  • 52) 0.292 322 745 875 935 652 293 264 865 875 244 140 625 004 115 320 803 667 476 48 × 2 = 0 + 0.584 645 491 751 871 304 586 529 731 750 488 281 250 008 230 641 607 334 952 96;
  • 53) 0.584 645 491 751 871 304 586 529 731 750 488 281 250 008 230 641 607 334 952 96 × 2 = 1 + 0.169 290 983 503 742 609 173 059 463 500 976 562 500 016 461 283 214 669 905 92;
  • 54) 0.169 290 983 503 742 609 173 059 463 500 976 562 500 016 461 283 214 669 905 92 × 2 = 0 + 0.338 581 967 007 485 218 346 118 927 001 953 125 000 032 922 566 429 339 811 84;
  • 55) 0.338 581 967 007 485 218 346 118 927 001 953 125 000 032 922 566 429 339 811 84 × 2 = 0 + 0.677 163 934 014 970 436 692 237 854 003 906 250 000 065 845 132 858 679 623 68;
  • 56) 0.677 163 934 014 970 436 692 237 854 003 906 250 000 065 845 132 858 679 623 68 × 2 = 1 + 0.354 327 868 029 940 873 384 475 708 007 812 500 000 131 690 265 717 359 247 36;
  • 57) 0.354 327 868 029 940 873 384 475 708 007 812 500 000 131 690 265 717 359 247 36 × 2 = 0 + 0.708 655 736 059 881 746 768 951 416 015 625 000 000 263 380 531 434 718 494 72;
  • 58) 0.708 655 736 059 881 746 768 951 416 015 625 000 000 263 380 531 434 718 494 72 × 2 = 1 + 0.417 311 472 119 763 493 537 902 832 031 250 000 000 526 761 062 869 436 989 44;
  • 59) 0.417 311 472 119 763 493 537 902 832 031 250 000 000 526 761 062 869 436 989 44 × 2 = 0 + 0.834 622 944 239 526 987 075 805 664 062 500 000 001 053 522 125 738 873 978 88;
  • 60) 0.834 622 944 239 526 987 075 805 664 062 500 000 001 053 522 125 738 873 978 88 × 2 = 1 + 0.669 245 888 479 053 974 151 611 328 125 000 000 002 107 044 251 477 747 957 76;
  • 61) 0.669 245 888 479 053 974 151 611 328 125 000 000 002 107 044 251 477 747 957 76 × 2 = 1 + 0.338 491 776 958 107 948 303 222 656 250 000 000 004 214 088 502 955 495 915 52;
  • 62) 0.338 491 776 958 107 948 303 222 656 250 000 000 004 214 088 502 955 495 915 52 × 2 = 0 + 0.676 983 553 916 215 896 606 445 312 500 000 000 008 428 177 005 910 991 831 04;
  • 63) 0.676 983 553 916 215 896 606 445 312 500 000 000 008 428 177 005 910 991 831 04 × 2 = 1 + 0.353 967 107 832 431 793 212 890 625 000 000 000 016 856 354 011 821 983 662 08;
  • 64) 0.353 967 107 832 431 793 212 890 625 000 000 000 016 856 354 011 821 983 662 08 × 2 = 0 + 0.707 934 215 664 863 586 425 781 250 000 000 000 033 712 708 023 643 967 324 16;
  • 65) 0.707 934 215 664 863 586 425 781 250 000 000 000 033 712 708 023 643 967 324 16 × 2 = 1 + 0.415 868 431 329 727 172 851 562 500 000 000 000 067 425 416 047 287 934 648 32;
  • 66) 0.415 868 431 329 727 172 851 562 500 000 000 000 067 425 416 047 287 934 648 32 × 2 = 0 + 0.831 736 862 659 454 345 703 125 000 000 000 000 134 850 832 094 575 869 296 64;
  • 67) 0.831 736 862 659 454 345 703 125 000 000 000 000 134 850 832 094 575 869 296 64 × 2 = 1 + 0.663 473 725 318 908 691 406 250 000 000 000 000 269 701 664 189 151 738 593 28;
  • 68) 0.663 473 725 318 908 691 406 250 000 000 000 000 269 701 664 189 151 738 593 28 × 2 = 1 + 0.326 947 450 637 817 382 812 500 000 000 000 000 539 403 328 378 303 477 186 56;
  • 69) 0.326 947 450 637 817 382 812 500 000 000 000 000 539 403 328 378 303 477 186 56 × 2 = 0 + 0.653 894 901 275 634 765 625 000 000 000 000 001 078 806 656 756 606 954 373 12;
  • 70) 0.653 894 901 275 634 765 625 000 000 000 000 001 078 806 656 756 606 954 373 12 × 2 = 1 + 0.307 789 802 551 269 531 250 000 000 000 000 002 157 613 313 513 213 908 746 24;
  • 71) 0.307 789 802 551 269 531 250 000 000 000 000 002 157 613 313 513 213 908 746 24 × 2 = 0 + 0.615 579 605 102 539 062 500 000 000 000 000 004 315 226 627 026 427 817 492 48;
  • 72) 0.615 579 605 102 539 062 500 000 000 000 000 004 315 226 627 026 427 817 492 48 × 2 = 1 + 0.231 159 210 205 078 125 000 000 000 000 000 008 630 453 254 052 855 634 984 96;
  • 73) 0.231 159 210 205 078 125 000 000 000 000 000 008 630 453 254 052 855 634 984 96 × 2 = 0 + 0.462 318 420 410 156 250 000 000 000 000 000 017 260 906 508 105 711 269 969 92;
  • 74) 0.462 318 420 410 156 250 000 000 000 000 000 017 260 906 508 105 711 269 969 92 × 2 = 0 + 0.924 636 840 820 312 500 000 000 000 000 000 034 521 813 016 211 422 539 939 84;
  • 75) 0.924 636 840 820 312 500 000 000 000 000 000 034 521 813 016 211 422 539 939 84 × 2 = 1 + 0.849 273 681 640 625 000 000 000 000 000 000 069 043 626 032 422 845 079 879 68;
  • 76) 0.849 273 681 640 625 000 000 000 000 000 000 069 043 626 032 422 845 079 879 68 × 2 = 1 + 0.698 547 363 281 250 000 000 000 000 000 000 138 087 252 064 845 690 159 759 36;
  • 77) 0.698 547 363 281 250 000 000 000 000 000 000 138 087 252 064 845 690 159 759 36 × 2 = 1 + 0.397 094 726 562 500 000 000 000 000 000 000 276 174 504 129 691 380 319 518 72;
  • 78) 0.397 094 726 562 500 000 000 000 000 000 000 276 174 504 129 691 380 319 518 72 × 2 = 0 + 0.794 189 453 125 000 000 000 000 000 000 000 552 349 008 259 382 760 639 037 44;
  • 79) 0.794 189 453 125 000 000 000 000 000 000 000 552 349 008 259 382 760 639 037 44 × 2 = 1 + 0.588 378 906 250 000 000 000 000 000 000 001 104 698 016 518 765 521 278 074 88;
  • 80) 0.588 378 906 250 000 000 000 000 000 000 001 104 698 016 518 765 521 278 074 88 × 2 = 1 + 0.176 757 812 500 000 000 000 000 000 000 002 209 396 033 037 531 042 556 149 76;
  • 81) 0.176 757 812 500 000 000 000 000 000 000 002 209 396 033 037 531 042 556 149 76 × 2 = 0 + 0.353 515 625 000 000 000 000 000 000 000 004 418 792 066 075 062 085 112 299 52;
  • 82) 0.353 515 625 000 000 000 000 000 000 000 004 418 792 066 075 062 085 112 299 52 × 2 = 0 + 0.707 031 250 000 000 000 000 000 000 000 008 837 584 132 150 124 170 224 599 04;
  • 83) 0.707 031 250 000 000 000 000 000 000 000 008 837 584 132 150 124 170 224 599 04 × 2 = 1 + 0.414 062 500 000 000 000 000 000 000 000 017 675 168 264 300 248 340 449 198 08;
  • 84) 0.414 062 500 000 000 000 000 000 000 000 017 675 168 264 300 248 340 449 198 08 × 2 = 0 + 0.828 125 000 000 000 000 000 000 000 000 035 350 336 528 600 496 680 898 396 16;
  • 85) 0.828 125 000 000 000 000 000 000 000 000 035 350 336 528 600 496 680 898 396 16 × 2 = 1 + 0.656 250 000 000 000 000 000 000 000 000 070 700 673 057 200 993 361 796 792 32;
  • 86) 0.656 250 000 000 000 000 000 000 000 000 070 700 673 057 200 993 361 796 792 32 × 2 = 1 + 0.312 500 000 000 000 000 000 000 000 000 141 401 346 114 401 986 723 593 584 64;
  • 87) 0.312 500 000 000 000 000 000 000 000 000 141 401 346 114 401 986 723 593 584 64 × 2 = 0 + 0.625 000 000 000 000 000 000 000 000 000 282 802 692 228 803 973 447 187 169 28;
  • 88) 0.625 000 000 000 000 000 000 000 000 000 282 802 692 228 803 973 447 187 169 28 × 2 = 1 + 0.250 000 000 000 000 000 000 000 000 000 565 605 384 457 607 946 894 374 338 56;
  • 89) 0.250 000 000 000 000 000 000 000 000 000 565 605 384 457 607 946 894 374 338 56 × 2 = 0 + 0.500 000 000 000 000 000 000 000 000 001 131 210 768 915 215 893 788 748 677 12;
  • 90) 0.500 000 000 000 000 000 000 000 000 001 131 210 768 915 215 893 788 748 677 12 × 2 = 1 + 0.000 000 000 000 000 000 000 000 000 002 262 421 537 830 431 787 577 497 354 24;
  • 91) 0.000 000 000 000 000 000 000 000 000 002 262 421 537 830 431 787 577 497 354 24 × 2 = 0 + 0.000 000 000 000 000 000 000 000 000 004 524 843 075 660 863 575 154 994 708 48;

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:


0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 031 76(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:

0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 031 76(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:


0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 031 76(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 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 031 76 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