0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 029 932 430 436 019 785 695 2 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 029 932 430 436 019 785 695 2(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
0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 029 932 430 436 019 785 695 2(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: 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 029 932 430 436 019 785 695 2.

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) 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 029 932 430 436 019 785 695 2 × 2 = 0 + 0.000 000 000 007 269 999 999 999 999 516 522 114 064 601 356 670 305 976 059 864 860 872 039 571 390 4;
  • 2) 0.000 000 000 007 269 999 999 999 999 516 522 114 064 601 356 670 305 976 059 864 860 872 039 571 390 4 × 2 = 0 + 0.000 000 000 014 539 999 999 999 999 033 044 228 129 202 713 340 611 952 119 729 721 744 079 142 780 8;
  • 3) 0.000 000 000 014 539 999 999 999 999 033 044 228 129 202 713 340 611 952 119 729 721 744 079 142 780 8 × 2 = 0 + 0.000 000 000 029 079 999 999 999 998 066 088 456 258 405 426 681 223 904 239 459 443 488 158 285 561 6;
  • 4) 0.000 000 000 029 079 999 999 999 998 066 088 456 258 405 426 681 223 904 239 459 443 488 158 285 561 6 × 2 = 0 + 0.000 000 000 058 159 999 999 999 996 132 176 912 516 810 853 362 447 808 478 918 886 976 316 571 123 2;
  • 5) 0.000 000 000 058 159 999 999 999 996 132 176 912 516 810 853 362 447 808 478 918 886 976 316 571 123 2 × 2 = 0 + 0.000 000 000 116 319 999 999 999 992 264 353 825 033 621 706 724 895 616 957 837 773 952 633 142 246 4;
  • 6) 0.000 000 000 116 319 999 999 999 992 264 353 825 033 621 706 724 895 616 957 837 773 952 633 142 246 4 × 2 = 0 + 0.000 000 000 232 639 999 999 999 984 528 707 650 067 243 413 449 791 233 915 675 547 905 266 284 492 8;
  • 7) 0.000 000 000 232 639 999 999 999 984 528 707 650 067 243 413 449 791 233 915 675 547 905 266 284 492 8 × 2 = 0 + 0.000 000 000 465 279 999 999 999 969 057 415 300 134 486 826 899 582 467 831 351 095 810 532 568 985 6;
  • 8) 0.000 000 000 465 279 999 999 999 969 057 415 300 134 486 826 899 582 467 831 351 095 810 532 568 985 6 × 2 = 0 + 0.000 000 000 930 559 999 999 999 938 114 830 600 268 973 653 799 164 935 662 702 191 621 065 137 971 2;
  • 9) 0.000 000 000 930 559 999 999 999 938 114 830 600 268 973 653 799 164 935 662 702 191 621 065 137 971 2 × 2 = 0 + 0.000 000 001 861 119 999 999 999 876 229 661 200 537 947 307 598 329 871 325 404 383 242 130 275 942 4;
  • 10) 0.000 000 001 861 119 999 999 999 876 229 661 200 537 947 307 598 329 871 325 404 383 242 130 275 942 4 × 2 = 0 + 0.000 000 003 722 239 999 999 999 752 459 322 401 075 894 615 196 659 742 650 808 766 484 260 551 884 8;
  • 11) 0.000 000 003 722 239 999 999 999 752 459 322 401 075 894 615 196 659 742 650 808 766 484 260 551 884 8 × 2 = 0 + 0.000 000 007 444 479 999 999 999 504 918 644 802 151 789 230 393 319 485 301 617 532 968 521 103 769 6;
  • 12) 0.000 000 007 444 479 999 999 999 504 918 644 802 151 789 230 393 319 485 301 617 532 968 521 103 769 6 × 2 = 0 + 0.000 000 014 888 959 999 999 999 009 837 289 604 303 578 460 786 638 970 603 235 065 937 042 207 539 2;
  • 13) 0.000 000 014 888 959 999 999 999 009 837 289 604 303 578 460 786 638 970 603 235 065 937 042 207 539 2 × 2 = 0 + 0.000 000 029 777 919 999 999 998 019 674 579 208 607 156 921 573 277 941 206 470 131 874 084 415 078 4;
  • 14) 0.000 000 029 777 919 999 999 998 019 674 579 208 607 156 921 573 277 941 206 470 131 874 084 415 078 4 × 2 = 0 + 0.000 000 059 555 839 999 999 996 039 349 158 417 214 313 843 146 555 882 412 940 263 748 168 830 156 8;
  • 15) 0.000 000 059 555 839 999 999 996 039 349 158 417 214 313 843 146 555 882 412 940 263 748 168 830 156 8 × 2 = 0 + 0.000 000 119 111 679 999 999 992 078 698 316 834 428 627 686 293 111 764 825 880 527 496 337 660 313 6;
  • 16) 0.000 000 119 111 679 999 999 992 078 698 316 834 428 627 686 293 111 764 825 880 527 496 337 660 313 6 × 2 = 0 + 0.000 000 238 223 359 999 999 984 157 396 633 668 857 255 372 586 223 529 651 761 054 992 675 320 627 2;
  • 17) 0.000 000 238 223 359 999 999 984 157 396 633 668 857 255 372 586 223 529 651 761 054 992 675 320 627 2 × 2 = 0 + 0.000 000 476 446 719 999 999 968 314 793 267 337 714 510 745 172 447 059 303 522 109 985 350 641 254 4;
  • 18) 0.000 000 476 446 719 999 999 968 314 793 267 337 714 510 745 172 447 059 303 522 109 985 350 641 254 4 × 2 = 0 + 0.000 000 952 893 439 999 999 936 629 586 534 675 429 021 490 344 894 118 607 044 219 970 701 282 508 8;
  • 19) 0.000 000 952 893 439 999 999 936 629 586 534 675 429 021 490 344 894 118 607 044 219 970 701 282 508 8 × 2 = 0 + 0.000 001 905 786 879 999 999 873 259 173 069 350 858 042 980 689 788 237 214 088 439 941 402 565 017 6;
  • 20) 0.000 001 905 786 879 999 999 873 259 173 069 350 858 042 980 689 788 237 214 088 439 941 402 565 017 6 × 2 = 0 + 0.000 003 811 573 759 999 999 746 518 346 138 701 716 085 961 379 576 474 428 176 879 882 805 130 035 2;
  • 21) 0.000 003 811 573 759 999 999 746 518 346 138 701 716 085 961 379 576 474 428 176 879 882 805 130 035 2 × 2 = 0 + 0.000 007 623 147 519 999 999 493 036 692 277 403 432 171 922 759 152 948 856 353 759 765 610 260 070 4;
  • 22) 0.000 007 623 147 519 999 999 493 036 692 277 403 432 171 922 759 152 948 856 353 759 765 610 260 070 4 × 2 = 0 + 0.000 015 246 295 039 999 998 986 073 384 554 806 864 343 845 518 305 897 712 707 519 531 220 520 140 8;
  • 23) 0.000 015 246 295 039 999 998 986 073 384 554 806 864 343 845 518 305 897 712 707 519 531 220 520 140 8 × 2 = 0 + 0.000 030 492 590 079 999 997 972 146 769 109 613 728 687 691 036 611 795 425 415 039 062 441 040 281 6;
  • 24) 0.000 030 492 590 079 999 997 972 146 769 109 613 728 687 691 036 611 795 425 415 039 062 441 040 281 6 × 2 = 0 + 0.000 060 985 180 159 999 995 944 293 538 219 227 457 375 382 073 223 590 850 830 078 124 882 080 563 2;
  • 25) 0.000 060 985 180 159 999 995 944 293 538 219 227 457 375 382 073 223 590 850 830 078 124 882 080 563 2 × 2 = 0 + 0.000 121 970 360 319 999 991 888 587 076 438 454 914 750 764 146 447 181 701 660 156 249 764 161 126 4;
  • 26) 0.000 121 970 360 319 999 991 888 587 076 438 454 914 750 764 146 447 181 701 660 156 249 764 161 126 4 × 2 = 0 + 0.000 243 940 720 639 999 983 777 174 152 876 909 829 501 528 292 894 363 403 320 312 499 528 322 252 8;
  • 27) 0.000 243 940 720 639 999 983 777 174 152 876 909 829 501 528 292 894 363 403 320 312 499 528 322 252 8 × 2 = 0 + 0.000 487 881 441 279 999 967 554 348 305 753 819 659 003 056 585 788 726 806 640 624 999 056 644 505 6;
  • 28) 0.000 487 881 441 279 999 967 554 348 305 753 819 659 003 056 585 788 726 806 640 624 999 056 644 505 6 × 2 = 0 + 0.000 975 762 882 559 999 935 108 696 611 507 639 318 006 113 171 577 453 613 281 249 998 113 289 011 2;
  • 29) 0.000 975 762 882 559 999 935 108 696 611 507 639 318 006 113 171 577 453 613 281 249 998 113 289 011 2 × 2 = 0 + 0.001 951 525 765 119 999 870 217 393 223 015 278 636 012 226 343 154 907 226 562 499 996 226 578 022 4;
  • 30) 0.001 951 525 765 119 999 870 217 393 223 015 278 636 012 226 343 154 907 226 562 499 996 226 578 022 4 × 2 = 0 + 0.003 903 051 530 239 999 740 434 786 446 030 557 272 024 452 686 309 814 453 124 999 992 453 156 044 8;
  • 31) 0.003 903 051 530 239 999 740 434 786 446 030 557 272 024 452 686 309 814 453 124 999 992 453 156 044 8 × 2 = 0 + 0.007 806 103 060 479 999 480 869 572 892 061 114 544 048 905 372 619 628 906 249 999 984 906 312 089 6;
  • 32) 0.007 806 103 060 479 999 480 869 572 892 061 114 544 048 905 372 619 628 906 249 999 984 906 312 089 6 × 2 = 0 + 0.015 612 206 120 959 998 961 739 145 784 122 229 088 097 810 745 239 257 812 499 999 969 812 624 179 2;
  • 33) 0.015 612 206 120 959 998 961 739 145 784 122 229 088 097 810 745 239 257 812 499 999 969 812 624 179 2 × 2 = 0 + 0.031 224 412 241 919 997 923 478 291 568 244 458 176 195 621 490 478 515 624 999 999 939 625 248 358 4;
  • 34) 0.031 224 412 241 919 997 923 478 291 568 244 458 176 195 621 490 478 515 624 999 999 939 625 248 358 4 × 2 = 0 + 0.062 448 824 483 839 995 846 956 583 136 488 916 352 391 242 980 957 031 249 999 999 879 250 496 716 8;
  • 35) 0.062 448 824 483 839 995 846 956 583 136 488 916 352 391 242 980 957 031 249 999 999 879 250 496 716 8 × 2 = 0 + 0.124 897 648 967 679 991 693 913 166 272 977 832 704 782 485 961 914 062 499 999 999 758 500 993 433 6;
  • 36) 0.124 897 648 967 679 991 693 913 166 272 977 832 704 782 485 961 914 062 499 999 999 758 500 993 433 6 × 2 = 0 + 0.249 795 297 935 359 983 387 826 332 545 955 665 409 564 971 923 828 124 999 999 999 517 001 986 867 2;
  • 37) 0.249 795 297 935 359 983 387 826 332 545 955 665 409 564 971 923 828 124 999 999 999 517 001 986 867 2 × 2 = 0 + 0.499 590 595 870 719 966 775 652 665 091 911 330 819 129 943 847 656 249 999 999 999 034 003 973 734 4;
  • 38) 0.499 590 595 870 719 966 775 652 665 091 911 330 819 129 943 847 656 249 999 999 999 034 003 973 734 4 × 2 = 0 + 0.999 181 191 741 439 933 551 305 330 183 822 661 638 259 887 695 312 499 999 999 998 068 007 947 468 8;
  • 39) 0.999 181 191 741 439 933 551 305 330 183 822 661 638 259 887 695 312 499 999 999 998 068 007 947 468 8 × 2 = 1 + 0.998 362 383 482 879 867 102 610 660 367 645 323 276 519 775 390 624 999 999 999 996 136 015 894 937 6;
  • 40) 0.998 362 383 482 879 867 102 610 660 367 645 323 276 519 775 390 624 999 999 999 996 136 015 894 937 6 × 2 = 1 + 0.996 724 766 965 759 734 205 221 320 735 290 646 553 039 550 781 249 999 999 999 992 272 031 789 875 2;
  • 41) 0.996 724 766 965 759 734 205 221 320 735 290 646 553 039 550 781 249 999 999 999 992 272 031 789 875 2 × 2 = 1 + 0.993 449 533 931 519 468 410 442 641 470 581 293 106 079 101 562 499 999 999 999 984 544 063 579 750 4;
  • 42) 0.993 449 533 931 519 468 410 442 641 470 581 293 106 079 101 562 499 999 999 999 984 544 063 579 750 4 × 2 = 1 + 0.986 899 067 863 038 936 820 885 282 941 162 586 212 158 203 124 999 999 999 999 969 088 127 159 500 8;
  • 43) 0.986 899 067 863 038 936 820 885 282 941 162 586 212 158 203 124 999 999 999 999 969 088 127 159 500 8 × 2 = 1 + 0.973 798 135 726 077 873 641 770 565 882 325 172 424 316 406 249 999 999 999 999 938 176 254 319 001 6;
  • 44) 0.973 798 135 726 077 873 641 770 565 882 325 172 424 316 406 249 999 999 999 999 938 176 254 319 001 6 × 2 = 1 + 0.947 596 271 452 155 747 283 541 131 764 650 344 848 632 812 499 999 999 999 999 876 352 508 638 003 2;
  • 45) 0.947 596 271 452 155 747 283 541 131 764 650 344 848 632 812 499 999 999 999 999 876 352 508 638 003 2 × 2 = 1 + 0.895 192 542 904 311 494 567 082 263 529 300 689 697 265 624 999 999 999 999 999 752 705 017 276 006 4;
  • 46) 0.895 192 542 904 311 494 567 082 263 529 300 689 697 265 624 999 999 999 999 999 752 705 017 276 006 4 × 2 = 1 + 0.790 385 085 808 622 989 134 164 527 058 601 379 394 531 249 999 999 999 999 999 505 410 034 552 012 8;
  • 47) 0.790 385 085 808 622 989 134 164 527 058 601 379 394 531 249 999 999 999 999 999 505 410 034 552 012 8 × 2 = 1 + 0.580 770 171 617 245 978 268 329 054 117 202 758 789 062 499 999 999 999 999 999 010 820 069 104 025 6;
  • 48) 0.580 770 171 617 245 978 268 329 054 117 202 758 789 062 499 999 999 999 999 999 010 820 069 104 025 6 × 2 = 1 + 0.161 540 343 234 491 956 536 658 108 234 405 517 578 124 999 999 999 999 999 998 021 640 138 208 051 2;
  • 49) 0.161 540 343 234 491 956 536 658 108 234 405 517 578 124 999 999 999 999 999 998 021 640 138 208 051 2 × 2 = 0 + 0.323 080 686 468 983 913 073 316 216 468 811 035 156 249 999 999 999 999 999 996 043 280 276 416 102 4;
  • 50) 0.323 080 686 468 983 913 073 316 216 468 811 035 156 249 999 999 999 999 999 996 043 280 276 416 102 4 × 2 = 0 + 0.646 161 372 937 967 826 146 632 432 937 622 070 312 499 999 999 999 999 999 992 086 560 552 832 204 8;
  • 51) 0.646 161 372 937 967 826 146 632 432 937 622 070 312 499 999 999 999 999 999 992 086 560 552 832 204 8 × 2 = 1 + 0.292 322 745 875 935 652 293 264 865 875 244 140 624 999 999 999 999 999 999 984 173 121 105 664 409 6;
  • 52) 0.292 322 745 875 935 652 293 264 865 875 244 140 624 999 999 999 999 999 999 984 173 121 105 664 409 6 × 2 = 0 + 0.584 645 491 751 871 304 586 529 731 750 488 281 249 999 999 999 999 999 999 968 346 242 211 328 819 2;
  • 53) 0.584 645 491 751 871 304 586 529 731 750 488 281 249 999 999 999 999 999 999 968 346 242 211 328 819 2 × 2 = 1 + 0.169 290 983 503 742 609 173 059 463 500 976 562 499 999 999 999 999 999 999 936 692 484 422 657 638 4;
  • 54) 0.169 290 983 503 742 609 173 059 463 500 976 562 499 999 999 999 999 999 999 936 692 484 422 657 638 4 × 2 = 0 + 0.338 581 967 007 485 218 346 118 927 001 953 124 999 999 999 999 999 999 999 873 384 968 845 315 276 8;
  • 55) 0.338 581 967 007 485 218 346 118 927 001 953 124 999 999 999 999 999 999 999 873 384 968 845 315 276 8 × 2 = 0 + 0.677 163 934 014 970 436 692 237 854 003 906 249 999 999 999 999 999 999 999 746 769 937 690 630 553 6;
  • 56) 0.677 163 934 014 970 436 692 237 854 003 906 249 999 999 999 999 999 999 999 746 769 937 690 630 553 6 × 2 = 1 + 0.354 327 868 029 940 873 384 475 708 007 812 499 999 999 999 999 999 999 999 493 539 875 381 261 107 2;
  • 57) 0.354 327 868 029 940 873 384 475 708 007 812 499 999 999 999 999 999 999 999 493 539 875 381 261 107 2 × 2 = 0 + 0.708 655 736 059 881 746 768 951 416 015 624 999 999 999 999 999 999 999 998 987 079 750 762 522 214 4;
  • 58) 0.708 655 736 059 881 746 768 951 416 015 624 999 999 999 999 999 999 999 998 987 079 750 762 522 214 4 × 2 = 1 + 0.417 311 472 119 763 493 537 902 832 031 249 999 999 999 999 999 999 999 997 974 159 501 525 044 428 8;
  • 59) 0.417 311 472 119 763 493 537 902 832 031 249 999 999 999 999 999 999 999 997 974 159 501 525 044 428 8 × 2 = 0 + 0.834 622 944 239 526 987 075 805 664 062 499 999 999 999 999 999 999 999 995 948 319 003 050 088 857 6;
  • 60) 0.834 622 944 239 526 987 075 805 664 062 499 999 999 999 999 999 999 999 995 948 319 003 050 088 857 6 × 2 = 1 + 0.669 245 888 479 053 974 151 611 328 124 999 999 999 999 999 999 999 999 991 896 638 006 100 177 715 2;
  • 61) 0.669 245 888 479 053 974 151 611 328 124 999 999 999 999 999 999 999 999 991 896 638 006 100 177 715 2 × 2 = 1 + 0.338 491 776 958 107 948 303 222 656 249 999 999 999 999 999 999 999 999 983 793 276 012 200 355 430 4;
  • 62) 0.338 491 776 958 107 948 303 222 656 249 999 999 999 999 999 999 999 999 983 793 276 012 200 355 430 4 × 2 = 0 + 0.676 983 553 916 215 896 606 445 312 499 999 999 999 999 999 999 999 999 967 586 552 024 400 710 860 8;
  • 63) 0.676 983 553 916 215 896 606 445 312 499 999 999 999 999 999 999 999 999 967 586 552 024 400 710 860 8 × 2 = 1 + 0.353 967 107 832 431 793 212 890 624 999 999 999 999 999 999 999 999 999 935 173 104 048 801 421 721 6;
  • 64) 0.353 967 107 832 431 793 212 890 624 999 999 999 999 999 999 999 999 999 935 173 104 048 801 421 721 6 × 2 = 0 + 0.707 934 215 664 863 586 425 781 249 999 999 999 999 999 999 999 999 999 870 346 208 097 602 843 443 2;
  • 65) 0.707 934 215 664 863 586 425 781 249 999 999 999 999 999 999 999 999 999 870 346 208 097 602 843 443 2 × 2 = 1 + 0.415 868 431 329 727 172 851 562 499 999 999 999 999 999 999 999 999 999 740 692 416 195 205 686 886 4;
  • 66) 0.415 868 431 329 727 172 851 562 499 999 999 999 999 999 999 999 999 999 740 692 416 195 205 686 886 4 × 2 = 0 + 0.831 736 862 659 454 345 703 124 999 999 999 999 999 999 999 999 999 999 481 384 832 390 411 373 772 8;
  • 67) 0.831 736 862 659 454 345 703 124 999 999 999 999 999 999 999 999 999 999 481 384 832 390 411 373 772 8 × 2 = 1 + 0.663 473 725 318 908 691 406 249 999 999 999 999 999 999 999 999 999 998 962 769 664 780 822 747 545 6;
  • 68) 0.663 473 725 318 908 691 406 249 999 999 999 999 999 999 999 999 999 998 962 769 664 780 822 747 545 6 × 2 = 1 + 0.326 947 450 637 817 382 812 499 999 999 999 999 999 999 999 999 999 997 925 539 329 561 645 495 091 2;
  • 69) 0.326 947 450 637 817 382 812 499 999 999 999 999 999 999 999 999 999 997 925 539 329 561 645 495 091 2 × 2 = 0 + 0.653 894 901 275 634 765 624 999 999 999 999 999 999 999 999 999 999 995 851 078 659 123 290 990 182 4;
  • 70) 0.653 894 901 275 634 765 624 999 999 999 999 999 999 999 999 999 999 995 851 078 659 123 290 990 182 4 × 2 = 1 + 0.307 789 802 551 269 531 249 999 999 999 999 999 999 999 999 999 999 991 702 157 318 246 581 980 364 8;
  • 71) 0.307 789 802 551 269 531 249 999 999 999 999 999 999 999 999 999 999 991 702 157 318 246 581 980 364 8 × 2 = 0 + 0.615 579 605 102 539 062 499 999 999 999 999 999 999 999 999 999 999 983 404 314 636 493 163 960 729 6;
  • 72) 0.615 579 605 102 539 062 499 999 999 999 999 999 999 999 999 999 999 983 404 314 636 493 163 960 729 6 × 2 = 1 + 0.231 159 210 205 078 124 999 999 999 999 999 999 999 999 999 999 999 966 808 629 272 986 327 921 459 2;
  • 73) 0.231 159 210 205 078 124 999 999 999 999 999 999 999 999 999 999 999 966 808 629 272 986 327 921 459 2 × 2 = 0 + 0.462 318 420 410 156 249 999 999 999 999 999 999 999 999 999 999 999 933 617 258 545 972 655 842 918 4;
  • 74) 0.462 318 420 410 156 249 999 999 999 999 999 999 999 999 999 999 999 933 617 258 545 972 655 842 918 4 × 2 = 0 + 0.924 636 840 820 312 499 999 999 999 999 999 999 999 999 999 999 999 867 234 517 091 945 311 685 836 8;
  • 75) 0.924 636 840 820 312 499 999 999 999 999 999 999 999 999 999 999 999 867 234 517 091 945 311 685 836 8 × 2 = 1 + 0.849 273 681 640 624 999 999 999 999 999 999 999 999 999 999 999 999 734 469 034 183 890 623 371 673 6;
  • 76) 0.849 273 681 640 624 999 999 999 999 999 999 999 999 999 999 999 999 734 469 034 183 890 623 371 673 6 × 2 = 1 + 0.698 547 363 281 249 999 999 999 999 999 999 999 999 999 999 999 999 468 938 068 367 781 246 743 347 2;
  • 77) 0.698 547 363 281 249 999 999 999 999 999 999 999 999 999 999 999 999 468 938 068 367 781 246 743 347 2 × 2 = 1 + 0.397 094 726 562 499 999 999 999 999 999 999 999 999 999 999 999 998 937 876 136 735 562 493 486 694 4;
  • 78) 0.397 094 726 562 499 999 999 999 999 999 999 999 999 999 999 999 998 937 876 136 735 562 493 486 694 4 × 2 = 0 + 0.794 189 453 124 999 999 999 999 999 999 999 999 999 999 999 999 997 875 752 273 471 124 986 973 388 8;
  • 79) 0.794 189 453 124 999 999 999 999 999 999 999 999 999 999 999 999 997 875 752 273 471 124 986 973 388 8 × 2 = 1 + 0.588 378 906 249 999 999 999 999 999 999 999 999 999 999 999 999 995 751 504 546 942 249 973 946 777 6;
  • 80) 0.588 378 906 249 999 999 999 999 999 999 999 999 999 999 999 999 995 751 504 546 942 249 973 946 777 6 × 2 = 1 + 0.176 757 812 499 999 999 999 999 999 999 999 999 999 999 999 999 991 503 009 093 884 499 947 893 555 2;
  • 81) 0.176 757 812 499 999 999 999 999 999 999 999 999 999 999 999 999 991 503 009 093 884 499 947 893 555 2 × 2 = 0 + 0.353 515 624 999 999 999 999 999 999 999 999 999 999 999 999 999 983 006 018 187 768 999 895 787 110 4;
  • 82) 0.353 515 624 999 999 999 999 999 999 999 999 999 999 999 999 999 983 006 018 187 768 999 895 787 110 4 × 2 = 0 + 0.707 031 249 999 999 999 999 999 999 999 999 999 999 999 999 999 966 012 036 375 537 999 791 574 220 8;
  • 83) 0.707 031 249 999 999 999 999 999 999 999 999 999 999 999 999 999 966 012 036 375 537 999 791 574 220 8 × 2 = 1 + 0.414 062 499 999 999 999 999 999 999 999 999 999 999 999 999 999 932 024 072 751 075 999 583 148 441 6;
  • 84) 0.414 062 499 999 999 999 999 999 999 999 999 999 999 999 999 999 932 024 072 751 075 999 583 148 441 6 × 2 = 0 + 0.828 124 999 999 999 999 999 999 999 999 999 999 999 999 999 999 864 048 145 502 151 999 166 296 883 2;
  • 85) 0.828 124 999 999 999 999 999 999 999 999 999 999 999 999 999 999 864 048 145 502 151 999 166 296 883 2 × 2 = 1 + 0.656 249 999 999 999 999 999 999 999 999 999 999 999 999 999 999 728 096 291 004 303 998 332 593 766 4;
  • 86) 0.656 249 999 999 999 999 999 999 999 999 999 999 999 999 999 999 728 096 291 004 303 998 332 593 766 4 × 2 = 1 + 0.312 499 999 999 999 999 999 999 999 999 999 999 999 999 999 999 456 192 582 008 607 996 665 187 532 8;
  • 87) 0.312 499 999 999 999 999 999 999 999 999 999 999 999 999 999 999 456 192 582 008 607 996 665 187 532 8 × 2 = 0 + 0.624 999 999 999 999 999 999 999 999 999 999 999 999 999 999 998 912 385 164 017 215 993 330 375 065 6;
  • 88) 0.624 999 999 999 999 999 999 999 999 999 999 999 999 999 999 998 912 385 164 017 215 993 330 375 065 6 × 2 = 1 + 0.249 999 999 999 999 999 999 999 999 999 999 999 999 999 999 997 824 770 328 034 431 986 660 750 131 2;
  • 89) 0.249 999 999 999 999 999 999 999 999 999 999 999 999 999 999 997 824 770 328 034 431 986 660 750 131 2 × 2 = 0 + 0.499 999 999 999 999 999 999 999 999 999 999 999 999 999 999 995 649 540 656 068 863 973 321 500 262 4;
  • 90) 0.499 999 999 999 999 999 999 999 999 999 999 999 999 999 999 995 649 540 656 068 863 973 321 500 262 4 × 2 = 0 + 0.999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 991 299 081 312 137 727 946 643 000 524 8;
  • 91) 0.999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 991 299 081 312 137 727 946 643 000 524 8 × 2 = 1 + 0.999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 982 598 162 624 275 455 893 286 001 049 6;

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:


0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 029 932 430 436 019 785 695 2(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:

0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 029 932 430 436 019 785 695 2(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:


0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 029 932 430 436 019 785 695 2(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 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 029 932 430 436 019 785 695 2 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