0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 029 932 430 436 019 785 703 01 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 029 932 430 436 019 785 703 01(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
0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 029 932 430 436 019 785 703 01(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: 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 029 932 430 436 019 785 703 01.

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) 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 029 932 430 436 019 785 703 01 × 2 = 0 + 0.000 000 000 007 269 999 999 999 999 516 522 114 064 601 356 670 305 976 059 864 860 872 039 571 406 02;
  • 2) 0.000 000 000 007 269 999 999 999 999 516 522 114 064 601 356 670 305 976 059 864 860 872 039 571 406 02 × 2 = 0 + 0.000 000 000 014 539 999 999 999 999 033 044 228 129 202 713 340 611 952 119 729 721 744 079 142 812 04;
  • 3) 0.000 000 000 014 539 999 999 999 999 033 044 228 129 202 713 340 611 952 119 729 721 744 079 142 812 04 × 2 = 0 + 0.000 000 000 029 079 999 999 999 998 066 088 456 258 405 426 681 223 904 239 459 443 488 158 285 624 08;
  • 4) 0.000 000 000 029 079 999 999 999 998 066 088 456 258 405 426 681 223 904 239 459 443 488 158 285 624 08 × 2 = 0 + 0.000 000 000 058 159 999 999 999 996 132 176 912 516 810 853 362 447 808 478 918 886 976 316 571 248 16;
  • 5) 0.000 000 000 058 159 999 999 999 996 132 176 912 516 810 853 362 447 808 478 918 886 976 316 571 248 16 × 2 = 0 + 0.000 000 000 116 319 999 999 999 992 264 353 825 033 621 706 724 895 616 957 837 773 952 633 142 496 32;
  • 6) 0.000 000 000 116 319 999 999 999 992 264 353 825 033 621 706 724 895 616 957 837 773 952 633 142 496 32 × 2 = 0 + 0.000 000 000 232 639 999 999 999 984 528 707 650 067 243 413 449 791 233 915 675 547 905 266 284 992 64;
  • 7) 0.000 000 000 232 639 999 999 999 984 528 707 650 067 243 413 449 791 233 915 675 547 905 266 284 992 64 × 2 = 0 + 0.000 000 000 465 279 999 999 999 969 057 415 300 134 486 826 899 582 467 831 351 095 810 532 569 985 28;
  • 8) 0.000 000 000 465 279 999 999 999 969 057 415 300 134 486 826 899 582 467 831 351 095 810 532 569 985 28 × 2 = 0 + 0.000 000 000 930 559 999 999 999 938 114 830 600 268 973 653 799 164 935 662 702 191 621 065 139 970 56;
  • 9) 0.000 000 000 930 559 999 999 999 938 114 830 600 268 973 653 799 164 935 662 702 191 621 065 139 970 56 × 2 = 0 + 0.000 000 001 861 119 999 999 999 876 229 661 200 537 947 307 598 329 871 325 404 383 242 130 279 941 12;
  • 10) 0.000 000 001 861 119 999 999 999 876 229 661 200 537 947 307 598 329 871 325 404 383 242 130 279 941 12 × 2 = 0 + 0.000 000 003 722 239 999 999 999 752 459 322 401 075 894 615 196 659 742 650 808 766 484 260 559 882 24;
  • 11) 0.000 000 003 722 239 999 999 999 752 459 322 401 075 894 615 196 659 742 650 808 766 484 260 559 882 24 × 2 = 0 + 0.000 000 007 444 479 999 999 999 504 918 644 802 151 789 230 393 319 485 301 617 532 968 521 119 764 48;
  • 12) 0.000 000 007 444 479 999 999 999 504 918 644 802 151 789 230 393 319 485 301 617 532 968 521 119 764 48 × 2 = 0 + 0.000 000 014 888 959 999 999 999 009 837 289 604 303 578 460 786 638 970 603 235 065 937 042 239 528 96;
  • 13) 0.000 000 014 888 959 999 999 999 009 837 289 604 303 578 460 786 638 970 603 235 065 937 042 239 528 96 × 2 = 0 + 0.000 000 029 777 919 999 999 998 019 674 579 208 607 156 921 573 277 941 206 470 131 874 084 479 057 92;
  • 14) 0.000 000 029 777 919 999 999 998 019 674 579 208 607 156 921 573 277 941 206 470 131 874 084 479 057 92 × 2 = 0 + 0.000 000 059 555 839 999 999 996 039 349 158 417 214 313 843 146 555 882 412 940 263 748 168 958 115 84;
  • 15) 0.000 000 059 555 839 999 999 996 039 349 158 417 214 313 843 146 555 882 412 940 263 748 168 958 115 84 × 2 = 0 + 0.000 000 119 111 679 999 999 992 078 698 316 834 428 627 686 293 111 764 825 880 527 496 337 916 231 68;
  • 16) 0.000 000 119 111 679 999 999 992 078 698 316 834 428 627 686 293 111 764 825 880 527 496 337 916 231 68 × 2 = 0 + 0.000 000 238 223 359 999 999 984 157 396 633 668 857 255 372 586 223 529 651 761 054 992 675 832 463 36;
  • 17) 0.000 000 238 223 359 999 999 984 157 396 633 668 857 255 372 586 223 529 651 761 054 992 675 832 463 36 × 2 = 0 + 0.000 000 476 446 719 999 999 968 314 793 267 337 714 510 745 172 447 059 303 522 109 985 351 664 926 72;
  • 18) 0.000 000 476 446 719 999 999 968 314 793 267 337 714 510 745 172 447 059 303 522 109 985 351 664 926 72 × 2 = 0 + 0.000 000 952 893 439 999 999 936 629 586 534 675 429 021 490 344 894 118 607 044 219 970 703 329 853 44;
  • 19) 0.000 000 952 893 439 999 999 936 629 586 534 675 429 021 490 344 894 118 607 044 219 970 703 329 853 44 × 2 = 0 + 0.000 001 905 786 879 999 999 873 259 173 069 350 858 042 980 689 788 237 214 088 439 941 406 659 706 88;
  • 20) 0.000 001 905 786 879 999 999 873 259 173 069 350 858 042 980 689 788 237 214 088 439 941 406 659 706 88 × 2 = 0 + 0.000 003 811 573 759 999 999 746 518 346 138 701 716 085 961 379 576 474 428 176 879 882 813 319 413 76;
  • 21) 0.000 003 811 573 759 999 999 746 518 346 138 701 716 085 961 379 576 474 428 176 879 882 813 319 413 76 × 2 = 0 + 0.000 007 623 147 519 999 999 493 036 692 277 403 432 171 922 759 152 948 856 353 759 765 626 638 827 52;
  • 22) 0.000 007 623 147 519 999 999 493 036 692 277 403 432 171 922 759 152 948 856 353 759 765 626 638 827 52 × 2 = 0 + 0.000 015 246 295 039 999 998 986 073 384 554 806 864 343 845 518 305 897 712 707 519 531 253 277 655 04;
  • 23) 0.000 015 246 295 039 999 998 986 073 384 554 806 864 343 845 518 305 897 712 707 519 531 253 277 655 04 × 2 = 0 + 0.000 030 492 590 079 999 997 972 146 769 109 613 728 687 691 036 611 795 425 415 039 062 506 555 310 08;
  • 24) 0.000 030 492 590 079 999 997 972 146 769 109 613 728 687 691 036 611 795 425 415 039 062 506 555 310 08 × 2 = 0 + 0.000 060 985 180 159 999 995 944 293 538 219 227 457 375 382 073 223 590 850 830 078 125 013 110 620 16;
  • 25) 0.000 060 985 180 159 999 995 944 293 538 219 227 457 375 382 073 223 590 850 830 078 125 013 110 620 16 × 2 = 0 + 0.000 121 970 360 319 999 991 888 587 076 438 454 914 750 764 146 447 181 701 660 156 250 026 221 240 32;
  • 26) 0.000 121 970 360 319 999 991 888 587 076 438 454 914 750 764 146 447 181 701 660 156 250 026 221 240 32 × 2 = 0 + 0.000 243 940 720 639 999 983 777 174 152 876 909 829 501 528 292 894 363 403 320 312 500 052 442 480 64;
  • 27) 0.000 243 940 720 639 999 983 777 174 152 876 909 829 501 528 292 894 363 403 320 312 500 052 442 480 64 × 2 = 0 + 0.000 487 881 441 279 999 967 554 348 305 753 819 659 003 056 585 788 726 806 640 625 000 104 884 961 28;
  • 28) 0.000 487 881 441 279 999 967 554 348 305 753 819 659 003 056 585 788 726 806 640 625 000 104 884 961 28 × 2 = 0 + 0.000 975 762 882 559 999 935 108 696 611 507 639 318 006 113 171 577 453 613 281 250 000 209 769 922 56;
  • 29) 0.000 975 762 882 559 999 935 108 696 611 507 639 318 006 113 171 577 453 613 281 250 000 209 769 922 56 × 2 = 0 + 0.001 951 525 765 119 999 870 217 393 223 015 278 636 012 226 343 154 907 226 562 500 000 419 539 845 12;
  • 30) 0.001 951 525 765 119 999 870 217 393 223 015 278 636 012 226 343 154 907 226 562 500 000 419 539 845 12 × 2 = 0 + 0.003 903 051 530 239 999 740 434 786 446 030 557 272 024 452 686 309 814 453 125 000 000 839 079 690 24;
  • 31) 0.003 903 051 530 239 999 740 434 786 446 030 557 272 024 452 686 309 814 453 125 000 000 839 079 690 24 × 2 = 0 + 0.007 806 103 060 479 999 480 869 572 892 061 114 544 048 905 372 619 628 906 250 000 001 678 159 380 48;
  • 32) 0.007 806 103 060 479 999 480 869 572 892 061 114 544 048 905 372 619 628 906 250 000 001 678 159 380 48 × 2 = 0 + 0.015 612 206 120 959 998 961 739 145 784 122 229 088 097 810 745 239 257 812 500 000 003 356 318 760 96;
  • 33) 0.015 612 206 120 959 998 961 739 145 784 122 229 088 097 810 745 239 257 812 500 000 003 356 318 760 96 × 2 = 0 + 0.031 224 412 241 919 997 923 478 291 568 244 458 176 195 621 490 478 515 625 000 000 006 712 637 521 92;
  • 34) 0.031 224 412 241 919 997 923 478 291 568 244 458 176 195 621 490 478 515 625 000 000 006 712 637 521 92 × 2 = 0 + 0.062 448 824 483 839 995 846 956 583 136 488 916 352 391 242 980 957 031 250 000 000 013 425 275 043 84;
  • 35) 0.062 448 824 483 839 995 846 956 583 136 488 916 352 391 242 980 957 031 250 000 000 013 425 275 043 84 × 2 = 0 + 0.124 897 648 967 679 991 693 913 166 272 977 832 704 782 485 961 914 062 500 000 000 026 850 550 087 68;
  • 36) 0.124 897 648 967 679 991 693 913 166 272 977 832 704 782 485 961 914 062 500 000 000 026 850 550 087 68 × 2 = 0 + 0.249 795 297 935 359 983 387 826 332 545 955 665 409 564 971 923 828 125 000 000 000 053 701 100 175 36;
  • 37) 0.249 795 297 935 359 983 387 826 332 545 955 665 409 564 971 923 828 125 000 000 000 053 701 100 175 36 × 2 = 0 + 0.499 590 595 870 719 966 775 652 665 091 911 330 819 129 943 847 656 250 000 000 000 107 402 200 350 72;
  • 38) 0.499 590 595 870 719 966 775 652 665 091 911 330 819 129 943 847 656 250 000 000 000 107 402 200 350 72 × 2 = 0 + 0.999 181 191 741 439 933 551 305 330 183 822 661 638 259 887 695 312 500 000 000 000 214 804 400 701 44;
  • 39) 0.999 181 191 741 439 933 551 305 330 183 822 661 638 259 887 695 312 500 000 000 000 214 804 400 701 44 × 2 = 1 + 0.998 362 383 482 879 867 102 610 660 367 645 323 276 519 775 390 625 000 000 000 000 429 608 801 402 88;
  • 40) 0.998 362 383 482 879 867 102 610 660 367 645 323 276 519 775 390 625 000 000 000 000 429 608 801 402 88 × 2 = 1 + 0.996 724 766 965 759 734 205 221 320 735 290 646 553 039 550 781 250 000 000 000 000 859 217 602 805 76;
  • 41) 0.996 724 766 965 759 734 205 221 320 735 290 646 553 039 550 781 250 000 000 000 000 859 217 602 805 76 × 2 = 1 + 0.993 449 533 931 519 468 410 442 641 470 581 293 106 079 101 562 500 000 000 000 001 718 435 205 611 52;
  • 42) 0.993 449 533 931 519 468 410 442 641 470 581 293 106 079 101 562 500 000 000 000 001 718 435 205 611 52 × 2 = 1 + 0.986 899 067 863 038 936 820 885 282 941 162 586 212 158 203 125 000 000 000 000 003 436 870 411 223 04;
  • 43) 0.986 899 067 863 038 936 820 885 282 941 162 586 212 158 203 125 000 000 000 000 003 436 870 411 223 04 × 2 = 1 + 0.973 798 135 726 077 873 641 770 565 882 325 172 424 316 406 250 000 000 000 000 006 873 740 822 446 08;
  • 44) 0.973 798 135 726 077 873 641 770 565 882 325 172 424 316 406 250 000 000 000 000 006 873 740 822 446 08 × 2 = 1 + 0.947 596 271 452 155 747 283 541 131 764 650 344 848 632 812 500 000 000 000 000 013 747 481 644 892 16;
  • 45) 0.947 596 271 452 155 747 283 541 131 764 650 344 848 632 812 500 000 000 000 000 013 747 481 644 892 16 × 2 = 1 + 0.895 192 542 904 311 494 567 082 263 529 300 689 697 265 625 000 000 000 000 000 027 494 963 289 784 32;
  • 46) 0.895 192 542 904 311 494 567 082 263 529 300 689 697 265 625 000 000 000 000 000 027 494 963 289 784 32 × 2 = 1 + 0.790 385 085 808 622 989 134 164 527 058 601 379 394 531 250 000 000 000 000 000 054 989 926 579 568 64;
  • 47) 0.790 385 085 808 622 989 134 164 527 058 601 379 394 531 250 000 000 000 000 000 054 989 926 579 568 64 × 2 = 1 + 0.580 770 171 617 245 978 268 329 054 117 202 758 789 062 500 000 000 000 000 000 109 979 853 159 137 28;
  • 48) 0.580 770 171 617 245 978 268 329 054 117 202 758 789 062 500 000 000 000 000 000 109 979 853 159 137 28 × 2 = 1 + 0.161 540 343 234 491 956 536 658 108 234 405 517 578 125 000 000 000 000 000 000 219 959 706 318 274 56;
  • 49) 0.161 540 343 234 491 956 536 658 108 234 405 517 578 125 000 000 000 000 000 000 219 959 706 318 274 56 × 2 = 0 + 0.323 080 686 468 983 913 073 316 216 468 811 035 156 250 000 000 000 000 000 000 439 919 412 636 549 12;
  • 50) 0.323 080 686 468 983 913 073 316 216 468 811 035 156 250 000 000 000 000 000 000 439 919 412 636 549 12 × 2 = 0 + 0.646 161 372 937 967 826 146 632 432 937 622 070 312 500 000 000 000 000 000 000 879 838 825 273 098 24;
  • 51) 0.646 161 372 937 967 826 146 632 432 937 622 070 312 500 000 000 000 000 000 000 879 838 825 273 098 24 × 2 = 1 + 0.292 322 745 875 935 652 293 264 865 875 244 140 625 000 000 000 000 000 000 001 759 677 650 546 196 48;
  • 52) 0.292 322 745 875 935 652 293 264 865 875 244 140 625 000 000 000 000 000 000 001 759 677 650 546 196 48 × 2 = 0 + 0.584 645 491 751 871 304 586 529 731 750 488 281 250 000 000 000 000 000 000 003 519 355 301 092 392 96;
  • 53) 0.584 645 491 751 871 304 586 529 731 750 488 281 250 000 000 000 000 000 000 003 519 355 301 092 392 96 × 2 = 1 + 0.169 290 983 503 742 609 173 059 463 500 976 562 500 000 000 000 000 000 000 007 038 710 602 184 785 92;
  • 54) 0.169 290 983 503 742 609 173 059 463 500 976 562 500 000 000 000 000 000 000 007 038 710 602 184 785 92 × 2 = 0 + 0.338 581 967 007 485 218 346 118 927 001 953 125 000 000 000 000 000 000 000 014 077 421 204 369 571 84;
  • 55) 0.338 581 967 007 485 218 346 118 927 001 953 125 000 000 000 000 000 000 000 014 077 421 204 369 571 84 × 2 = 0 + 0.677 163 934 014 970 436 692 237 854 003 906 250 000 000 000 000 000 000 000 028 154 842 408 739 143 68;
  • 56) 0.677 163 934 014 970 436 692 237 854 003 906 250 000 000 000 000 000 000 000 028 154 842 408 739 143 68 × 2 = 1 + 0.354 327 868 029 940 873 384 475 708 007 812 500 000 000 000 000 000 000 000 056 309 684 817 478 287 36;
  • 57) 0.354 327 868 029 940 873 384 475 708 007 812 500 000 000 000 000 000 000 000 056 309 684 817 478 287 36 × 2 = 0 + 0.708 655 736 059 881 746 768 951 416 015 625 000 000 000 000 000 000 000 000 112 619 369 634 956 574 72;
  • 58) 0.708 655 736 059 881 746 768 951 416 015 625 000 000 000 000 000 000 000 000 112 619 369 634 956 574 72 × 2 = 1 + 0.417 311 472 119 763 493 537 902 832 031 250 000 000 000 000 000 000 000 000 225 238 739 269 913 149 44;
  • 59) 0.417 311 472 119 763 493 537 902 832 031 250 000 000 000 000 000 000 000 000 225 238 739 269 913 149 44 × 2 = 0 + 0.834 622 944 239 526 987 075 805 664 062 500 000 000 000 000 000 000 000 000 450 477 478 539 826 298 88;
  • 60) 0.834 622 944 239 526 987 075 805 664 062 500 000 000 000 000 000 000 000 000 450 477 478 539 826 298 88 × 2 = 1 + 0.669 245 888 479 053 974 151 611 328 125 000 000 000 000 000 000 000 000 000 900 954 957 079 652 597 76;
  • 61) 0.669 245 888 479 053 974 151 611 328 125 000 000 000 000 000 000 000 000 000 900 954 957 079 652 597 76 × 2 = 1 + 0.338 491 776 958 107 948 303 222 656 250 000 000 000 000 000 000 000 000 001 801 909 914 159 305 195 52;
  • 62) 0.338 491 776 958 107 948 303 222 656 250 000 000 000 000 000 000 000 000 001 801 909 914 159 305 195 52 × 2 = 0 + 0.676 983 553 916 215 896 606 445 312 500 000 000 000 000 000 000 000 000 003 603 819 828 318 610 391 04;
  • 63) 0.676 983 553 916 215 896 606 445 312 500 000 000 000 000 000 000 000 000 003 603 819 828 318 610 391 04 × 2 = 1 + 0.353 967 107 832 431 793 212 890 625 000 000 000 000 000 000 000 000 000 007 207 639 656 637 220 782 08;
  • 64) 0.353 967 107 832 431 793 212 890 625 000 000 000 000 000 000 000 000 000 007 207 639 656 637 220 782 08 × 2 = 0 + 0.707 934 215 664 863 586 425 781 250 000 000 000 000 000 000 000 000 000 014 415 279 313 274 441 564 16;
  • 65) 0.707 934 215 664 863 586 425 781 250 000 000 000 000 000 000 000 000 000 014 415 279 313 274 441 564 16 × 2 = 1 + 0.415 868 431 329 727 172 851 562 500 000 000 000 000 000 000 000 000 000 028 830 558 626 548 883 128 32;
  • 66) 0.415 868 431 329 727 172 851 562 500 000 000 000 000 000 000 000 000 000 028 830 558 626 548 883 128 32 × 2 = 0 + 0.831 736 862 659 454 345 703 125 000 000 000 000 000 000 000 000 000 000 057 661 117 253 097 766 256 64;
  • 67) 0.831 736 862 659 454 345 703 125 000 000 000 000 000 000 000 000 000 000 057 661 117 253 097 766 256 64 × 2 = 1 + 0.663 473 725 318 908 691 406 250 000 000 000 000 000 000 000 000 000 000 115 322 234 506 195 532 513 28;
  • 68) 0.663 473 725 318 908 691 406 250 000 000 000 000 000 000 000 000 000 000 115 322 234 506 195 532 513 28 × 2 = 1 + 0.326 947 450 637 817 382 812 500 000 000 000 000 000 000 000 000 000 000 230 644 469 012 391 065 026 56;
  • 69) 0.326 947 450 637 817 382 812 500 000 000 000 000 000 000 000 000 000 000 230 644 469 012 391 065 026 56 × 2 = 0 + 0.653 894 901 275 634 765 625 000 000 000 000 000 000 000 000 000 000 000 461 288 938 024 782 130 053 12;
  • 70) 0.653 894 901 275 634 765 625 000 000 000 000 000 000 000 000 000 000 000 461 288 938 024 782 130 053 12 × 2 = 1 + 0.307 789 802 551 269 531 250 000 000 000 000 000 000 000 000 000 000 000 922 577 876 049 564 260 106 24;
  • 71) 0.307 789 802 551 269 531 250 000 000 000 000 000 000 000 000 000 000 000 922 577 876 049 564 260 106 24 × 2 = 0 + 0.615 579 605 102 539 062 500 000 000 000 000 000 000 000 000 000 000 001 845 155 752 099 128 520 212 48;
  • 72) 0.615 579 605 102 539 062 500 000 000 000 000 000 000 000 000 000 000 001 845 155 752 099 128 520 212 48 × 2 = 1 + 0.231 159 210 205 078 125 000 000 000 000 000 000 000 000 000 000 000 003 690 311 504 198 257 040 424 96;
  • 73) 0.231 159 210 205 078 125 000 000 000 000 000 000 000 000 000 000 000 003 690 311 504 198 257 040 424 96 × 2 = 0 + 0.462 318 420 410 156 250 000 000 000 000 000 000 000 000 000 000 000 007 380 623 008 396 514 080 849 92;
  • 74) 0.462 318 420 410 156 250 000 000 000 000 000 000 000 000 000 000 000 007 380 623 008 396 514 080 849 92 × 2 = 0 + 0.924 636 840 820 312 500 000 000 000 000 000 000 000 000 000 000 000 014 761 246 016 793 028 161 699 84;
  • 75) 0.924 636 840 820 312 500 000 000 000 000 000 000 000 000 000 000 000 014 761 246 016 793 028 161 699 84 × 2 = 1 + 0.849 273 681 640 625 000 000 000 000 000 000 000 000 000 000 000 000 029 522 492 033 586 056 323 399 68;
  • 76) 0.849 273 681 640 625 000 000 000 000 000 000 000 000 000 000 000 000 029 522 492 033 586 056 323 399 68 × 2 = 1 + 0.698 547 363 281 250 000 000 000 000 000 000 000 000 000 000 000 000 059 044 984 067 172 112 646 799 36;
  • 77) 0.698 547 363 281 250 000 000 000 000 000 000 000 000 000 000 000 000 059 044 984 067 172 112 646 799 36 × 2 = 1 + 0.397 094 726 562 500 000 000 000 000 000 000 000 000 000 000 000 000 118 089 968 134 344 225 293 598 72;
  • 78) 0.397 094 726 562 500 000 000 000 000 000 000 000 000 000 000 000 000 118 089 968 134 344 225 293 598 72 × 2 = 0 + 0.794 189 453 125 000 000 000 000 000 000 000 000 000 000 000 000 000 236 179 936 268 688 450 587 197 44;
  • 79) 0.794 189 453 125 000 000 000 000 000 000 000 000 000 000 000 000 000 236 179 936 268 688 450 587 197 44 × 2 = 1 + 0.588 378 906 250 000 000 000 000 000 000 000 000 000 000 000 000 000 472 359 872 537 376 901 174 394 88;
  • 80) 0.588 378 906 250 000 000 000 000 000 000 000 000 000 000 000 000 000 472 359 872 537 376 901 174 394 88 × 2 = 1 + 0.176 757 812 500 000 000 000 000 000 000 000 000 000 000 000 000 000 944 719 745 074 753 802 348 789 76;
  • 81) 0.176 757 812 500 000 000 000 000 000 000 000 000 000 000 000 000 000 944 719 745 074 753 802 348 789 76 × 2 = 0 + 0.353 515 625 000 000 000 000 000 000 000 000 000 000 000 000 000 001 889 439 490 149 507 604 697 579 52;
  • 82) 0.353 515 625 000 000 000 000 000 000 000 000 000 000 000 000 000 001 889 439 490 149 507 604 697 579 52 × 2 = 0 + 0.707 031 250 000 000 000 000 000 000 000 000 000 000 000 000 000 003 778 878 980 299 015 209 395 159 04;
  • 83) 0.707 031 250 000 000 000 000 000 000 000 000 000 000 000 000 000 003 778 878 980 299 015 209 395 159 04 × 2 = 1 + 0.414 062 500 000 000 000 000 000 000 000 000 000 000 000 000 000 007 557 757 960 598 030 418 790 318 08;
  • 84) 0.414 062 500 000 000 000 000 000 000 000 000 000 000 000 000 000 007 557 757 960 598 030 418 790 318 08 × 2 = 0 + 0.828 125 000 000 000 000 000 000 000 000 000 000 000 000 000 000 015 115 515 921 196 060 837 580 636 16;
  • 85) 0.828 125 000 000 000 000 000 000 000 000 000 000 000 000 000 000 015 115 515 921 196 060 837 580 636 16 × 2 = 1 + 0.656 250 000 000 000 000 000 000 000 000 000 000 000 000 000 000 030 231 031 842 392 121 675 161 272 32;
  • 86) 0.656 250 000 000 000 000 000 000 000 000 000 000 000 000 000 000 030 231 031 842 392 121 675 161 272 32 × 2 = 1 + 0.312 500 000 000 000 000 000 000 000 000 000 000 000 000 000 000 060 462 063 684 784 243 350 322 544 64;
  • 87) 0.312 500 000 000 000 000 000 000 000 000 000 000 000 000 000 000 060 462 063 684 784 243 350 322 544 64 × 2 = 0 + 0.625 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 120 924 127 369 568 486 700 645 089 28;
  • 88) 0.625 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 120 924 127 369 568 486 700 645 089 28 × 2 = 1 + 0.250 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 241 848 254 739 136 973 401 290 178 56;
  • 89) 0.250 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 241 848 254 739 136 973 401 290 178 56 × 2 = 0 + 0.500 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 483 696 509 478 273 946 802 580 357 12;
  • 90) 0.500 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 483 696 509 478 273 946 802 580 357 12 × 2 = 1 + 0.000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 967 393 018 956 547 893 605 160 714 24;
  • 91) 0.000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 967 393 018 956 547 893 605 160 714 24 × 2 = 0 + 0.000 000 000 000 000 000 000 000 000 000 000 000 000 000 000 001 934 786 037 913 095 787 210 321 428 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:


0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 029 932 430 436 019 785 703 01(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:

0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 029 932 430 436 019 785 703 01(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:


0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 029 932 430 436 019 785 703 01(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 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 029 932 430 436 019 785 703 01 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