0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 029 932 430 436 019 785 656 5 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 029 932 430 436 019 785 656 5(10) to 64 bit double precision IEEE 754 binary floating point representation standard (1 bit for sign, 11 bits for exponent, 52 bits for mantissa)

What are the steps to convert decimal number
0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 029 932 430 436 019 785 656 5(10) to 64 bit double precision IEEE 754 binary floating point representation (1 bit for sign, 11 bits for exponent, 52 bits for mantissa)

1. First, convert to binary (in base 2) the integer part: 0.
Divide the number repeatedly by 2.

Keep track of each remainder.

We stop when we get a quotient that is equal to zero.


  • division = quotient + remainder;
  • 0 ÷ 2 = 0 + 0;

2. Construct the base 2 representation of the integer part of the number.

Take all the remainders starting from the bottom of the list constructed above.

0(10) =

0(2)


3. Convert to binary (base 2) the fractional part: 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 029 932 430 436 019 785 656 5.

Multiply it repeatedly by 2.

Keep track of each integer part of the results.

Stop when we get a fractional part that is equal to zero.


  • #) multiplying = integer + fractional part;
  • 1) 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 029 932 430 436 019 785 656 5 × 2 = 0 + 0.000 000 000 007 269 999 999 999 999 516 522 114 064 601 356 670 305 976 059 864 860 872 039 571 313;
  • 2) 0.000 000 000 007 269 999 999 999 999 516 522 114 064 601 356 670 305 976 059 864 860 872 039 571 313 × 2 = 0 + 0.000 000 000 014 539 999 999 999 999 033 044 228 129 202 713 340 611 952 119 729 721 744 079 142 626;
  • 3) 0.000 000 000 014 539 999 999 999 999 033 044 228 129 202 713 340 611 952 119 729 721 744 079 142 626 × 2 = 0 + 0.000 000 000 029 079 999 999 999 998 066 088 456 258 405 426 681 223 904 239 459 443 488 158 285 252;
  • 4) 0.000 000 000 029 079 999 999 999 998 066 088 456 258 405 426 681 223 904 239 459 443 488 158 285 252 × 2 = 0 + 0.000 000 000 058 159 999 999 999 996 132 176 912 516 810 853 362 447 808 478 918 886 976 316 570 504;
  • 5) 0.000 000 000 058 159 999 999 999 996 132 176 912 516 810 853 362 447 808 478 918 886 976 316 570 504 × 2 = 0 + 0.000 000 000 116 319 999 999 999 992 264 353 825 033 621 706 724 895 616 957 837 773 952 633 141 008;
  • 6) 0.000 000 000 116 319 999 999 999 992 264 353 825 033 621 706 724 895 616 957 837 773 952 633 141 008 × 2 = 0 + 0.000 000 000 232 639 999 999 999 984 528 707 650 067 243 413 449 791 233 915 675 547 905 266 282 016;
  • 7) 0.000 000 000 232 639 999 999 999 984 528 707 650 067 243 413 449 791 233 915 675 547 905 266 282 016 × 2 = 0 + 0.000 000 000 465 279 999 999 999 969 057 415 300 134 486 826 899 582 467 831 351 095 810 532 564 032;
  • 8) 0.000 000 000 465 279 999 999 999 969 057 415 300 134 486 826 899 582 467 831 351 095 810 532 564 032 × 2 = 0 + 0.000 000 000 930 559 999 999 999 938 114 830 600 268 973 653 799 164 935 662 702 191 621 065 128 064;
  • 9) 0.000 000 000 930 559 999 999 999 938 114 830 600 268 973 653 799 164 935 662 702 191 621 065 128 064 × 2 = 0 + 0.000 000 001 861 119 999 999 999 876 229 661 200 537 947 307 598 329 871 325 404 383 242 130 256 128;
  • 10) 0.000 000 001 861 119 999 999 999 876 229 661 200 537 947 307 598 329 871 325 404 383 242 130 256 128 × 2 = 0 + 0.000 000 003 722 239 999 999 999 752 459 322 401 075 894 615 196 659 742 650 808 766 484 260 512 256;
  • 11) 0.000 000 003 722 239 999 999 999 752 459 322 401 075 894 615 196 659 742 650 808 766 484 260 512 256 × 2 = 0 + 0.000 000 007 444 479 999 999 999 504 918 644 802 151 789 230 393 319 485 301 617 532 968 521 024 512;
  • 12) 0.000 000 007 444 479 999 999 999 504 918 644 802 151 789 230 393 319 485 301 617 532 968 521 024 512 × 2 = 0 + 0.000 000 014 888 959 999 999 999 009 837 289 604 303 578 460 786 638 970 603 235 065 937 042 049 024;
  • 13) 0.000 000 014 888 959 999 999 999 009 837 289 604 303 578 460 786 638 970 603 235 065 937 042 049 024 × 2 = 0 + 0.000 000 029 777 919 999 999 998 019 674 579 208 607 156 921 573 277 941 206 470 131 874 084 098 048;
  • 14) 0.000 000 029 777 919 999 999 998 019 674 579 208 607 156 921 573 277 941 206 470 131 874 084 098 048 × 2 = 0 + 0.000 000 059 555 839 999 999 996 039 349 158 417 214 313 843 146 555 882 412 940 263 748 168 196 096;
  • 15) 0.000 000 059 555 839 999 999 996 039 349 158 417 214 313 843 146 555 882 412 940 263 748 168 196 096 × 2 = 0 + 0.000 000 119 111 679 999 999 992 078 698 316 834 428 627 686 293 111 764 825 880 527 496 336 392 192;
  • 16) 0.000 000 119 111 679 999 999 992 078 698 316 834 428 627 686 293 111 764 825 880 527 496 336 392 192 × 2 = 0 + 0.000 000 238 223 359 999 999 984 157 396 633 668 857 255 372 586 223 529 651 761 054 992 672 784 384;
  • 17) 0.000 000 238 223 359 999 999 984 157 396 633 668 857 255 372 586 223 529 651 761 054 992 672 784 384 × 2 = 0 + 0.000 000 476 446 719 999 999 968 314 793 267 337 714 510 745 172 447 059 303 522 109 985 345 568 768;
  • 18) 0.000 000 476 446 719 999 999 968 314 793 267 337 714 510 745 172 447 059 303 522 109 985 345 568 768 × 2 = 0 + 0.000 000 952 893 439 999 999 936 629 586 534 675 429 021 490 344 894 118 607 044 219 970 691 137 536;
  • 19) 0.000 000 952 893 439 999 999 936 629 586 534 675 429 021 490 344 894 118 607 044 219 970 691 137 536 × 2 = 0 + 0.000 001 905 786 879 999 999 873 259 173 069 350 858 042 980 689 788 237 214 088 439 941 382 275 072;
  • 20) 0.000 001 905 786 879 999 999 873 259 173 069 350 858 042 980 689 788 237 214 088 439 941 382 275 072 × 2 = 0 + 0.000 003 811 573 759 999 999 746 518 346 138 701 716 085 961 379 576 474 428 176 879 882 764 550 144;
  • 21) 0.000 003 811 573 759 999 999 746 518 346 138 701 716 085 961 379 576 474 428 176 879 882 764 550 144 × 2 = 0 + 0.000 007 623 147 519 999 999 493 036 692 277 403 432 171 922 759 152 948 856 353 759 765 529 100 288;
  • 22) 0.000 007 623 147 519 999 999 493 036 692 277 403 432 171 922 759 152 948 856 353 759 765 529 100 288 × 2 = 0 + 0.000 015 246 295 039 999 998 986 073 384 554 806 864 343 845 518 305 897 712 707 519 531 058 200 576;
  • 23) 0.000 015 246 295 039 999 998 986 073 384 554 806 864 343 845 518 305 897 712 707 519 531 058 200 576 × 2 = 0 + 0.000 030 492 590 079 999 997 972 146 769 109 613 728 687 691 036 611 795 425 415 039 062 116 401 152;
  • 24) 0.000 030 492 590 079 999 997 972 146 769 109 613 728 687 691 036 611 795 425 415 039 062 116 401 152 × 2 = 0 + 0.000 060 985 180 159 999 995 944 293 538 219 227 457 375 382 073 223 590 850 830 078 124 232 802 304;
  • 25) 0.000 060 985 180 159 999 995 944 293 538 219 227 457 375 382 073 223 590 850 830 078 124 232 802 304 × 2 = 0 + 0.000 121 970 360 319 999 991 888 587 076 438 454 914 750 764 146 447 181 701 660 156 248 465 604 608;
  • 26) 0.000 121 970 360 319 999 991 888 587 076 438 454 914 750 764 146 447 181 701 660 156 248 465 604 608 × 2 = 0 + 0.000 243 940 720 639 999 983 777 174 152 876 909 829 501 528 292 894 363 403 320 312 496 931 209 216;
  • 27) 0.000 243 940 720 639 999 983 777 174 152 876 909 829 501 528 292 894 363 403 320 312 496 931 209 216 × 2 = 0 + 0.000 487 881 441 279 999 967 554 348 305 753 819 659 003 056 585 788 726 806 640 624 993 862 418 432;
  • 28) 0.000 487 881 441 279 999 967 554 348 305 753 819 659 003 056 585 788 726 806 640 624 993 862 418 432 × 2 = 0 + 0.000 975 762 882 559 999 935 108 696 611 507 639 318 006 113 171 577 453 613 281 249 987 724 836 864;
  • 29) 0.000 975 762 882 559 999 935 108 696 611 507 639 318 006 113 171 577 453 613 281 249 987 724 836 864 × 2 = 0 + 0.001 951 525 765 119 999 870 217 393 223 015 278 636 012 226 343 154 907 226 562 499 975 449 673 728;
  • 30) 0.001 951 525 765 119 999 870 217 393 223 015 278 636 012 226 343 154 907 226 562 499 975 449 673 728 × 2 = 0 + 0.003 903 051 530 239 999 740 434 786 446 030 557 272 024 452 686 309 814 453 124 999 950 899 347 456;
  • 31) 0.003 903 051 530 239 999 740 434 786 446 030 557 272 024 452 686 309 814 453 124 999 950 899 347 456 × 2 = 0 + 0.007 806 103 060 479 999 480 869 572 892 061 114 544 048 905 372 619 628 906 249 999 901 798 694 912;
  • 32) 0.007 806 103 060 479 999 480 869 572 892 061 114 544 048 905 372 619 628 906 249 999 901 798 694 912 × 2 = 0 + 0.015 612 206 120 959 998 961 739 145 784 122 229 088 097 810 745 239 257 812 499 999 803 597 389 824;
  • 33) 0.015 612 206 120 959 998 961 739 145 784 122 229 088 097 810 745 239 257 812 499 999 803 597 389 824 × 2 = 0 + 0.031 224 412 241 919 997 923 478 291 568 244 458 176 195 621 490 478 515 624 999 999 607 194 779 648;
  • 34) 0.031 224 412 241 919 997 923 478 291 568 244 458 176 195 621 490 478 515 624 999 999 607 194 779 648 × 2 = 0 + 0.062 448 824 483 839 995 846 956 583 136 488 916 352 391 242 980 957 031 249 999 999 214 389 559 296;
  • 35) 0.062 448 824 483 839 995 846 956 583 136 488 916 352 391 242 980 957 031 249 999 999 214 389 559 296 × 2 = 0 + 0.124 897 648 967 679 991 693 913 166 272 977 832 704 782 485 961 914 062 499 999 998 428 779 118 592;
  • 36) 0.124 897 648 967 679 991 693 913 166 272 977 832 704 782 485 961 914 062 499 999 998 428 779 118 592 × 2 = 0 + 0.249 795 297 935 359 983 387 826 332 545 955 665 409 564 971 923 828 124 999 999 996 857 558 237 184;
  • 37) 0.249 795 297 935 359 983 387 826 332 545 955 665 409 564 971 923 828 124 999 999 996 857 558 237 184 × 2 = 0 + 0.499 590 595 870 719 966 775 652 665 091 911 330 819 129 943 847 656 249 999 999 993 715 116 474 368;
  • 38) 0.499 590 595 870 719 966 775 652 665 091 911 330 819 129 943 847 656 249 999 999 993 715 116 474 368 × 2 = 0 + 0.999 181 191 741 439 933 551 305 330 183 822 661 638 259 887 695 312 499 999 999 987 430 232 948 736;
  • 39) 0.999 181 191 741 439 933 551 305 330 183 822 661 638 259 887 695 312 499 999 999 987 430 232 948 736 × 2 = 1 + 0.998 362 383 482 879 867 102 610 660 367 645 323 276 519 775 390 624 999 999 999 974 860 465 897 472;
  • 40) 0.998 362 383 482 879 867 102 610 660 367 645 323 276 519 775 390 624 999 999 999 974 860 465 897 472 × 2 = 1 + 0.996 724 766 965 759 734 205 221 320 735 290 646 553 039 550 781 249 999 999 999 949 720 931 794 944;
  • 41) 0.996 724 766 965 759 734 205 221 320 735 290 646 553 039 550 781 249 999 999 999 949 720 931 794 944 × 2 = 1 + 0.993 449 533 931 519 468 410 442 641 470 581 293 106 079 101 562 499 999 999 999 899 441 863 589 888;
  • 42) 0.993 449 533 931 519 468 410 442 641 470 581 293 106 079 101 562 499 999 999 999 899 441 863 589 888 × 2 = 1 + 0.986 899 067 863 038 936 820 885 282 941 162 586 212 158 203 124 999 999 999 999 798 883 727 179 776;
  • 43) 0.986 899 067 863 038 936 820 885 282 941 162 586 212 158 203 124 999 999 999 999 798 883 727 179 776 × 2 = 1 + 0.973 798 135 726 077 873 641 770 565 882 325 172 424 316 406 249 999 999 999 999 597 767 454 359 552;
  • 44) 0.973 798 135 726 077 873 641 770 565 882 325 172 424 316 406 249 999 999 999 999 597 767 454 359 552 × 2 = 1 + 0.947 596 271 452 155 747 283 541 131 764 650 344 848 632 812 499 999 999 999 999 195 534 908 719 104;
  • 45) 0.947 596 271 452 155 747 283 541 131 764 650 344 848 632 812 499 999 999 999 999 195 534 908 719 104 × 2 = 1 + 0.895 192 542 904 311 494 567 082 263 529 300 689 697 265 624 999 999 999 999 998 391 069 817 438 208;
  • 46) 0.895 192 542 904 311 494 567 082 263 529 300 689 697 265 624 999 999 999 999 998 391 069 817 438 208 × 2 = 1 + 0.790 385 085 808 622 989 134 164 527 058 601 379 394 531 249 999 999 999 999 996 782 139 634 876 416;
  • 47) 0.790 385 085 808 622 989 134 164 527 058 601 379 394 531 249 999 999 999 999 996 782 139 634 876 416 × 2 = 1 + 0.580 770 171 617 245 978 268 329 054 117 202 758 789 062 499 999 999 999 999 993 564 279 269 752 832;
  • 48) 0.580 770 171 617 245 978 268 329 054 117 202 758 789 062 499 999 999 999 999 993 564 279 269 752 832 × 2 = 1 + 0.161 540 343 234 491 956 536 658 108 234 405 517 578 124 999 999 999 999 999 987 128 558 539 505 664;
  • 49) 0.161 540 343 234 491 956 536 658 108 234 405 517 578 124 999 999 999 999 999 987 128 558 539 505 664 × 2 = 0 + 0.323 080 686 468 983 913 073 316 216 468 811 035 156 249 999 999 999 999 999 974 257 117 079 011 328;
  • 50) 0.323 080 686 468 983 913 073 316 216 468 811 035 156 249 999 999 999 999 999 974 257 117 079 011 328 × 2 = 0 + 0.646 161 372 937 967 826 146 632 432 937 622 070 312 499 999 999 999 999 999 948 514 234 158 022 656;
  • 51) 0.646 161 372 937 967 826 146 632 432 937 622 070 312 499 999 999 999 999 999 948 514 234 158 022 656 × 2 = 1 + 0.292 322 745 875 935 652 293 264 865 875 244 140 624 999 999 999 999 999 999 897 028 468 316 045 312;
  • 52) 0.292 322 745 875 935 652 293 264 865 875 244 140 624 999 999 999 999 999 999 897 028 468 316 045 312 × 2 = 0 + 0.584 645 491 751 871 304 586 529 731 750 488 281 249 999 999 999 999 999 999 794 056 936 632 090 624;
  • 53) 0.584 645 491 751 871 304 586 529 731 750 488 281 249 999 999 999 999 999 999 794 056 936 632 090 624 × 2 = 1 + 0.169 290 983 503 742 609 173 059 463 500 976 562 499 999 999 999 999 999 999 588 113 873 264 181 248;
  • 54) 0.169 290 983 503 742 609 173 059 463 500 976 562 499 999 999 999 999 999 999 588 113 873 264 181 248 × 2 = 0 + 0.338 581 967 007 485 218 346 118 927 001 953 124 999 999 999 999 999 999 999 176 227 746 528 362 496;
  • 55) 0.338 581 967 007 485 218 346 118 927 001 953 124 999 999 999 999 999 999 999 176 227 746 528 362 496 × 2 = 0 + 0.677 163 934 014 970 436 692 237 854 003 906 249 999 999 999 999 999 999 998 352 455 493 056 724 992;
  • 56) 0.677 163 934 014 970 436 692 237 854 003 906 249 999 999 999 999 999 999 998 352 455 493 056 724 992 × 2 = 1 + 0.354 327 868 029 940 873 384 475 708 007 812 499 999 999 999 999 999 999 996 704 910 986 113 449 984;
  • 57) 0.354 327 868 029 940 873 384 475 708 007 812 499 999 999 999 999 999 999 996 704 910 986 113 449 984 × 2 = 0 + 0.708 655 736 059 881 746 768 951 416 015 624 999 999 999 999 999 999 999 993 409 821 972 226 899 968;
  • 58) 0.708 655 736 059 881 746 768 951 416 015 624 999 999 999 999 999 999 999 993 409 821 972 226 899 968 × 2 = 1 + 0.417 311 472 119 763 493 537 902 832 031 249 999 999 999 999 999 999 999 986 819 643 944 453 799 936;
  • 59) 0.417 311 472 119 763 493 537 902 832 031 249 999 999 999 999 999 999 999 986 819 643 944 453 799 936 × 2 = 0 + 0.834 622 944 239 526 987 075 805 664 062 499 999 999 999 999 999 999 999 973 639 287 888 907 599 872;
  • 60) 0.834 622 944 239 526 987 075 805 664 062 499 999 999 999 999 999 999 999 973 639 287 888 907 599 872 × 2 = 1 + 0.669 245 888 479 053 974 151 611 328 124 999 999 999 999 999 999 999 999 947 278 575 777 815 199 744;
  • 61) 0.669 245 888 479 053 974 151 611 328 124 999 999 999 999 999 999 999 999 947 278 575 777 815 199 744 × 2 = 1 + 0.338 491 776 958 107 948 303 222 656 249 999 999 999 999 999 999 999 999 894 557 151 555 630 399 488;
  • 62) 0.338 491 776 958 107 948 303 222 656 249 999 999 999 999 999 999 999 999 894 557 151 555 630 399 488 × 2 = 0 + 0.676 983 553 916 215 896 606 445 312 499 999 999 999 999 999 999 999 999 789 114 303 111 260 798 976;
  • 63) 0.676 983 553 916 215 896 606 445 312 499 999 999 999 999 999 999 999 999 789 114 303 111 260 798 976 × 2 = 1 + 0.353 967 107 832 431 793 212 890 624 999 999 999 999 999 999 999 999 999 578 228 606 222 521 597 952;
  • 64) 0.353 967 107 832 431 793 212 890 624 999 999 999 999 999 999 999 999 999 578 228 606 222 521 597 952 × 2 = 0 + 0.707 934 215 664 863 586 425 781 249 999 999 999 999 999 999 999 999 999 156 457 212 445 043 195 904;
  • 65) 0.707 934 215 664 863 586 425 781 249 999 999 999 999 999 999 999 999 999 156 457 212 445 043 195 904 × 2 = 1 + 0.415 868 431 329 727 172 851 562 499 999 999 999 999 999 999 999 999 998 312 914 424 890 086 391 808;
  • 66) 0.415 868 431 329 727 172 851 562 499 999 999 999 999 999 999 999 999 998 312 914 424 890 086 391 808 × 2 = 0 + 0.831 736 862 659 454 345 703 124 999 999 999 999 999 999 999 999 999 996 625 828 849 780 172 783 616;
  • 67) 0.831 736 862 659 454 345 703 124 999 999 999 999 999 999 999 999 999 996 625 828 849 780 172 783 616 × 2 = 1 + 0.663 473 725 318 908 691 406 249 999 999 999 999 999 999 999 999 999 993 251 657 699 560 345 567 232;
  • 68) 0.663 473 725 318 908 691 406 249 999 999 999 999 999 999 999 999 999 993 251 657 699 560 345 567 232 × 2 = 1 + 0.326 947 450 637 817 382 812 499 999 999 999 999 999 999 999 999 999 986 503 315 399 120 691 134 464;
  • 69) 0.326 947 450 637 817 382 812 499 999 999 999 999 999 999 999 999 999 986 503 315 399 120 691 134 464 × 2 = 0 + 0.653 894 901 275 634 765 624 999 999 999 999 999 999 999 999 999 999 973 006 630 798 241 382 268 928;
  • 70) 0.653 894 901 275 634 765 624 999 999 999 999 999 999 999 999 999 999 973 006 630 798 241 382 268 928 × 2 = 1 + 0.307 789 802 551 269 531 249 999 999 999 999 999 999 999 999 999 999 946 013 261 596 482 764 537 856;
  • 71) 0.307 789 802 551 269 531 249 999 999 999 999 999 999 999 999 999 999 946 013 261 596 482 764 537 856 × 2 = 0 + 0.615 579 605 102 539 062 499 999 999 999 999 999 999 999 999 999 999 892 026 523 192 965 529 075 712;
  • 72) 0.615 579 605 102 539 062 499 999 999 999 999 999 999 999 999 999 999 892 026 523 192 965 529 075 712 × 2 = 1 + 0.231 159 210 205 078 124 999 999 999 999 999 999 999 999 999 999 999 784 053 046 385 931 058 151 424;
  • 73) 0.231 159 210 205 078 124 999 999 999 999 999 999 999 999 999 999 999 784 053 046 385 931 058 151 424 × 2 = 0 + 0.462 318 420 410 156 249 999 999 999 999 999 999 999 999 999 999 999 568 106 092 771 862 116 302 848;
  • 74) 0.462 318 420 410 156 249 999 999 999 999 999 999 999 999 999 999 999 568 106 092 771 862 116 302 848 × 2 = 0 + 0.924 636 840 820 312 499 999 999 999 999 999 999 999 999 999 999 999 136 212 185 543 724 232 605 696;
  • 75) 0.924 636 840 820 312 499 999 999 999 999 999 999 999 999 999 999 999 136 212 185 543 724 232 605 696 × 2 = 1 + 0.849 273 681 640 624 999 999 999 999 999 999 999 999 999 999 999 998 272 424 371 087 448 465 211 392;
  • 76) 0.849 273 681 640 624 999 999 999 999 999 999 999 999 999 999 999 998 272 424 371 087 448 465 211 392 × 2 = 1 + 0.698 547 363 281 249 999 999 999 999 999 999 999 999 999 999 999 996 544 848 742 174 896 930 422 784;
  • 77) 0.698 547 363 281 249 999 999 999 999 999 999 999 999 999 999 999 996 544 848 742 174 896 930 422 784 × 2 = 1 + 0.397 094 726 562 499 999 999 999 999 999 999 999 999 999 999 999 993 089 697 484 349 793 860 845 568;
  • 78) 0.397 094 726 562 499 999 999 999 999 999 999 999 999 999 999 999 993 089 697 484 349 793 860 845 568 × 2 = 0 + 0.794 189 453 124 999 999 999 999 999 999 999 999 999 999 999 999 986 179 394 968 699 587 721 691 136;
  • 79) 0.794 189 453 124 999 999 999 999 999 999 999 999 999 999 999 999 986 179 394 968 699 587 721 691 136 × 2 = 1 + 0.588 378 906 249 999 999 999 999 999 999 999 999 999 999 999 999 972 358 789 937 399 175 443 382 272;
  • 80) 0.588 378 906 249 999 999 999 999 999 999 999 999 999 999 999 999 972 358 789 937 399 175 443 382 272 × 2 = 1 + 0.176 757 812 499 999 999 999 999 999 999 999 999 999 999 999 999 944 717 579 874 798 350 886 764 544;
  • 81) 0.176 757 812 499 999 999 999 999 999 999 999 999 999 999 999 999 944 717 579 874 798 350 886 764 544 × 2 = 0 + 0.353 515 624 999 999 999 999 999 999 999 999 999 999 999 999 999 889 435 159 749 596 701 773 529 088;
  • 82) 0.353 515 624 999 999 999 999 999 999 999 999 999 999 999 999 999 889 435 159 749 596 701 773 529 088 × 2 = 0 + 0.707 031 249 999 999 999 999 999 999 999 999 999 999 999 999 999 778 870 319 499 193 403 547 058 176;
  • 83) 0.707 031 249 999 999 999 999 999 999 999 999 999 999 999 999 999 778 870 319 499 193 403 547 058 176 × 2 = 1 + 0.414 062 499 999 999 999 999 999 999 999 999 999 999 999 999 999 557 740 638 998 386 807 094 116 352;
  • 84) 0.414 062 499 999 999 999 999 999 999 999 999 999 999 999 999 999 557 740 638 998 386 807 094 116 352 × 2 = 0 + 0.828 124 999 999 999 999 999 999 999 999 999 999 999 999 999 999 115 481 277 996 773 614 188 232 704;
  • 85) 0.828 124 999 999 999 999 999 999 999 999 999 999 999 999 999 999 115 481 277 996 773 614 188 232 704 × 2 = 1 + 0.656 249 999 999 999 999 999 999 999 999 999 999 999 999 999 998 230 962 555 993 547 228 376 465 408;
  • 86) 0.656 249 999 999 999 999 999 999 999 999 999 999 999 999 999 998 230 962 555 993 547 228 376 465 408 × 2 = 1 + 0.312 499 999 999 999 999 999 999 999 999 999 999 999 999 999 996 461 925 111 987 094 456 752 930 816;
  • 87) 0.312 499 999 999 999 999 999 999 999 999 999 999 999 999 999 996 461 925 111 987 094 456 752 930 816 × 2 = 0 + 0.624 999 999 999 999 999 999 999 999 999 999 999 999 999 999 992 923 850 223 974 188 913 505 861 632;
  • 88) 0.624 999 999 999 999 999 999 999 999 999 999 999 999 999 999 992 923 850 223 974 188 913 505 861 632 × 2 = 1 + 0.249 999 999 999 999 999 999 999 999 999 999 999 999 999 999 985 847 700 447 948 377 827 011 723 264;
  • 89) 0.249 999 999 999 999 999 999 999 999 999 999 999 999 999 999 985 847 700 447 948 377 827 011 723 264 × 2 = 0 + 0.499 999 999 999 999 999 999 999 999 999 999 999 999 999 999 971 695 400 895 896 755 654 023 446 528;
  • 90) 0.499 999 999 999 999 999 999 999 999 999 999 999 999 999 999 971 695 400 895 896 755 654 023 446 528 × 2 = 0 + 0.999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 943 390 801 791 793 511 308 046 893 056;
  • 91) 0.999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 943 390 801 791 793 511 308 046 893 056 × 2 = 1 + 0.999 999 999 999 999 999 999 999 999 999 999 999 999 999 999 886 781 603 583 587 022 616 093 786 112;

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:


0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 029 932 430 436 019 785 656 5(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0011 1111 1111 0010 1001 0101 1010 1011 0101 0011 1011 0010 1101 001(2)

5. Positive number before normalization:

0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 029 932 430 436 019 785 656 5(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0011 1111 1111 0010 1001 0101 1010 1011 0101 0011 1011 0010 1101 001(2)

6. Normalize the binary representation of the number.

Shift the decimal mark 39 positions to the right, so that only one non zero digit remains to the left of it:


0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 029 932 430 436 019 785 656 5(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0011 1111 1111 0010 1001 0101 1010 1011 0101 0011 1011 0010 1101 001(2) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0011 1111 1111 0010 1001 0101 1010 1011 0101 0011 1011 0010 1101 001(2) × 20 =

1.1111 1111 1001 0100 1010 1101 0101 1010 1001 1101 1001 0110 1001(2) × 2-39


7. Up to this moment, there are the following elements that would feed into the 64 bit double precision IEEE 754 binary floating point representation:

Sign 0 (a positive number)

Exponent (unadjusted): -39

Mantissa (not normalized):
1.1111 1111 1001 0100 1010 1101 0101 1010 1001 1101 1001 0110 1001


8. Adjust the exponent.

Use the 11 bit excess/bias notation:


Exponent (adjusted) =

Exponent (unadjusted) + 2(11-1) - 1 =

-39 + 2(11-1) - 1 =

(-39 + 1 023)(10) =

984(10)


9. Convert the adjusted exponent from the decimal (base 10) to 11 bit binary.

Use the same technique of repeatedly dividing by 2:


  • division = quotient + remainder;
  • 984 ÷ 2 = 492 + 0;
  • 492 ÷ 2 = 246 + 0;
  • 246 ÷ 2 = 123 + 0;
  • 123 ÷ 2 = 61 + 1;
  • 61 ÷ 2 = 30 + 1;
  • 30 ÷ 2 = 15 + 0;
  • 15 ÷ 2 = 7 + 1;
  • 7 ÷ 2 = 3 + 1;
  • 3 ÷ 2 = 1 + 1;
  • 1 ÷ 2 = 0 + 1;

10. Construct the base 2 representation of the adjusted exponent.

Take all the remainders starting from the bottom of the list constructed above.


Exponent (adjusted) =

984(10) =

011 1101 1000(2)


11. Normalize the mantissa.

a) Remove the leading (the leftmost) bit, since it's allways 1, and the decimal point, if the case.

b) Adjust its length to 52 bits, only if necessary (not the case here).


Mantissa (normalized) =

1. 1111 1111 1001 0100 1010 1101 0101 1010 1001 1101 1001 0110 1001 =

1111 1111 1001 0100 1010 1101 0101 1010 1001 1101 1001 0110 1001


12. The three elements that make up the number's 64 bit double precision IEEE 754 binary floating point representation:

Sign (1 bit) =
0 (a positive number)

Exponent (11 bits) =
011 1101 1000

Mantissa (52 bits) =
1111 1111 1001 0100 1010 1101 0101 1010 1001 1101 1001 0110 1001


Decimal number 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 029 932 430 436 019 785 656 5 converted to 64 bit double precision IEEE 754 binary floating point representation:

0 - 011 1101 1000 - 1111 1111 1001 0100 1010 1101 0101 1010 1001 1101 1001 0110 1001

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How to convert numbers from the decimal system (base ten) to 64 bit double precision IEEE 754 binary floating point standard

Follow the steps below to convert a base 10 decimal number to 64 bit double precision IEEE 754 binary floating point:

  • 1. If the number to be converted is negative, start with its the positive version.
  • 2. First convert the integer part. Divide repeatedly by 2 the positive representation of the integer number that is to be converted to binary, until we get a quotient that is equal to zero, keeping track of each remainder.
  • 3. Construct the base 2 representation of the positive integer part of the number, by taking all the remainders from the previous operations, starting from the bottom of the list constructed above. Thus, the last remainder of the divisions becomes the first symbol (the leftmost) of the base two number, while the first remainder becomes the last symbol (the rightmost).
  • 4. Then convert the fractional part. Multiply the number repeatedly by 2, until we get a fractional part that is equal to zero, keeping track of each integer part of the results.
  • 5. Construct the base 2 representation of the fractional part of the number, by taking all the integer parts of the multiplying operations, starting from the top of the list constructed above (they should appear in the binary representation, from left to right, in the order they have been calculated).
  • 6. Normalize the binary representation of the number, shifting the decimal mark (the decimal point) "n" positions either to the left, or to the right, so that only one non zero digit remains to the left of the decimal mark.
  • 7. Adjust the exponent in 11 bit excess/bias notation and then convert it from decimal (base 10) to 11 bit binary, by using the same technique of repeatedly dividing by 2, as shown above:
    Exponent (adjusted) = Exponent (unadjusted) + 2(11-1) - 1
  • 8. Normalize mantissa, remove the leading (leftmost) bit, since it's allways '1' (and the decimal mark, if the case) and adjust its length to 52 bits, either by removing the excess bits from the right (losing precision...) or by adding extra bits set on '0' to the right.
  • 9. Sign (it takes 1 bit) is either 1 for a negative or 0 for a positive number.

Example: convert the negative number -31.640 215 from the decimal system (base ten) to 64 bit double precision IEEE 754 binary floating point:

  • 1. Start with the positive version of the number:

    |-31.640 215| = 31.640 215

  • 2. First convert the integer part, 31. Divide it repeatedly by 2, keeping track of each remainder, until we get a quotient that is equal to zero:
    • division = quotient + remainder;
    • 31 ÷ 2 = 15 + 1;
    • 15 ÷ 2 = 7 + 1;
    • 7 ÷ 2 = 3 + 1;
    • 3 ÷ 2 = 1 + 1;
    • 1 ÷ 2 = 0 + 1;
    • We have encountered a quotient that is ZERO => FULL STOP
  • 3. Construct the base 2 representation of the integer part of the number by taking all the remainders of the previous dividing operations, starting from the bottom of the list constructed above:

    31(10) = 1 1111(2)

  • 4. Then, convert the fractional part, 0.640 215. Multiply repeatedly by 2, keeping track of each integer part of the results, until we get a fractional part that is equal to zero:
    • #) multiplying = integer + fractional part;
    • 1) 0.640 215 × 2 = 1 + 0.280 43;
    • 2) 0.280 43 × 2 = 0 + 0.560 86;
    • 3) 0.560 86 × 2 = 1 + 0.121 72;
    • 4) 0.121 72 × 2 = 0 + 0.243 44;
    • 5) 0.243 44 × 2 = 0 + 0.486 88;
    • 6) 0.486 88 × 2 = 0 + 0.973 76;
    • 7) 0.973 76 × 2 = 1 + 0.947 52;
    • 8) 0.947 52 × 2 = 1 + 0.895 04;
    • 9) 0.895 04 × 2 = 1 + 0.790 08;
    • 10) 0.790 08 × 2 = 1 + 0.580 16;
    • 11) 0.580 16 × 2 = 1 + 0.160 32;
    • 12) 0.160 32 × 2 = 0 + 0.320 64;
    • 13) 0.320 64 × 2 = 0 + 0.641 28;
    • 14) 0.641 28 × 2 = 1 + 0.282 56;
    • 15) 0.282 56 × 2 = 0 + 0.565 12;
    • 16) 0.565 12 × 2 = 1 + 0.130 24;
    • 17) 0.130 24 × 2 = 0 + 0.260 48;
    • 18) 0.260 48 × 2 = 0 + 0.520 96;
    • 19) 0.520 96 × 2 = 1 + 0.041 92;
    • 20) 0.041 92 × 2 = 0 + 0.083 84;
    • 21) 0.083 84 × 2 = 0 + 0.167 68;
    • 22) 0.167 68 × 2 = 0 + 0.335 36;
    • 23) 0.335 36 × 2 = 0 + 0.670 72;
    • 24) 0.670 72 × 2 = 1 + 0.341 44;
    • 25) 0.341 44 × 2 = 0 + 0.682 88;
    • 26) 0.682 88 × 2 = 1 + 0.365 76;
    • 27) 0.365 76 × 2 = 0 + 0.731 52;
    • 28) 0.731 52 × 2 = 1 + 0.463 04;
    • 29) 0.463 04 × 2 = 0 + 0.926 08;
    • 30) 0.926 08 × 2 = 1 + 0.852 16;
    • 31) 0.852 16 × 2 = 1 + 0.704 32;
    • 32) 0.704 32 × 2 = 1 + 0.408 64;
    • 33) 0.408 64 × 2 = 0 + 0.817 28;
    • 34) 0.817 28 × 2 = 1 + 0.634 56;
    • 35) 0.634 56 × 2 = 1 + 0.269 12;
    • 36) 0.269 12 × 2 = 0 + 0.538 24;
    • 37) 0.538 24 × 2 = 1 + 0.076 48;
    • 38) 0.076 48 × 2 = 0 + 0.152 96;
    • 39) 0.152 96 × 2 = 0 + 0.305 92;
    • 40) 0.305 92 × 2 = 0 + 0.611 84;
    • 41) 0.611 84 × 2 = 1 + 0.223 68;
    • 42) 0.223 68 × 2 = 0 + 0.447 36;
    • 43) 0.447 36 × 2 = 0 + 0.894 72;
    • 44) 0.894 72 × 2 = 1 + 0.789 44;
    • 45) 0.789 44 × 2 = 1 + 0.578 88;
    • 46) 0.578 88 × 2 = 1 + 0.157 76;
    • 47) 0.157 76 × 2 = 0 + 0.315 52;
    • 48) 0.315 52 × 2 = 0 + 0.631 04;
    • 49) 0.631 04 × 2 = 1 + 0.262 08;
    • 50) 0.262 08 × 2 = 0 + 0.524 16;
    • 51) 0.524 16 × 2 = 1 + 0.048 32;
    • 52) 0.048 32 × 2 = 0 + 0.096 64;
    • 53) 0.096 64 × 2 = 0 + 0.193 28;
    • We didn't get any fractional part that was equal to zero. But we had enough iterations (over Mantissa limit = 52) and at least one integer part that was different from zero => FULL STOP (losing precision...).
  • 5. Construct the base 2 representation of the fractional part of the number, by taking all the integer parts of the previous multiplying operations, starting from the top of the constructed list above:

    0.640 215(10) = 0.1010 0011 1110 0101 0010 0001 0101 0111 0110 1000 1001 1100 1010 0(2)

  • 6. Summarizing - the positive number before normalization:

    31.640 215(10) = 1 1111.1010 0011 1110 0101 0010 0001 0101 0111 0110 1000 1001 1100 1010 0(2)

  • 7. Normalize the binary representation of the number, shifting the decimal mark 4 positions to the left so that only one non-zero digit stays to the left of the decimal mark:

    31.640 215(10) =
    1 1111.1010 0011 1110 0101 0010 0001 0101 0111 0110 1000 1001 1100 1010 0(2) =
    1 1111.1010 0011 1110 0101 0010 0001 0101 0111 0110 1000 1001 1100 1010 0(2) × 20 =
    1.1111 1010 0011 1110 0101 0010 0001 0101 0111 0110 1000 1001 1100 1010 0(2) × 24

  • 8. Up to this moment, there are the following elements that would feed into the 64 bit double precision IEEE 754 binary floating point representation:

    Sign: 1 (a negative number)

    Exponent (unadjusted): 4

    Mantissa (not-normalized): 1.1111 1010 0011 1110 0101 0010 0001 0101 0111 0110 1000 1001 1100 1010 0

  • 9. Adjust the exponent in 11 bit excess/bias notation and then convert it from decimal (base 10) to 11 bit binary (base 2), by using the same technique of repeatedly dividing it by 2, as shown above:

    Exponent (adjusted) = Exponent (unadjusted) + 2(11-1) - 1 = (4 + 1023)(10) = 1027(10) =
    100 0000 0011(2)

  • 10. Normalize mantissa, remove the leading (leftmost) bit, since it's allways '1' (and the decimal sign) and adjust its length to 52 bits, by removing the excess bits, from the right (losing precision...):

    Mantissa (not-normalized): 1.1111 1010 0011 1110 0101 0010 0001 0101 0111 0110 1000 1001 1100 1010 0

    Mantissa (normalized): 1111 1010 0011 1110 0101 0010 0001 0101 0111 0110 1000 1001 1100

  • Conclusion:

    Sign (1 bit) = 1 (a negative number)

    Exponent (8 bits) = 100 0000 0011

    Mantissa (52 bits) = 1111 1010 0011 1110 0101 0010 0001 0101 0111 0110 1000 1001 1100

  • Number -31.640 215, converted from decimal system (base 10) to 64 bit double precision IEEE 754 binary floating point =
    1 - 100 0000 0011 - 1111 1010 0011 1110 0101 0010 0001 0101 0111 0110 1000 1001 1100