0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 033 19 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 033 19(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
0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 033 19(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: 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 033 19.

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) 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 033 19 × 2 = 0 + 0.000 000 000 007 269 999 999 999 999 516 522 114 064 601 356 670 305 976 066 38;
  • 2) 0.000 000 000 007 269 999 999 999 999 516 522 114 064 601 356 670 305 976 066 38 × 2 = 0 + 0.000 000 000 014 539 999 999 999 999 033 044 228 129 202 713 340 611 952 132 76;
  • 3) 0.000 000 000 014 539 999 999 999 999 033 044 228 129 202 713 340 611 952 132 76 × 2 = 0 + 0.000 000 000 029 079 999 999 999 998 066 088 456 258 405 426 681 223 904 265 52;
  • 4) 0.000 000 000 029 079 999 999 999 998 066 088 456 258 405 426 681 223 904 265 52 × 2 = 0 + 0.000 000 000 058 159 999 999 999 996 132 176 912 516 810 853 362 447 808 531 04;
  • 5) 0.000 000 000 058 159 999 999 999 996 132 176 912 516 810 853 362 447 808 531 04 × 2 = 0 + 0.000 000 000 116 319 999 999 999 992 264 353 825 033 621 706 724 895 617 062 08;
  • 6) 0.000 000 000 116 319 999 999 999 992 264 353 825 033 621 706 724 895 617 062 08 × 2 = 0 + 0.000 000 000 232 639 999 999 999 984 528 707 650 067 243 413 449 791 234 124 16;
  • 7) 0.000 000 000 232 639 999 999 999 984 528 707 650 067 243 413 449 791 234 124 16 × 2 = 0 + 0.000 000 000 465 279 999 999 999 969 057 415 300 134 486 826 899 582 468 248 32;
  • 8) 0.000 000 000 465 279 999 999 999 969 057 415 300 134 486 826 899 582 468 248 32 × 2 = 0 + 0.000 000 000 930 559 999 999 999 938 114 830 600 268 973 653 799 164 936 496 64;
  • 9) 0.000 000 000 930 559 999 999 999 938 114 830 600 268 973 653 799 164 936 496 64 × 2 = 0 + 0.000 000 001 861 119 999 999 999 876 229 661 200 537 947 307 598 329 872 993 28;
  • 10) 0.000 000 001 861 119 999 999 999 876 229 661 200 537 947 307 598 329 872 993 28 × 2 = 0 + 0.000 000 003 722 239 999 999 999 752 459 322 401 075 894 615 196 659 745 986 56;
  • 11) 0.000 000 003 722 239 999 999 999 752 459 322 401 075 894 615 196 659 745 986 56 × 2 = 0 + 0.000 000 007 444 479 999 999 999 504 918 644 802 151 789 230 393 319 491 973 12;
  • 12) 0.000 000 007 444 479 999 999 999 504 918 644 802 151 789 230 393 319 491 973 12 × 2 = 0 + 0.000 000 014 888 959 999 999 999 009 837 289 604 303 578 460 786 638 983 946 24;
  • 13) 0.000 000 014 888 959 999 999 999 009 837 289 604 303 578 460 786 638 983 946 24 × 2 = 0 + 0.000 000 029 777 919 999 999 998 019 674 579 208 607 156 921 573 277 967 892 48;
  • 14) 0.000 000 029 777 919 999 999 998 019 674 579 208 607 156 921 573 277 967 892 48 × 2 = 0 + 0.000 000 059 555 839 999 999 996 039 349 158 417 214 313 843 146 555 935 784 96;
  • 15) 0.000 000 059 555 839 999 999 996 039 349 158 417 214 313 843 146 555 935 784 96 × 2 = 0 + 0.000 000 119 111 679 999 999 992 078 698 316 834 428 627 686 293 111 871 569 92;
  • 16) 0.000 000 119 111 679 999 999 992 078 698 316 834 428 627 686 293 111 871 569 92 × 2 = 0 + 0.000 000 238 223 359 999 999 984 157 396 633 668 857 255 372 586 223 743 139 84;
  • 17) 0.000 000 238 223 359 999 999 984 157 396 633 668 857 255 372 586 223 743 139 84 × 2 = 0 + 0.000 000 476 446 719 999 999 968 314 793 267 337 714 510 745 172 447 486 279 68;
  • 18) 0.000 000 476 446 719 999 999 968 314 793 267 337 714 510 745 172 447 486 279 68 × 2 = 0 + 0.000 000 952 893 439 999 999 936 629 586 534 675 429 021 490 344 894 972 559 36;
  • 19) 0.000 000 952 893 439 999 999 936 629 586 534 675 429 021 490 344 894 972 559 36 × 2 = 0 + 0.000 001 905 786 879 999 999 873 259 173 069 350 858 042 980 689 789 945 118 72;
  • 20) 0.000 001 905 786 879 999 999 873 259 173 069 350 858 042 980 689 789 945 118 72 × 2 = 0 + 0.000 003 811 573 759 999 999 746 518 346 138 701 716 085 961 379 579 890 237 44;
  • 21) 0.000 003 811 573 759 999 999 746 518 346 138 701 716 085 961 379 579 890 237 44 × 2 = 0 + 0.000 007 623 147 519 999 999 493 036 692 277 403 432 171 922 759 159 780 474 88;
  • 22) 0.000 007 623 147 519 999 999 493 036 692 277 403 432 171 922 759 159 780 474 88 × 2 = 0 + 0.000 015 246 295 039 999 998 986 073 384 554 806 864 343 845 518 319 560 949 76;
  • 23) 0.000 015 246 295 039 999 998 986 073 384 554 806 864 343 845 518 319 560 949 76 × 2 = 0 + 0.000 030 492 590 079 999 997 972 146 769 109 613 728 687 691 036 639 121 899 52;
  • 24) 0.000 030 492 590 079 999 997 972 146 769 109 613 728 687 691 036 639 121 899 52 × 2 = 0 + 0.000 060 985 180 159 999 995 944 293 538 219 227 457 375 382 073 278 243 799 04;
  • 25) 0.000 060 985 180 159 999 995 944 293 538 219 227 457 375 382 073 278 243 799 04 × 2 = 0 + 0.000 121 970 360 319 999 991 888 587 076 438 454 914 750 764 146 556 487 598 08;
  • 26) 0.000 121 970 360 319 999 991 888 587 076 438 454 914 750 764 146 556 487 598 08 × 2 = 0 + 0.000 243 940 720 639 999 983 777 174 152 876 909 829 501 528 293 112 975 196 16;
  • 27) 0.000 243 940 720 639 999 983 777 174 152 876 909 829 501 528 293 112 975 196 16 × 2 = 0 + 0.000 487 881 441 279 999 967 554 348 305 753 819 659 003 056 586 225 950 392 32;
  • 28) 0.000 487 881 441 279 999 967 554 348 305 753 819 659 003 056 586 225 950 392 32 × 2 = 0 + 0.000 975 762 882 559 999 935 108 696 611 507 639 318 006 113 172 451 900 784 64;
  • 29) 0.000 975 762 882 559 999 935 108 696 611 507 639 318 006 113 172 451 900 784 64 × 2 = 0 + 0.001 951 525 765 119 999 870 217 393 223 015 278 636 012 226 344 903 801 569 28;
  • 30) 0.001 951 525 765 119 999 870 217 393 223 015 278 636 012 226 344 903 801 569 28 × 2 = 0 + 0.003 903 051 530 239 999 740 434 786 446 030 557 272 024 452 689 807 603 138 56;
  • 31) 0.003 903 051 530 239 999 740 434 786 446 030 557 272 024 452 689 807 603 138 56 × 2 = 0 + 0.007 806 103 060 479 999 480 869 572 892 061 114 544 048 905 379 615 206 277 12;
  • 32) 0.007 806 103 060 479 999 480 869 572 892 061 114 544 048 905 379 615 206 277 12 × 2 = 0 + 0.015 612 206 120 959 998 961 739 145 784 122 229 088 097 810 759 230 412 554 24;
  • 33) 0.015 612 206 120 959 998 961 739 145 784 122 229 088 097 810 759 230 412 554 24 × 2 = 0 + 0.031 224 412 241 919 997 923 478 291 568 244 458 176 195 621 518 460 825 108 48;
  • 34) 0.031 224 412 241 919 997 923 478 291 568 244 458 176 195 621 518 460 825 108 48 × 2 = 0 + 0.062 448 824 483 839 995 846 956 583 136 488 916 352 391 243 036 921 650 216 96;
  • 35) 0.062 448 824 483 839 995 846 956 583 136 488 916 352 391 243 036 921 650 216 96 × 2 = 0 + 0.124 897 648 967 679 991 693 913 166 272 977 832 704 782 486 073 843 300 433 92;
  • 36) 0.124 897 648 967 679 991 693 913 166 272 977 832 704 782 486 073 843 300 433 92 × 2 = 0 + 0.249 795 297 935 359 983 387 826 332 545 955 665 409 564 972 147 686 600 867 84;
  • 37) 0.249 795 297 935 359 983 387 826 332 545 955 665 409 564 972 147 686 600 867 84 × 2 = 0 + 0.499 590 595 870 719 966 775 652 665 091 911 330 819 129 944 295 373 201 735 68;
  • 38) 0.499 590 595 870 719 966 775 652 665 091 911 330 819 129 944 295 373 201 735 68 × 2 = 0 + 0.999 181 191 741 439 933 551 305 330 183 822 661 638 259 888 590 746 403 471 36;
  • 39) 0.999 181 191 741 439 933 551 305 330 183 822 661 638 259 888 590 746 403 471 36 × 2 = 1 + 0.998 362 383 482 879 867 102 610 660 367 645 323 276 519 777 181 492 806 942 72;
  • 40) 0.998 362 383 482 879 867 102 610 660 367 645 323 276 519 777 181 492 806 942 72 × 2 = 1 + 0.996 724 766 965 759 734 205 221 320 735 290 646 553 039 554 362 985 613 885 44;
  • 41) 0.996 724 766 965 759 734 205 221 320 735 290 646 553 039 554 362 985 613 885 44 × 2 = 1 + 0.993 449 533 931 519 468 410 442 641 470 581 293 106 079 108 725 971 227 770 88;
  • 42) 0.993 449 533 931 519 468 410 442 641 470 581 293 106 079 108 725 971 227 770 88 × 2 = 1 + 0.986 899 067 863 038 936 820 885 282 941 162 586 212 158 217 451 942 455 541 76;
  • 43) 0.986 899 067 863 038 936 820 885 282 941 162 586 212 158 217 451 942 455 541 76 × 2 = 1 + 0.973 798 135 726 077 873 641 770 565 882 325 172 424 316 434 903 884 911 083 52;
  • 44) 0.973 798 135 726 077 873 641 770 565 882 325 172 424 316 434 903 884 911 083 52 × 2 = 1 + 0.947 596 271 452 155 747 283 541 131 764 650 344 848 632 869 807 769 822 167 04;
  • 45) 0.947 596 271 452 155 747 283 541 131 764 650 344 848 632 869 807 769 822 167 04 × 2 = 1 + 0.895 192 542 904 311 494 567 082 263 529 300 689 697 265 739 615 539 644 334 08;
  • 46) 0.895 192 542 904 311 494 567 082 263 529 300 689 697 265 739 615 539 644 334 08 × 2 = 1 + 0.790 385 085 808 622 989 134 164 527 058 601 379 394 531 479 231 079 288 668 16;
  • 47) 0.790 385 085 808 622 989 134 164 527 058 601 379 394 531 479 231 079 288 668 16 × 2 = 1 + 0.580 770 171 617 245 978 268 329 054 117 202 758 789 062 958 462 158 577 336 32;
  • 48) 0.580 770 171 617 245 978 268 329 054 117 202 758 789 062 958 462 158 577 336 32 × 2 = 1 + 0.161 540 343 234 491 956 536 658 108 234 405 517 578 125 916 924 317 154 672 64;
  • 49) 0.161 540 343 234 491 956 536 658 108 234 405 517 578 125 916 924 317 154 672 64 × 2 = 0 + 0.323 080 686 468 983 913 073 316 216 468 811 035 156 251 833 848 634 309 345 28;
  • 50) 0.323 080 686 468 983 913 073 316 216 468 811 035 156 251 833 848 634 309 345 28 × 2 = 0 + 0.646 161 372 937 967 826 146 632 432 937 622 070 312 503 667 697 268 618 690 56;
  • 51) 0.646 161 372 937 967 826 146 632 432 937 622 070 312 503 667 697 268 618 690 56 × 2 = 1 + 0.292 322 745 875 935 652 293 264 865 875 244 140 625 007 335 394 537 237 381 12;
  • 52) 0.292 322 745 875 935 652 293 264 865 875 244 140 625 007 335 394 537 237 381 12 × 2 = 0 + 0.584 645 491 751 871 304 586 529 731 750 488 281 250 014 670 789 074 474 762 24;
  • 53) 0.584 645 491 751 871 304 586 529 731 750 488 281 250 014 670 789 074 474 762 24 × 2 = 1 + 0.169 290 983 503 742 609 173 059 463 500 976 562 500 029 341 578 148 949 524 48;
  • 54) 0.169 290 983 503 742 609 173 059 463 500 976 562 500 029 341 578 148 949 524 48 × 2 = 0 + 0.338 581 967 007 485 218 346 118 927 001 953 125 000 058 683 156 297 899 048 96;
  • 55) 0.338 581 967 007 485 218 346 118 927 001 953 125 000 058 683 156 297 899 048 96 × 2 = 0 + 0.677 163 934 014 970 436 692 237 854 003 906 250 000 117 366 312 595 798 097 92;
  • 56) 0.677 163 934 014 970 436 692 237 854 003 906 250 000 117 366 312 595 798 097 92 × 2 = 1 + 0.354 327 868 029 940 873 384 475 708 007 812 500 000 234 732 625 191 596 195 84;
  • 57) 0.354 327 868 029 940 873 384 475 708 007 812 500 000 234 732 625 191 596 195 84 × 2 = 0 + 0.708 655 736 059 881 746 768 951 416 015 625 000 000 469 465 250 383 192 391 68;
  • 58) 0.708 655 736 059 881 746 768 951 416 015 625 000 000 469 465 250 383 192 391 68 × 2 = 1 + 0.417 311 472 119 763 493 537 902 832 031 250 000 000 938 930 500 766 384 783 36;
  • 59) 0.417 311 472 119 763 493 537 902 832 031 250 000 000 938 930 500 766 384 783 36 × 2 = 0 + 0.834 622 944 239 526 987 075 805 664 062 500 000 001 877 861 001 532 769 566 72;
  • 60) 0.834 622 944 239 526 987 075 805 664 062 500 000 001 877 861 001 532 769 566 72 × 2 = 1 + 0.669 245 888 479 053 974 151 611 328 125 000 000 003 755 722 003 065 539 133 44;
  • 61) 0.669 245 888 479 053 974 151 611 328 125 000 000 003 755 722 003 065 539 133 44 × 2 = 1 + 0.338 491 776 958 107 948 303 222 656 250 000 000 007 511 444 006 131 078 266 88;
  • 62) 0.338 491 776 958 107 948 303 222 656 250 000 000 007 511 444 006 131 078 266 88 × 2 = 0 + 0.676 983 553 916 215 896 606 445 312 500 000 000 015 022 888 012 262 156 533 76;
  • 63) 0.676 983 553 916 215 896 606 445 312 500 000 000 015 022 888 012 262 156 533 76 × 2 = 1 + 0.353 967 107 832 431 793 212 890 625 000 000 000 030 045 776 024 524 313 067 52;
  • 64) 0.353 967 107 832 431 793 212 890 625 000 000 000 030 045 776 024 524 313 067 52 × 2 = 0 + 0.707 934 215 664 863 586 425 781 250 000 000 000 060 091 552 049 048 626 135 04;
  • 65) 0.707 934 215 664 863 586 425 781 250 000 000 000 060 091 552 049 048 626 135 04 × 2 = 1 + 0.415 868 431 329 727 172 851 562 500 000 000 000 120 183 104 098 097 252 270 08;
  • 66) 0.415 868 431 329 727 172 851 562 500 000 000 000 120 183 104 098 097 252 270 08 × 2 = 0 + 0.831 736 862 659 454 345 703 125 000 000 000 000 240 366 208 196 194 504 540 16;
  • 67) 0.831 736 862 659 454 345 703 125 000 000 000 000 240 366 208 196 194 504 540 16 × 2 = 1 + 0.663 473 725 318 908 691 406 250 000 000 000 000 480 732 416 392 389 009 080 32;
  • 68) 0.663 473 725 318 908 691 406 250 000 000 000 000 480 732 416 392 389 009 080 32 × 2 = 1 + 0.326 947 450 637 817 382 812 500 000 000 000 000 961 464 832 784 778 018 160 64;
  • 69) 0.326 947 450 637 817 382 812 500 000 000 000 000 961 464 832 784 778 018 160 64 × 2 = 0 + 0.653 894 901 275 634 765 625 000 000 000 000 001 922 929 665 569 556 036 321 28;
  • 70) 0.653 894 901 275 634 765 625 000 000 000 000 001 922 929 665 569 556 036 321 28 × 2 = 1 + 0.307 789 802 551 269 531 250 000 000 000 000 003 845 859 331 139 112 072 642 56;
  • 71) 0.307 789 802 551 269 531 250 000 000 000 000 003 845 859 331 139 112 072 642 56 × 2 = 0 + 0.615 579 605 102 539 062 500 000 000 000 000 007 691 718 662 278 224 145 285 12;
  • 72) 0.615 579 605 102 539 062 500 000 000 000 000 007 691 718 662 278 224 145 285 12 × 2 = 1 + 0.231 159 210 205 078 125 000 000 000 000 000 015 383 437 324 556 448 290 570 24;
  • 73) 0.231 159 210 205 078 125 000 000 000 000 000 015 383 437 324 556 448 290 570 24 × 2 = 0 + 0.462 318 420 410 156 250 000 000 000 000 000 030 766 874 649 112 896 581 140 48;
  • 74) 0.462 318 420 410 156 250 000 000 000 000 000 030 766 874 649 112 896 581 140 48 × 2 = 0 + 0.924 636 840 820 312 500 000 000 000 000 000 061 533 749 298 225 793 162 280 96;
  • 75) 0.924 636 840 820 312 500 000 000 000 000 000 061 533 749 298 225 793 162 280 96 × 2 = 1 + 0.849 273 681 640 625 000 000 000 000 000 000 123 067 498 596 451 586 324 561 92;
  • 76) 0.849 273 681 640 625 000 000 000 000 000 000 123 067 498 596 451 586 324 561 92 × 2 = 1 + 0.698 547 363 281 250 000 000 000 000 000 000 246 134 997 192 903 172 649 123 84;
  • 77) 0.698 547 363 281 250 000 000 000 000 000 000 246 134 997 192 903 172 649 123 84 × 2 = 1 + 0.397 094 726 562 500 000 000 000 000 000 000 492 269 994 385 806 345 298 247 68;
  • 78) 0.397 094 726 562 500 000 000 000 000 000 000 492 269 994 385 806 345 298 247 68 × 2 = 0 + 0.794 189 453 125 000 000 000 000 000 000 000 984 539 988 771 612 690 596 495 36;
  • 79) 0.794 189 453 125 000 000 000 000 000 000 000 984 539 988 771 612 690 596 495 36 × 2 = 1 + 0.588 378 906 250 000 000 000 000 000 000 001 969 079 977 543 225 381 192 990 72;
  • 80) 0.588 378 906 250 000 000 000 000 000 000 001 969 079 977 543 225 381 192 990 72 × 2 = 1 + 0.176 757 812 500 000 000 000 000 000 000 003 938 159 955 086 450 762 385 981 44;
  • 81) 0.176 757 812 500 000 000 000 000 000 000 003 938 159 955 086 450 762 385 981 44 × 2 = 0 + 0.353 515 625 000 000 000 000 000 000 000 007 876 319 910 172 901 524 771 962 88;
  • 82) 0.353 515 625 000 000 000 000 000 000 000 007 876 319 910 172 901 524 771 962 88 × 2 = 0 + 0.707 031 250 000 000 000 000 000 000 000 015 752 639 820 345 803 049 543 925 76;
  • 83) 0.707 031 250 000 000 000 000 000 000 000 015 752 639 820 345 803 049 543 925 76 × 2 = 1 + 0.414 062 500 000 000 000 000 000 000 000 031 505 279 640 691 606 099 087 851 52;
  • 84) 0.414 062 500 000 000 000 000 000 000 000 031 505 279 640 691 606 099 087 851 52 × 2 = 0 + 0.828 125 000 000 000 000 000 000 000 000 063 010 559 281 383 212 198 175 703 04;
  • 85) 0.828 125 000 000 000 000 000 000 000 000 063 010 559 281 383 212 198 175 703 04 × 2 = 1 + 0.656 250 000 000 000 000 000 000 000 000 126 021 118 562 766 424 396 351 406 08;
  • 86) 0.656 250 000 000 000 000 000 000 000 000 126 021 118 562 766 424 396 351 406 08 × 2 = 1 + 0.312 500 000 000 000 000 000 000 000 000 252 042 237 125 532 848 792 702 812 16;
  • 87) 0.312 500 000 000 000 000 000 000 000 000 252 042 237 125 532 848 792 702 812 16 × 2 = 0 + 0.625 000 000 000 000 000 000 000 000 000 504 084 474 251 065 697 585 405 624 32;
  • 88) 0.625 000 000 000 000 000 000 000 000 000 504 084 474 251 065 697 585 405 624 32 × 2 = 1 + 0.250 000 000 000 000 000 000 000 000 001 008 168 948 502 131 395 170 811 248 64;
  • 89) 0.250 000 000 000 000 000 000 000 000 001 008 168 948 502 131 395 170 811 248 64 × 2 = 0 + 0.500 000 000 000 000 000 000 000 000 002 016 337 897 004 262 790 341 622 497 28;
  • 90) 0.500 000 000 000 000 000 000 000 000 002 016 337 897 004 262 790 341 622 497 28 × 2 = 1 + 0.000 000 000 000 000 000 000 000 000 004 032 675 794 008 525 580 683 244 994 56;
  • 91) 0.000 000 000 000 000 000 000 000 000 004 032 675 794 008 525 580 683 244 994 56 × 2 = 0 + 0.000 000 000 000 000 000 000 000 000 008 065 351 588 017 051 161 366 489 989 12;

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:


0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 033 19(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:

0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 033 19(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:


0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 033 19(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 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 033 19 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