0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 032 59 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 032 59(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
0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 032 59(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: 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 032 59.

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) 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 032 59 × 2 = 0 + 0.000 000 000 007 269 999 999 999 999 516 522 114 064 601 356 670 305 976 065 18;
  • 2) 0.000 000 000 007 269 999 999 999 999 516 522 114 064 601 356 670 305 976 065 18 × 2 = 0 + 0.000 000 000 014 539 999 999 999 999 033 044 228 129 202 713 340 611 952 130 36;
  • 3) 0.000 000 000 014 539 999 999 999 999 033 044 228 129 202 713 340 611 952 130 36 × 2 = 0 + 0.000 000 000 029 079 999 999 999 998 066 088 456 258 405 426 681 223 904 260 72;
  • 4) 0.000 000 000 029 079 999 999 999 998 066 088 456 258 405 426 681 223 904 260 72 × 2 = 0 + 0.000 000 000 058 159 999 999 999 996 132 176 912 516 810 853 362 447 808 521 44;
  • 5) 0.000 000 000 058 159 999 999 999 996 132 176 912 516 810 853 362 447 808 521 44 × 2 = 0 + 0.000 000 000 116 319 999 999 999 992 264 353 825 033 621 706 724 895 617 042 88;
  • 6) 0.000 000 000 116 319 999 999 999 992 264 353 825 033 621 706 724 895 617 042 88 × 2 = 0 + 0.000 000 000 232 639 999 999 999 984 528 707 650 067 243 413 449 791 234 085 76;
  • 7) 0.000 000 000 232 639 999 999 999 984 528 707 650 067 243 413 449 791 234 085 76 × 2 = 0 + 0.000 000 000 465 279 999 999 999 969 057 415 300 134 486 826 899 582 468 171 52;
  • 8) 0.000 000 000 465 279 999 999 999 969 057 415 300 134 486 826 899 582 468 171 52 × 2 = 0 + 0.000 000 000 930 559 999 999 999 938 114 830 600 268 973 653 799 164 936 343 04;
  • 9) 0.000 000 000 930 559 999 999 999 938 114 830 600 268 973 653 799 164 936 343 04 × 2 = 0 + 0.000 000 001 861 119 999 999 999 876 229 661 200 537 947 307 598 329 872 686 08;
  • 10) 0.000 000 001 861 119 999 999 999 876 229 661 200 537 947 307 598 329 872 686 08 × 2 = 0 + 0.000 000 003 722 239 999 999 999 752 459 322 401 075 894 615 196 659 745 372 16;
  • 11) 0.000 000 003 722 239 999 999 999 752 459 322 401 075 894 615 196 659 745 372 16 × 2 = 0 + 0.000 000 007 444 479 999 999 999 504 918 644 802 151 789 230 393 319 490 744 32;
  • 12) 0.000 000 007 444 479 999 999 999 504 918 644 802 151 789 230 393 319 490 744 32 × 2 = 0 + 0.000 000 014 888 959 999 999 999 009 837 289 604 303 578 460 786 638 981 488 64;
  • 13) 0.000 000 014 888 959 999 999 999 009 837 289 604 303 578 460 786 638 981 488 64 × 2 = 0 + 0.000 000 029 777 919 999 999 998 019 674 579 208 607 156 921 573 277 962 977 28;
  • 14) 0.000 000 029 777 919 999 999 998 019 674 579 208 607 156 921 573 277 962 977 28 × 2 = 0 + 0.000 000 059 555 839 999 999 996 039 349 158 417 214 313 843 146 555 925 954 56;
  • 15) 0.000 000 059 555 839 999 999 996 039 349 158 417 214 313 843 146 555 925 954 56 × 2 = 0 + 0.000 000 119 111 679 999 999 992 078 698 316 834 428 627 686 293 111 851 909 12;
  • 16) 0.000 000 119 111 679 999 999 992 078 698 316 834 428 627 686 293 111 851 909 12 × 2 = 0 + 0.000 000 238 223 359 999 999 984 157 396 633 668 857 255 372 586 223 703 818 24;
  • 17) 0.000 000 238 223 359 999 999 984 157 396 633 668 857 255 372 586 223 703 818 24 × 2 = 0 + 0.000 000 476 446 719 999 999 968 314 793 267 337 714 510 745 172 447 407 636 48;
  • 18) 0.000 000 476 446 719 999 999 968 314 793 267 337 714 510 745 172 447 407 636 48 × 2 = 0 + 0.000 000 952 893 439 999 999 936 629 586 534 675 429 021 490 344 894 815 272 96;
  • 19) 0.000 000 952 893 439 999 999 936 629 586 534 675 429 021 490 344 894 815 272 96 × 2 = 0 + 0.000 001 905 786 879 999 999 873 259 173 069 350 858 042 980 689 789 630 545 92;
  • 20) 0.000 001 905 786 879 999 999 873 259 173 069 350 858 042 980 689 789 630 545 92 × 2 = 0 + 0.000 003 811 573 759 999 999 746 518 346 138 701 716 085 961 379 579 261 091 84;
  • 21) 0.000 003 811 573 759 999 999 746 518 346 138 701 716 085 961 379 579 261 091 84 × 2 = 0 + 0.000 007 623 147 519 999 999 493 036 692 277 403 432 171 922 759 158 522 183 68;
  • 22) 0.000 007 623 147 519 999 999 493 036 692 277 403 432 171 922 759 158 522 183 68 × 2 = 0 + 0.000 015 246 295 039 999 998 986 073 384 554 806 864 343 845 518 317 044 367 36;
  • 23) 0.000 015 246 295 039 999 998 986 073 384 554 806 864 343 845 518 317 044 367 36 × 2 = 0 + 0.000 030 492 590 079 999 997 972 146 769 109 613 728 687 691 036 634 088 734 72;
  • 24) 0.000 030 492 590 079 999 997 972 146 769 109 613 728 687 691 036 634 088 734 72 × 2 = 0 + 0.000 060 985 180 159 999 995 944 293 538 219 227 457 375 382 073 268 177 469 44;
  • 25) 0.000 060 985 180 159 999 995 944 293 538 219 227 457 375 382 073 268 177 469 44 × 2 = 0 + 0.000 121 970 360 319 999 991 888 587 076 438 454 914 750 764 146 536 354 938 88;
  • 26) 0.000 121 970 360 319 999 991 888 587 076 438 454 914 750 764 146 536 354 938 88 × 2 = 0 + 0.000 243 940 720 639 999 983 777 174 152 876 909 829 501 528 293 072 709 877 76;
  • 27) 0.000 243 940 720 639 999 983 777 174 152 876 909 829 501 528 293 072 709 877 76 × 2 = 0 + 0.000 487 881 441 279 999 967 554 348 305 753 819 659 003 056 586 145 419 755 52;
  • 28) 0.000 487 881 441 279 999 967 554 348 305 753 819 659 003 056 586 145 419 755 52 × 2 = 0 + 0.000 975 762 882 559 999 935 108 696 611 507 639 318 006 113 172 290 839 511 04;
  • 29) 0.000 975 762 882 559 999 935 108 696 611 507 639 318 006 113 172 290 839 511 04 × 2 = 0 + 0.001 951 525 765 119 999 870 217 393 223 015 278 636 012 226 344 581 679 022 08;
  • 30) 0.001 951 525 765 119 999 870 217 393 223 015 278 636 012 226 344 581 679 022 08 × 2 = 0 + 0.003 903 051 530 239 999 740 434 786 446 030 557 272 024 452 689 163 358 044 16;
  • 31) 0.003 903 051 530 239 999 740 434 786 446 030 557 272 024 452 689 163 358 044 16 × 2 = 0 + 0.007 806 103 060 479 999 480 869 572 892 061 114 544 048 905 378 326 716 088 32;
  • 32) 0.007 806 103 060 479 999 480 869 572 892 061 114 544 048 905 378 326 716 088 32 × 2 = 0 + 0.015 612 206 120 959 998 961 739 145 784 122 229 088 097 810 756 653 432 176 64;
  • 33) 0.015 612 206 120 959 998 961 739 145 784 122 229 088 097 810 756 653 432 176 64 × 2 = 0 + 0.031 224 412 241 919 997 923 478 291 568 244 458 176 195 621 513 306 864 353 28;
  • 34) 0.031 224 412 241 919 997 923 478 291 568 244 458 176 195 621 513 306 864 353 28 × 2 = 0 + 0.062 448 824 483 839 995 846 956 583 136 488 916 352 391 243 026 613 728 706 56;
  • 35) 0.062 448 824 483 839 995 846 956 583 136 488 916 352 391 243 026 613 728 706 56 × 2 = 0 + 0.124 897 648 967 679 991 693 913 166 272 977 832 704 782 486 053 227 457 413 12;
  • 36) 0.124 897 648 967 679 991 693 913 166 272 977 832 704 782 486 053 227 457 413 12 × 2 = 0 + 0.249 795 297 935 359 983 387 826 332 545 955 665 409 564 972 106 454 914 826 24;
  • 37) 0.249 795 297 935 359 983 387 826 332 545 955 665 409 564 972 106 454 914 826 24 × 2 = 0 + 0.499 590 595 870 719 966 775 652 665 091 911 330 819 129 944 212 909 829 652 48;
  • 38) 0.499 590 595 870 719 966 775 652 665 091 911 330 819 129 944 212 909 829 652 48 × 2 = 0 + 0.999 181 191 741 439 933 551 305 330 183 822 661 638 259 888 425 819 659 304 96;
  • 39) 0.999 181 191 741 439 933 551 305 330 183 822 661 638 259 888 425 819 659 304 96 × 2 = 1 + 0.998 362 383 482 879 867 102 610 660 367 645 323 276 519 776 851 639 318 609 92;
  • 40) 0.998 362 383 482 879 867 102 610 660 367 645 323 276 519 776 851 639 318 609 92 × 2 = 1 + 0.996 724 766 965 759 734 205 221 320 735 290 646 553 039 553 703 278 637 219 84;
  • 41) 0.996 724 766 965 759 734 205 221 320 735 290 646 553 039 553 703 278 637 219 84 × 2 = 1 + 0.993 449 533 931 519 468 410 442 641 470 581 293 106 079 107 406 557 274 439 68;
  • 42) 0.993 449 533 931 519 468 410 442 641 470 581 293 106 079 107 406 557 274 439 68 × 2 = 1 + 0.986 899 067 863 038 936 820 885 282 941 162 586 212 158 214 813 114 548 879 36;
  • 43) 0.986 899 067 863 038 936 820 885 282 941 162 586 212 158 214 813 114 548 879 36 × 2 = 1 + 0.973 798 135 726 077 873 641 770 565 882 325 172 424 316 429 626 229 097 758 72;
  • 44) 0.973 798 135 726 077 873 641 770 565 882 325 172 424 316 429 626 229 097 758 72 × 2 = 1 + 0.947 596 271 452 155 747 283 541 131 764 650 344 848 632 859 252 458 195 517 44;
  • 45) 0.947 596 271 452 155 747 283 541 131 764 650 344 848 632 859 252 458 195 517 44 × 2 = 1 + 0.895 192 542 904 311 494 567 082 263 529 300 689 697 265 718 504 916 391 034 88;
  • 46) 0.895 192 542 904 311 494 567 082 263 529 300 689 697 265 718 504 916 391 034 88 × 2 = 1 + 0.790 385 085 808 622 989 134 164 527 058 601 379 394 531 437 009 832 782 069 76;
  • 47) 0.790 385 085 808 622 989 134 164 527 058 601 379 394 531 437 009 832 782 069 76 × 2 = 1 + 0.580 770 171 617 245 978 268 329 054 117 202 758 789 062 874 019 665 564 139 52;
  • 48) 0.580 770 171 617 245 978 268 329 054 117 202 758 789 062 874 019 665 564 139 52 × 2 = 1 + 0.161 540 343 234 491 956 536 658 108 234 405 517 578 125 748 039 331 128 279 04;
  • 49) 0.161 540 343 234 491 956 536 658 108 234 405 517 578 125 748 039 331 128 279 04 × 2 = 0 + 0.323 080 686 468 983 913 073 316 216 468 811 035 156 251 496 078 662 256 558 08;
  • 50) 0.323 080 686 468 983 913 073 316 216 468 811 035 156 251 496 078 662 256 558 08 × 2 = 0 + 0.646 161 372 937 967 826 146 632 432 937 622 070 312 502 992 157 324 513 116 16;
  • 51) 0.646 161 372 937 967 826 146 632 432 937 622 070 312 502 992 157 324 513 116 16 × 2 = 1 + 0.292 322 745 875 935 652 293 264 865 875 244 140 625 005 984 314 649 026 232 32;
  • 52) 0.292 322 745 875 935 652 293 264 865 875 244 140 625 005 984 314 649 026 232 32 × 2 = 0 + 0.584 645 491 751 871 304 586 529 731 750 488 281 250 011 968 629 298 052 464 64;
  • 53) 0.584 645 491 751 871 304 586 529 731 750 488 281 250 011 968 629 298 052 464 64 × 2 = 1 + 0.169 290 983 503 742 609 173 059 463 500 976 562 500 023 937 258 596 104 929 28;
  • 54) 0.169 290 983 503 742 609 173 059 463 500 976 562 500 023 937 258 596 104 929 28 × 2 = 0 + 0.338 581 967 007 485 218 346 118 927 001 953 125 000 047 874 517 192 209 858 56;
  • 55) 0.338 581 967 007 485 218 346 118 927 001 953 125 000 047 874 517 192 209 858 56 × 2 = 0 + 0.677 163 934 014 970 436 692 237 854 003 906 250 000 095 749 034 384 419 717 12;
  • 56) 0.677 163 934 014 970 436 692 237 854 003 906 250 000 095 749 034 384 419 717 12 × 2 = 1 + 0.354 327 868 029 940 873 384 475 708 007 812 500 000 191 498 068 768 839 434 24;
  • 57) 0.354 327 868 029 940 873 384 475 708 007 812 500 000 191 498 068 768 839 434 24 × 2 = 0 + 0.708 655 736 059 881 746 768 951 416 015 625 000 000 382 996 137 537 678 868 48;
  • 58) 0.708 655 736 059 881 746 768 951 416 015 625 000 000 382 996 137 537 678 868 48 × 2 = 1 + 0.417 311 472 119 763 493 537 902 832 031 250 000 000 765 992 275 075 357 736 96;
  • 59) 0.417 311 472 119 763 493 537 902 832 031 250 000 000 765 992 275 075 357 736 96 × 2 = 0 + 0.834 622 944 239 526 987 075 805 664 062 500 000 001 531 984 550 150 715 473 92;
  • 60) 0.834 622 944 239 526 987 075 805 664 062 500 000 001 531 984 550 150 715 473 92 × 2 = 1 + 0.669 245 888 479 053 974 151 611 328 125 000 000 003 063 969 100 301 430 947 84;
  • 61) 0.669 245 888 479 053 974 151 611 328 125 000 000 003 063 969 100 301 430 947 84 × 2 = 1 + 0.338 491 776 958 107 948 303 222 656 250 000 000 006 127 938 200 602 861 895 68;
  • 62) 0.338 491 776 958 107 948 303 222 656 250 000 000 006 127 938 200 602 861 895 68 × 2 = 0 + 0.676 983 553 916 215 896 606 445 312 500 000 000 012 255 876 401 205 723 791 36;
  • 63) 0.676 983 553 916 215 896 606 445 312 500 000 000 012 255 876 401 205 723 791 36 × 2 = 1 + 0.353 967 107 832 431 793 212 890 625 000 000 000 024 511 752 802 411 447 582 72;
  • 64) 0.353 967 107 832 431 793 212 890 625 000 000 000 024 511 752 802 411 447 582 72 × 2 = 0 + 0.707 934 215 664 863 586 425 781 250 000 000 000 049 023 505 604 822 895 165 44;
  • 65) 0.707 934 215 664 863 586 425 781 250 000 000 000 049 023 505 604 822 895 165 44 × 2 = 1 + 0.415 868 431 329 727 172 851 562 500 000 000 000 098 047 011 209 645 790 330 88;
  • 66) 0.415 868 431 329 727 172 851 562 500 000 000 000 098 047 011 209 645 790 330 88 × 2 = 0 + 0.831 736 862 659 454 345 703 125 000 000 000 000 196 094 022 419 291 580 661 76;
  • 67) 0.831 736 862 659 454 345 703 125 000 000 000 000 196 094 022 419 291 580 661 76 × 2 = 1 + 0.663 473 725 318 908 691 406 250 000 000 000 000 392 188 044 838 583 161 323 52;
  • 68) 0.663 473 725 318 908 691 406 250 000 000 000 000 392 188 044 838 583 161 323 52 × 2 = 1 + 0.326 947 450 637 817 382 812 500 000 000 000 000 784 376 089 677 166 322 647 04;
  • 69) 0.326 947 450 637 817 382 812 500 000 000 000 000 784 376 089 677 166 322 647 04 × 2 = 0 + 0.653 894 901 275 634 765 625 000 000 000 000 001 568 752 179 354 332 645 294 08;
  • 70) 0.653 894 901 275 634 765 625 000 000 000 000 001 568 752 179 354 332 645 294 08 × 2 = 1 + 0.307 789 802 551 269 531 250 000 000 000 000 003 137 504 358 708 665 290 588 16;
  • 71) 0.307 789 802 551 269 531 250 000 000 000 000 003 137 504 358 708 665 290 588 16 × 2 = 0 + 0.615 579 605 102 539 062 500 000 000 000 000 006 275 008 717 417 330 581 176 32;
  • 72) 0.615 579 605 102 539 062 500 000 000 000 000 006 275 008 717 417 330 581 176 32 × 2 = 1 + 0.231 159 210 205 078 125 000 000 000 000 000 012 550 017 434 834 661 162 352 64;
  • 73) 0.231 159 210 205 078 125 000 000 000 000 000 012 550 017 434 834 661 162 352 64 × 2 = 0 + 0.462 318 420 410 156 250 000 000 000 000 000 025 100 034 869 669 322 324 705 28;
  • 74) 0.462 318 420 410 156 250 000 000 000 000 000 025 100 034 869 669 322 324 705 28 × 2 = 0 + 0.924 636 840 820 312 500 000 000 000 000 000 050 200 069 739 338 644 649 410 56;
  • 75) 0.924 636 840 820 312 500 000 000 000 000 000 050 200 069 739 338 644 649 410 56 × 2 = 1 + 0.849 273 681 640 625 000 000 000 000 000 000 100 400 139 478 677 289 298 821 12;
  • 76) 0.849 273 681 640 625 000 000 000 000 000 000 100 400 139 478 677 289 298 821 12 × 2 = 1 + 0.698 547 363 281 250 000 000 000 000 000 000 200 800 278 957 354 578 597 642 24;
  • 77) 0.698 547 363 281 250 000 000 000 000 000 000 200 800 278 957 354 578 597 642 24 × 2 = 1 + 0.397 094 726 562 500 000 000 000 000 000 000 401 600 557 914 709 157 195 284 48;
  • 78) 0.397 094 726 562 500 000 000 000 000 000 000 401 600 557 914 709 157 195 284 48 × 2 = 0 + 0.794 189 453 125 000 000 000 000 000 000 000 803 201 115 829 418 314 390 568 96;
  • 79) 0.794 189 453 125 000 000 000 000 000 000 000 803 201 115 829 418 314 390 568 96 × 2 = 1 + 0.588 378 906 250 000 000 000 000 000 000 001 606 402 231 658 836 628 781 137 92;
  • 80) 0.588 378 906 250 000 000 000 000 000 000 001 606 402 231 658 836 628 781 137 92 × 2 = 1 + 0.176 757 812 500 000 000 000 000 000 000 003 212 804 463 317 673 257 562 275 84;
  • 81) 0.176 757 812 500 000 000 000 000 000 000 003 212 804 463 317 673 257 562 275 84 × 2 = 0 + 0.353 515 625 000 000 000 000 000 000 000 006 425 608 926 635 346 515 124 551 68;
  • 82) 0.353 515 625 000 000 000 000 000 000 000 006 425 608 926 635 346 515 124 551 68 × 2 = 0 + 0.707 031 250 000 000 000 000 000 000 000 012 851 217 853 270 693 030 249 103 36;
  • 83) 0.707 031 250 000 000 000 000 000 000 000 012 851 217 853 270 693 030 249 103 36 × 2 = 1 + 0.414 062 500 000 000 000 000 000 000 000 025 702 435 706 541 386 060 498 206 72;
  • 84) 0.414 062 500 000 000 000 000 000 000 000 025 702 435 706 541 386 060 498 206 72 × 2 = 0 + 0.828 125 000 000 000 000 000 000 000 000 051 404 871 413 082 772 120 996 413 44;
  • 85) 0.828 125 000 000 000 000 000 000 000 000 051 404 871 413 082 772 120 996 413 44 × 2 = 1 + 0.656 250 000 000 000 000 000 000 000 000 102 809 742 826 165 544 241 992 826 88;
  • 86) 0.656 250 000 000 000 000 000 000 000 000 102 809 742 826 165 544 241 992 826 88 × 2 = 1 + 0.312 500 000 000 000 000 000 000 000 000 205 619 485 652 331 088 483 985 653 76;
  • 87) 0.312 500 000 000 000 000 000 000 000 000 205 619 485 652 331 088 483 985 653 76 × 2 = 0 + 0.625 000 000 000 000 000 000 000 000 000 411 238 971 304 662 176 967 971 307 52;
  • 88) 0.625 000 000 000 000 000 000 000 000 000 411 238 971 304 662 176 967 971 307 52 × 2 = 1 + 0.250 000 000 000 000 000 000 000 000 000 822 477 942 609 324 353 935 942 615 04;
  • 89) 0.250 000 000 000 000 000 000 000 000 000 822 477 942 609 324 353 935 942 615 04 × 2 = 0 + 0.500 000 000 000 000 000 000 000 000 001 644 955 885 218 648 707 871 885 230 08;
  • 90) 0.500 000 000 000 000 000 000 000 000 001 644 955 885 218 648 707 871 885 230 08 × 2 = 1 + 0.000 000 000 000 000 000 000 000 000 003 289 911 770 437 297 415 743 770 460 16;
  • 91) 0.000 000 000 000 000 000 000 000 000 003 289 911 770 437 297 415 743 770 460 16 × 2 = 0 + 0.000 000 000 000 000 000 000 000 000 006 579 823 540 874 594 831 487 540 920 32;

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:


0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 032 59(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:

0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 032 59(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:


0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 032 59(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 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 032 59 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