0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 031 7 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 031 7(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
0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 031 7(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: 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 031 7.

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) 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 031 7 × 2 = 0 + 0.000 000 000 007 269 999 999 999 999 516 522 114 064 601 356 670 305 976 063 4;
  • 2) 0.000 000 000 007 269 999 999 999 999 516 522 114 064 601 356 670 305 976 063 4 × 2 = 0 + 0.000 000 000 014 539 999 999 999 999 033 044 228 129 202 713 340 611 952 126 8;
  • 3) 0.000 000 000 014 539 999 999 999 999 033 044 228 129 202 713 340 611 952 126 8 × 2 = 0 + 0.000 000 000 029 079 999 999 999 998 066 088 456 258 405 426 681 223 904 253 6;
  • 4) 0.000 000 000 029 079 999 999 999 998 066 088 456 258 405 426 681 223 904 253 6 × 2 = 0 + 0.000 000 000 058 159 999 999 999 996 132 176 912 516 810 853 362 447 808 507 2;
  • 5) 0.000 000 000 058 159 999 999 999 996 132 176 912 516 810 853 362 447 808 507 2 × 2 = 0 + 0.000 000 000 116 319 999 999 999 992 264 353 825 033 621 706 724 895 617 014 4;
  • 6) 0.000 000 000 116 319 999 999 999 992 264 353 825 033 621 706 724 895 617 014 4 × 2 = 0 + 0.000 000 000 232 639 999 999 999 984 528 707 650 067 243 413 449 791 234 028 8;
  • 7) 0.000 000 000 232 639 999 999 999 984 528 707 650 067 243 413 449 791 234 028 8 × 2 = 0 + 0.000 000 000 465 279 999 999 999 969 057 415 300 134 486 826 899 582 468 057 6;
  • 8) 0.000 000 000 465 279 999 999 999 969 057 415 300 134 486 826 899 582 468 057 6 × 2 = 0 + 0.000 000 000 930 559 999 999 999 938 114 830 600 268 973 653 799 164 936 115 2;
  • 9) 0.000 000 000 930 559 999 999 999 938 114 830 600 268 973 653 799 164 936 115 2 × 2 = 0 + 0.000 000 001 861 119 999 999 999 876 229 661 200 537 947 307 598 329 872 230 4;
  • 10) 0.000 000 001 861 119 999 999 999 876 229 661 200 537 947 307 598 329 872 230 4 × 2 = 0 + 0.000 000 003 722 239 999 999 999 752 459 322 401 075 894 615 196 659 744 460 8;
  • 11) 0.000 000 003 722 239 999 999 999 752 459 322 401 075 894 615 196 659 744 460 8 × 2 = 0 + 0.000 000 007 444 479 999 999 999 504 918 644 802 151 789 230 393 319 488 921 6;
  • 12) 0.000 000 007 444 479 999 999 999 504 918 644 802 151 789 230 393 319 488 921 6 × 2 = 0 + 0.000 000 014 888 959 999 999 999 009 837 289 604 303 578 460 786 638 977 843 2;
  • 13) 0.000 000 014 888 959 999 999 999 009 837 289 604 303 578 460 786 638 977 843 2 × 2 = 0 + 0.000 000 029 777 919 999 999 998 019 674 579 208 607 156 921 573 277 955 686 4;
  • 14) 0.000 000 029 777 919 999 999 998 019 674 579 208 607 156 921 573 277 955 686 4 × 2 = 0 + 0.000 000 059 555 839 999 999 996 039 349 158 417 214 313 843 146 555 911 372 8;
  • 15) 0.000 000 059 555 839 999 999 996 039 349 158 417 214 313 843 146 555 911 372 8 × 2 = 0 + 0.000 000 119 111 679 999 999 992 078 698 316 834 428 627 686 293 111 822 745 6;
  • 16) 0.000 000 119 111 679 999 999 992 078 698 316 834 428 627 686 293 111 822 745 6 × 2 = 0 + 0.000 000 238 223 359 999 999 984 157 396 633 668 857 255 372 586 223 645 491 2;
  • 17) 0.000 000 238 223 359 999 999 984 157 396 633 668 857 255 372 586 223 645 491 2 × 2 = 0 + 0.000 000 476 446 719 999 999 968 314 793 267 337 714 510 745 172 447 290 982 4;
  • 18) 0.000 000 476 446 719 999 999 968 314 793 267 337 714 510 745 172 447 290 982 4 × 2 = 0 + 0.000 000 952 893 439 999 999 936 629 586 534 675 429 021 490 344 894 581 964 8;
  • 19) 0.000 000 952 893 439 999 999 936 629 586 534 675 429 021 490 344 894 581 964 8 × 2 = 0 + 0.000 001 905 786 879 999 999 873 259 173 069 350 858 042 980 689 789 163 929 6;
  • 20) 0.000 001 905 786 879 999 999 873 259 173 069 350 858 042 980 689 789 163 929 6 × 2 = 0 + 0.000 003 811 573 759 999 999 746 518 346 138 701 716 085 961 379 578 327 859 2;
  • 21) 0.000 003 811 573 759 999 999 746 518 346 138 701 716 085 961 379 578 327 859 2 × 2 = 0 + 0.000 007 623 147 519 999 999 493 036 692 277 403 432 171 922 759 156 655 718 4;
  • 22) 0.000 007 623 147 519 999 999 493 036 692 277 403 432 171 922 759 156 655 718 4 × 2 = 0 + 0.000 015 246 295 039 999 998 986 073 384 554 806 864 343 845 518 313 311 436 8;
  • 23) 0.000 015 246 295 039 999 998 986 073 384 554 806 864 343 845 518 313 311 436 8 × 2 = 0 + 0.000 030 492 590 079 999 997 972 146 769 109 613 728 687 691 036 626 622 873 6;
  • 24) 0.000 030 492 590 079 999 997 972 146 769 109 613 728 687 691 036 626 622 873 6 × 2 = 0 + 0.000 060 985 180 159 999 995 944 293 538 219 227 457 375 382 073 253 245 747 2;
  • 25) 0.000 060 985 180 159 999 995 944 293 538 219 227 457 375 382 073 253 245 747 2 × 2 = 0 + 0.000 121 970 360 319 999 991 888 587 076 438 454 914 750 764 146 506 491 494 4;
  • 26) 0.000 121 970 360 319 999 991 888 587 076 438 454 914 750 764 146 506 491 494 4 × 2 = 0 + 0.000 243 940 720 639 999 983 777 174 152 876 909 829 501 528 293 012 982 988 8;
  • 27) 0.000 243 940 720 639 999 983 777 174 152 876 909 829 501 528 293 012 982 988 8 × 2 = 0 + 0.000 487 881 441 279 999 967 554 348 305 753 819 659 003 056 586 025 965 977 6;
  • 28) 0.000 487 881 441 279 999 967 554 348 305 753 819 659 003 056 586 025 965 977 6 × 2 = 0 + 0.000 975 762 882 559 999 935 108 696 611 507 639 318 006 113 172 051 931 955 2;
  • 29) 0.000 975 762 882 559 999 935 108 696 611 507 639 318 006 113 172 051 931 955 2 × 2 = 0 + 0.001 951 525 765 119 999 870 217 393 223 015 278 636 012 226 344 103 863 910 4;
  • 30) 0.001 951 525 765 119 999 870 217 393 223 015 278 636 012 226 344 103 863 910 4 × 2 = 0 + 0.003 903 051 530 239 999 740 434 786 446 030 557 272 024 452 688 207 727 820 8;
  • 31) 0.003 903 051 530 239 999 740 434 786 446 030 557 272 024 452 688 207 727 820 8 × 2 = 0 + 0.007 806 103 060 479 999 480 869 572 892 061 114 544 048 905 376 415 455 641 6;
  • 32) 0.007 806 103 060 479 999 480 869 572 892 061 114 544 048 905 376 415 455 641 6 × 2 = 0 + 0.015 612 206 120 959 998 961 739 145 784 122 229 088 097 810 752 830 911 283 2;
  • 33) 0.015 612 206 120 959 998 961 739 145 784 122 229 088 097 810 752 830 911 283 2 × 2 = 0 + 0.031 224 412 241 919 997 923 478 291 568 244 458 176 195 621 505 661 822 566 4;
  • 34) 0.031 224 412 241 919 997 923 478 291 568 244 458 176 195 621 505 661 822 566 4 × 2 = 0 + 0.062 448 824 483 839 995 846 956 583 136 488 916 352 391 243 011 323 645 132 8;
  • 35) 0.062 448 824 483 839 995 846 956 583 136 488 916 352 391 243 011 323 645 132 8 × 2 = 0 + 0.124 897 648 967 679 991 693 913 166 272 977 832 704 782 486 022 647 290 265 6;
  • 36) 0.124 897 648 967 679 991 693 913 166 272 977 832 704 782 486 022 647 290 265 6 × 2 = 0 + 0.249 795 297 935 359 983 387 826 332 545 955 665 409 564 972 045 294 580 531 2;
  • 37) 0.249 795 297 935 359 983 387 826 332 545 955 665 409 564 972 045 294 580 531 2 × 2 = 0 + 0.499 590 595 870 719 966 775 652 665 091 911 330 819 129 944 090 589 161 062 4;
  • 38) 0.499 590 595 870 719 966 775 652 665 091 911 330 819 129 944 090 589 161 062 4 × 2 = 0 + 0.999 181 191 741 439 933 551 305 330 183 822 661 638 259 888 181 178 322 124 8;
  • 39) 0.999 181 191 741 439 933 551 305 330 183 822 661 638 259 888 181 178 322 124 8 × 2 = 1 + 0.998 362 383 482 879 867 102 610 660 367 645 323 276 519 776 362 356 644 249 6;
  • 40) 0.998 362 383 482 879 867 102 610 660 367 645 323 276 519 776 362 356 644 249 6 × 2 = 1 + 0.996 724 766 965 759 734 205 221 320 735 290 646 553 039 552 724 713 288 499 2;
  • 41) 0.996 724 766 965 759 734 205 221 320 735 290 646 553 039 552 724 713 288 499 2 × 2 = 1 + 0.993 449 533 931 519 468 410 442 641 470 581 293 106 079 105 449 426 576 998 4;
  • 42) 0.993 449 533 931 519 468 410 442 641 470 581 293 106 079 105 449 426 576 998 4 × 2 = 1 + 0.986 899 067 863 038 936 820 885 282 941 162 586 212 158 210 898 853 153 996 8;
  • 43) 0.986 899 067 863 038 936 820 885 282 941 162 586 212 158 210 898 853 153 996 8 × 2 = 1 + 0.973 798 135 726 077 873 641 770 565 882 325 172 424 316 421 797 706 307 993 6;
  • 44) 0.973 798 135 726 077 873 641 770 565 882 325 172 424 316 421 797 706 307 993 6 × 2 = 1 + 0.947 596 271 452 155 747 283 541 131 764 650 344 848 632 843 595 412 615 987 2;
  • 45) 0.947 596 271 452 155 747 283 541 131 764 650 344 848 632 843 595 412 615 987 2 × 2 = 1 + 0.895 192 542 904 311 494 567 082 263 529 300 689 697 265 687 190 825 231 974 4;
  • 46) 0.895 192 542 904 311 494 567 082 263 529 300 689 697 265 687 190 825 231 974 4 × 2 = 1 + 0.790 385 085 808 622 989 134 164 527 058 601 379 394 531 374 381 650 463 948 8;
  • 47) 0.790 385 085 808 622 989 134 164 527 058 601 379 394 531 374 381 650 463 948 8 × 2 = 1 + 0.580 770 171 617 245 978 268 329 054 117 202 758 789 062 748 763 300 927 897 6;
  • 48) 0.580 770 171 617 245 978 268 329 054 117 202 758 789 062 748 763 300 927 897 6 × 2 = 1 + 0.161 540 343 234 491 956 536 658 108 234 405 517 578 125 497 526 601 855 795 2;
  • 49) 0.161 540 343 234 491 956 536 658 108 234 405 517 578 125 497 526 601 855 795 2 × 2 = 0 + 0.323 080 686 468 983 913 073 316 216 468 811 035 156 250 995 053 203 711 590 4;
  • 50) 0.323 080 686 468 983 913 073 316 216 468 811 035 156 250 995 053 203 711 590 4 × 2 = 0 + 0.646 161 372 937 967 826 146 632 432 937 622 070 312 501 990 106 407 423 180 8;
  • 51) 0.646 161 372 937 967 826 146 632 432 937 622 070 312 501 990 106 407 423 180 8 × 2 = 1 + 0.292 322 745 875 935 652 293 264 865 875 244 140 625 003 980 212 814 846 361 6;
  • 52) 0.292 322 745 875 935 652 293 264 865 875 244 140 625 003 980 212 814 846 361 6 × 2 = 0 + 0.584 645 491 751 871 304 586 529 731 750 488 281 250 007 960 425 629 692 723 2;
  • 53) 0.584 645 491 751 871 304 586 529 731 750 488 281 250 007 960 425 629 692 723 2 × 2 = 1 + 0.169 290 983 503 742 609 173 059 463 500 976 562 500 015 920 851 259 385 446 4;
  • 54) 0.169 290 983 503 742 609 173 059 463 500 976 562 500 015 920 851 259 385 446 4 × 2 = 0 + 0.338 581 967 007 485 218 346 118 927 001 953 125 000 031 841 702 518 770 892 8;
  • 55) 0.338 581 967 007 485 218 346 118 927 001 953 125 000 031 841 702 518 770 892 8 × 2 = 0 + 0.677 163 934 014 970 436 692 237 854 003 906 250 000 063 683 405 037 541 785 6;
  • 56) 0.677 163 934 014 970 436 692 237 854 003 906 250 000 063 683 405 037 541 785 6 × 2 = 1 + 0.354 327 868 029 940 873 384 475 708 007 812 500 000 127 366 810 075 083 571 2;
  • 57) 0.354 327 868 029 940 873 384 475 708 007 812 500 000 127 366 810 075 083 571 2 × 2 = 0 + 0.708 655 736 059 881 746 768 951 416 015 625 000 000 254 733 620 150 167 142 4;
  • 58) 0.708 655 736 059 881 746 768 951 416 015 625 000 000 254 733 620 150 167 142 4 × 2 = 1 + 0.417 311 472 119 763 493 537 902 832 031 250 000 000 509 467 240 300 334 284 8;
  • 59) 0.417 311 472 119 763 493 537 902 832 031 250 000 000 509 467 240 300 334 284 8 × 2 = 0 + 0.834 622 944 239 526 987 075 805 664 062 500 000 001 018 934 480 600 668 569 6;
  • 60) 0.834 622 944 239 526 987 075 805 664 062 500 000 001 018 934 480 600 668 569 6 × 2 = 1 + 0.669 245 888 479 053 974 151 611 328 125 000 000 002 037 868 961 201 337 139 2;
  • 61) 0.669 245 888 479 053 974 151 611 328 125 000 000 002 037 868 961 201 337 139 2 × 2 = 1 + 0.338 491 776 958 107 948 303 222 656 250 000 000 004 075 737 922 402 674 278 4;
  • 62) 0.338 491 776 958 107 948 303 222 656 250 000 000 004 075 737 922 402 674 278 4 × 2 = 0 + 0.676 983 553 916 215 896 606 445 312 500 000 000 008 151 475 844 805 348 556 8;
  • 63) 0.676 983 553 916 215 896 606 445 312 500 000 000 008 151 475 844 805 348 556 8 × 2 = 1 + 0.353 967 107 832 431 793 212 890 625 000 000 000 016 302 951 689 610 697 113 6;
  • 64) 0.353 967 107 832 431 793 212 890 625 000 000 000 016 302 951 689 610 697 113 6 × 2 = 0 + 0.707 934 215 664 863 586 425 781 250 000 000 000 032 605 903 379 221 394 227 2;
  • 65) 0.707 934 215 664 863 586 425 781 250 000 000 000 032 605 903 379 221 394 227 2 × 2 = 1 + 0.415 868 431 329 727 172 851 562 500 000 000 000 065 211 806 758 442 788 454 4;
  • 66) 0.415 868 431 329 727 172 851 562 500 000 000 000 065 211 806 758 442 788 454 4 × 2 = 0 + 0.831 736 862 659 454 345 703 125 000 000 000 000 130 423 613 516 885 576 908 8;
  • 67) 0.831 736 862 659 454 345 703 125 000 000 000 000 130 423 613 516 885 576 908 8 × 2 = 1 + 0.663 473 725 318 908 691 406 250 000 000 000 000 260 847 227 033 771 153 817 6;
  • 68) 0.663 473 725 318 908 691 406 250 000 000 000 000 260 847 227 033 771 153 817 6 × 2 = 1 + 0.326 947 450 637 817 382 812 500 000 000 000 000 521 694 454 067 542 307 635 2;
  • 69) 0.326 947 450 637 817 382 812 500 000 000 000 000 521 694 454 067 542 307 635 2 × 2 = 0 + 0.653 894 901 275 634 765 625 000 000 000 000 001 043 388 908 135 084 615 270 4;
  • 70) 0.653 894 901 275 634 765 625 000 000 000 000 001 043 388 908 135 084 615 270 4 × 2 = 1 + 0.307 789 802 551 269 531 250 000 000 000 000 002 086 777 816 270 169 230 540 8;
  • 71) 0.307 789 802 551 269 531 250 000 000 000 000 002 086 777 816 270 169 230 540 8 × 2 = 0 + 0.615 579 605 102 539 062 500 000 000 000 000 004 173 555 632 540 338 461 081 6;
  • 72) 0.615 579 605 102 539 062 500 000 000 000 000 004 173 555 632 540 338 461 081 6 × 2 = 1 + 0.231 159 210 205 078 125 000 000 000 000 000 008 347 111 265 080 676 922 163 2;
  • 73) 0.231 159 210 205 078 125 000 000 000 000 000 008 347 111 265 080 676 922 163 2 × 2 = 0 + 0.462 318 420 410 156 250 000 000 000 000 000 016 694 222 530 161 353 844 326 4;
  • 74) 0.462 318 420 410 156 250 000 000 000 000 000 016 694 222 530 161 353 844 326 4 × 2 = 0 + 0.924 636 840 820 312 500 000 000 000 000 000 033 388 445 060 322 707 688 652 8;
  • 75) 0.924 636 840 820 312 500 000 000 000 000 000 033 388 445 060 322 707 688 652 8 × 2 = 1 + 0.849 273 681 640 625 000 000 000 000 000 000 066 776 890 120 645 415 377 305 6;
  • 76) 0.849 273 681 640 625 000 000 000 000 000 000 066 776 890 120 645 415 377 305 6 × 2 = 1 + 0.698 547 363 281 250 000 000 000 000 000 000 133 553 780 241 290 830 754 611 2;
  • 77) 0.698 547 363 281 250 000 000 000 000 000 000 133 553 780 241 290 830 754 611 2 × 2 = 1 + 0.397 094 726 562 500 000 000 000 000 000 000 267 107 560 482 581 661 509 222 4;
  • 78) 0.397 094 726 562 500 000 000 000 000 000 000 267 107 560 482 581 661 509 222 4 × 2 = 0 + 0.794 189 453 125 000 000 000 000 000 000 000 534 215 120 965 163 323 018 444 8;
  • 79) 0.794 189 453 125 000 000 000 000 000 000 000 534 215 120 965 163 323 018 444 8 × 2 = 1 + 0.588 378 906 250 000 000 000 000 000 000 001 068 430 241 930 326 646 036 889 6;
  • 80) 0.588 378 906 250 000 000 000 000 000 000 001 068 430 241 930 326 646 036 889 6 × 2 = 1 + 0.176 757 812 500 000 000 000 000 000 000 002 136 860 483 860 653 292 073 779 2;
  • 81) 0.176 757 812 500 000 000 000 000 000 000 002 136 860 483 860 653 292 073 779 2 × 2 = 0 + 0.353 515 625 000 000 000 000 000 000 000 004 273 720 967 721 306 584 147 558 4;
  • 82) 0.353 515 625 000 000 000 000 000 000 000 004 273 720 967 721 306 584 147 558 4 × 2 = 0 + 0.707 031 250 000 000 000 000 000 000 000 008 547 441 935 442 613 168 295 116 8;
  • 83) 0.707 031 250 000 000 000 000 000 000 000 008 547 441 935 442 613 168 295 116 8 × 2 = 1 + 0.414 062 500 000 000 000 000 000 000 000 017 094 883 870 885 226 336 590 233 6;
  • 84) 0.414 062 500 000 000 000 000 000 000 000 017 094 883 870 885 226 336 590 233 6 × 2 = 0 + 0.828 125 000 000 000 000 000 000 000 000 034 189 767 741 770 452 673 180 467 2;
  • 85) 0.828 125 000 000 000 000 000 000 000 000 034 189 767 741 770 452 673 180 467 2 × 2 = 1 + 0.656 250 000 000 000 000 000 000 000 000 068 379 535 483 540 905 346 360 934 4;
  • 86) 0.656 250 000 000 000 000 000 000 000 000 068 379 535 483 540 905 346 360 934 4 × 2 = 1 + 0.312 500 000 000 000 000 000 000 000 000 136 759 070 967 081 810 692 721 868 8;
  • 87) 0.312 500 000 000 000 000 000 000 000 000 136 759 070 967 081 810 692 721 868 8 × 2 = 0 + 0.625 000 000 000 000 000 000 000 000 000 273 518 141 934 163 621 385 443 737 6;
  • 88) 0.625 000 000 000 000 000 000 000 000 000 273 518 141 934 163 621 385 443 737 6 × 2 = 1 + 0.250 000 000 000 000 000 000 000 000 000 547 036 283 868 327 242 770 887 475 2;
  • 89) 0.250 000 000 000 000 000 000 000 000 000 547 036 283 868 327 242 770 887 475 2 × 2 = 0 + 0.500 000 000 000 000 000 000 000 000 001 094 072 567 736 654 485 541 774 950 4;
  • 90) 0.500 000 000 000 000 000 000 000 000 001 094 072 567 736 654 485 541 774 950 4 × 2 = 1 + 0.000 000 000 000 000 000 000 000 000 002 188 145 135 473 308 971 083 549 900 8;
  • 91) 0.000 000 000 000 000 000 000 000 000 002 188 145 135 473 308 971 083 549 900 8 × 2 = 0 + 0.000 000 000 000 000 000 000 000 000 004 376 290 270 946 617 942 167 099 801 6;

We didn't get any fractional part that was equal to zero. But we had enough iterations (over Mantissa limit) and at least one integer that was different from zero => FULL STOP (Losing precision - the converted number we get in the end will be just a very good approximation of the initial one).


4. Construct the base 2 representation of the fractional part of the number.

Take all the integer parts of the multiplying operations, starting from the top of the constructed list above:


0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 031 7(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:

0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 031 7(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:


0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 031 7(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 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 031 7 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