0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 147 6 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 147 6(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
0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 147 6(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: 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 147 6.

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) 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 147 6 × 2 = 0 + 0.000 000 000 007 269 999 999 999 999 516 522 114 064 601 356 670 295 2;
  • 2) 0.000 000 000 007 269 999 999 999 999 516 522 114 064 601 356 670 295 2 × 2 = 0 + 0.000 000 000 014 539 999 999 999 999 033 044 228 129 202 713 340 590 4;
  • 3) 0.000 000 000 014 539 999 999 999 999 033 044 228 129 202 713 340 590 4 × 2 = 0 + 0.000 000 000 029 079 999 999 999 998 066 088 456 258 405 426 681 180 8;
  • 4) 0.000 000 000 029 079 999 999 999 998 066 088 456 258 405 426 681 180 8 × 2 = 0 + 0.000 000 000 058 159 999 999 999 996 132 176 912 516 810 853 362 361 6;
  • 5) 0.000 000 000 058 159 999 999 999 996 132 176 912 516 810 853 362 361 6 × 2 = 0 + 0.000 000 000 116 319 999 999 999 992 264 353 825 033 621 706 724 723 2;
  • 6) 0.000 000 000 116 319 999 999 999 992 264 353 825 033 621 706 724 723 2 × 2 = 0 + 0.000 000 000 232 639 999 999 999 984 528 707 650 067 243 413 449 446 4;
  • 7) 0.000 000 000 232 639 999 999 999 984 528 707 650 067 243 413 449 446 4 × 2 = 0 + 0.000 000 000 465 279 999 999 999 969 057 415 300 134 486 826 898 892 8;
  • 8) 0.000 000 000 465 279 999 999 999 969 057 415 300 134 486 826 898 892 8 × 2 = 0 + 0.000 000 000 930 559 999 999 999 938 114 830 600 268 973 653 797 785 6;
  • 9) 0.000 000 000 930 559 999 999 999 938 114 830 600 268 973 653 797 785 6 × 2 = 0 + 0.000 000 001 861 119 999 999 999 876 229 661 200 537 947 307 595 571 2;
  • 10) 0.000 000 001 861 119 999 999 999 876 229 661 200 537 947 307 595 571 2 × 2 = 0 + 0.000 000 003 722 239 999 999 999 752 459 322 401 075 894 615 191 142 4;
  • 11) 0.000 000 003 722 239 999 999 999 752 459 322 401 075 894 615 191 142 4 × 2 = 0 + 0.000 000 007 444 479 999 999 999 504 918 644 802 151 789 230 382 284 8;
  • 12) 0.000 000 007 444 479 999 999 999 504 918 644 802 151 789 230 382 284 8 × 2 = 0 + 0.000 000 014 888 959 999 999 999 009 837 289 604 303 578 460 764 569 6;
  • 13) 0.000 000 014 888 959 999 999 999 009 837 289 604 303 578 460 764 569 6 × 2 = 0 + 0.000 000 029 777 919 999 999 998 019 674 579 208 607 156 921 529 139 2;
  • 14) 0.000 000 029 777 919 999 999 998 019 674 579 208 607 156 921 529 139 2 × 2 = 0 + 0.000 000 059 555 839 999 999 996 039 349 158 417 214 313 843 058 278 4;
  • 15) 0.000 000 059 555 839 999 999 996 039 349 158 417 214 313 843 058 278 4 × 2 = 0 + 0.000 000 119 111 679 999 999 992 078 698 316 834 428 627 686 116 556 8;
  • 16) 0.000 000 119 111 679 999 999 992 078 698 316 834 428 627 686 116 556 8 × 2 = 0 + 0.000 000 238 223 359 999 999 984 157 396 633 668 857 255 372 233 113 6;
  • 17) 0.000 000 238 223 359 999 999 984 157 396 633 668 857 255 372 233 113 6 × 2 = 0 + 0.000 000 476 446 719 999 999 968 314 793 267 337 714 510 744 466 227 2;
  • 18) 0.000 000 476 446 719 999 999 968 314 793 267 337 714 510 744 466 227 2 × 2 = 0 + 0.000 000 952 893 439 999 999 936 629 586 534 675 429 021 488 932 454 4;
  • 19) 0.000 000 952 893 439 999 999 936 629 586 534 675 429 021 488 932 454 4 × 2 = 0 + 0.000 001 905 786 879 999 999 873 259 173 069 350 858 042 977 864 908 8;
  • 20) 0.000 001 905 786 879 999 999 873 259 173 069 350 858 042 977 864 908 8 × 2 = 0 + 0.000 003 811 573 759 999 999 746 518 346 138 701 716 085 955 729 817 6;
  • 21) 0.000 003 811 573 759 999 999 746 518 346 138 701 716 085 955 729 817 6 × 2 = 0 + 0.000 007 623 147 519 999 999 493 036 692 277 403 432 171 911 459 635 2;
  • 22) 0.000 007 623 147 519 999 999 493 036 692 277 403 432 171 911 459 635 2 × 2 = 0 + 0.000 015 246 295 039 999 998 986 073 384 554 806 864 343 822 919 270 4;
  • 23) 0.000 015 246 295 039 999 998 986 073 384 554 806 864 343 822 919 270 4 × 2 = 0 + 0.000 030 492 590 079 999 997 972 146 769 109 613 728 687 645 838 540 8;
  • 24) 0.000 030 492 590 079 999 997 972 146 769 109 613 728 687 645 838 540 8 × 2 = 0 + 0.000 060 985 180 159 999 995 944 293 538 219 227 457 375 291 677 081 6;
  • 25) 0.000 060 985 180 159 999 995 944 293 538 219 227 457 375 291 677 081 6 × 2 = 0 + 0.000 121 970 360 319 999 991 888 587 076 438 454 914 750 583 354 163 2;
  • 26) 0.000 121 970 360 319 999 991 888 587 076 438 454 914 750 583 354 163 2 × 2 = 0 + 0.000 243 940 720 639 999 983 777 174 152 876 909 829 501 166 708 326 4;
  • 27) 0.000 243 940 720 639 999 983 777 174 152 876 909 829 501 166 708 326 4 × 2 = 0 + 0.000 487 881 441 279 999 967 554 348 305 753 819 659 002 333 416 652 8;
  • 28) 0.000 487 881 441 279 999 967 554 348 305 753 819 659 002 333 416 652 8 × 2 = 0 + 0.000 975 762 882 559 999 935 108 696 611 507 639 318 004 666 833 305 6;
  • 29) 0.000 975 762 882 559 999 935 108 696 611 507 639 318 004 666 833 305 6 × 2 = 0 + 0.001 951 525 765 119 999 870 217 393 223 015 278 636 009 333 666 611 2;
  • 30) 0.001 951 525 765 119 999 870 217 393 223 015 278 636 009 333 666 611 2 × 2 = 0 + 0.003 903 051 530 239 999 740 434 786 446 030 557 272 018 667 333 222 4;
  • 31) 0.003 903 051 530 239 999 740 434 786 446 030 557 272 018 667 333 222 4 × 2 = 0 + 0.007 806 103 060 479 999 480 869 572 892 061 114 544 037 334 666 444 8;
  • 32) 0.007 806 103 060 479 999 480 869 572 892 061 114 544 037 334 666 444 8 × 2 = 0 + 0.015 612 206 120 959 998 961 739 145 784 122 229 088 074 669 332 889 6;
  • 33) 0.015 612 206 120 959 998 961 739 145 784 122 229 088 074 669 332 889 6 × 2 = 0 + 0.031 224 412 241 919 997 923 478 291 568 244 458 176 149 338 665 779 2;
  • 34) 0.031 224 412 241 919 997 923 478 291 568 244 458 176 149 338 665 779 2 × 2 = 0 + 0.062 448 824 483 839 995 846 956 583 136 488 916 352 298 677 331 558 4;
  • 35) 0.062 448 824 483 839 995 846 956 583 136 488 916 352 298 677 331 558 4 × 2 = 0 + 0.124 897 648 967 679 991 693 913 166 272 977 832 704 597 354 663 116 8;
  • 36) 0.124 897 648 967 679 991 693 913 166 272 977 832 704 597 354 663 116 8 × 2 = 0 + 0.249 795 297 935 359 983 387 826 332 545 955 665 409 194 709 326 233 6;
  • 37) 0.249 795 297 935 359 983 387 826 332 545 955 665 409 194 709 326 233 6 × 2 = 0 + 0.499 590 595 870 719 966 775 652 665 091 911 330 818 389 418 652 467 2;
  • 38) 0.499 590 595 870 719 966 775 652 665 091 911 330 818 389 418 652 467 2 × 2 = 0 + 0.999 181 191 741 439 933 551 305 330 183 822 661 636 778 837 304 934 4;
  • 39) 0.999 181 191 741 439 933 551 305 330 183 822 661 636 778 837 304 934 4 × 2 = 1 + 0.998 362 383 482 879 867 102 610 660 367 645 323 273 557 674 609 868 8;
  • 40) 0.998 362 383 482 879 867 102 610 660 367 645 323 273 557 674 609 868 8 × 2 = 1 + 0.996 724 766 965 759 734 205 221 320 735 290 646 547 115 349 219 737 6;
  • 41) 0.996 724 766 965 759 734 205 221 320 735 290 646 547 115 349 219 737 6 × 2 = 1 + 0.993 449 533 931 519 468 410 442 641 470 581 293 094 230 698 439 475 2;
  • 42) 0.993 449 533 931 519 468 410 442 641 470 581 293 094 230 698 439 475 2 × 2 = 1 + 0.986 899 067 863 038 936 820 885 282 941 162 586 188 461 396 878 950 4;
  • 43) 0.986 899 067 863 038 936 820 885 282 941 162 586 188 461 396 878 950 4 × 2 = 1 + 0.973 798 135 726 077 873 641 770 565 882 325 172 376 922 793 757 900 8;
  • 44) 0.973 798 135 726 077 873 641 770 565 882 325 172 376 922 793 757 900 8 × 2 = 1 + 0.947 596 271 452 155 747 283 541 131 764 650 344 753 845 587 515 801 6;
  • 45) 0.947 596 271 452 155 747 283 541 131 764 650 344 753 845 587 515 801 6 × 2 = 1 + 0.895 192 542 904 311 494 567 082 263 529 300 689 507 691 175 031 603 2;
  • 46) 0.895 192 542 904 311 494 567 082 263 529 300 689 507 691 175 031 603 2 × 2 = 1 + 0.790 385 085 808 622 989 134 164 527 058 601 379 015 382 350 063 206 4;
  • 47) 0.790 385 085 808 622 989 134 164 527 058 601 379 015 382 350 063 206 4 × 2 = 1 + 0.580 770 171 617 245 978 268 329 054 117 202 758 030 764 700 126 412 8;
  • 48) 0.580 770 171 617 245 978 268 329 054 117 202 758 030 764 700 126 412 8 × 2 = 1 + 0.161 540 343 234 491 956 536 658 108 234 405 516 061 529 400 252 825 6;
  • 49) 0.161 540 343 234 491 956 536 658 108 234 405 516 061 529 400 252 825 6 × 2 = 0 + 0.323 080 686 468 983 913 073 316 216 468 811 032 123 058 800 505 651 2;
  • 50) 0.323 080 686 468 983 913 073 316 216 468 811 032 123 058 800 505 651 2 × 2 = 0 + 0.646 161 372 937 967 826 146 632 432 937 622 064 246 117 601 011 302 4;
  • 51) 0.646 161 372 937 967 826 146 632 432 937 622 064 246 117 601 011 302 4 × 2 = 1 + 0.292 322 745 875 935 652 293 264 865 875 244 128 492 235 202 022 604 8;
  • 52) 0.292 322 745 875 935 652 293 264 865 875 244 128 492 235 202 022 604 8 × 2 = 0 + 0.584 645 491 751 871 304 586 529 731 750 488 256 984 470 404 045 209 6;
  • 53) 0.584 645 491 751 871 304 586 529 731 750 488 256 984 470 404 045 209 6 × 2 = 1 + 0.169 290 983 503 742 609 173 059 463 500 976 513 968 940 808 090 419 2;
  • 54) 0.169 290 983 503 742 609 173 059 463 500 976 513 968 940 808 090 419 2 × 2 = 0 + 0.338 581 967 007 485 218 346 118 927 001 953 027 937 881 616 180 838 4;
  • 55) 0.338 581 967 007 485 218 346 118 927 001 953 027 937 881 616 180 838 4 × 2 = 0 + 0.677 163 934 014 970 436 692 237 854 003 906 055 875 763 232 361 676 8;
  • 56) 0.677 163 934 014 970 436 692 237 854 003 906 055 875 763 232 361 676 8 × 2 = 1 + 0.354 327 868 029 940 873 384 475 708 007 812 111 751 526 464 723 353 6;
  • 57) 0.354 327 868 029 940 873 384 475 708 007 812 111 751 526 464 723 353 6 × 2 = 0 + 0.708 655 736 059 881 746 768 951 416 015 624 223 503 052 929 446 707 2;
  • 58) 0.708 655 736 059 881 746 768 951 416 015 624 223 503 052 929 446 707 2 × 2 = 1 + 0.417 311 472 119 763 493 537 902 832 031 248 447 006 105 858 893 414 4;
  • 59) 0.417 311 472 119 763 493 537 902 832 031 248 447 006 105 858 893 414 4 × 2 = 0 + 0.834 622 944 239 526 987 075 805 664 062 496 894 012 211 717 786 828 8;
  • 60) 0.834 622 944 239 526 987 075 805 664 062 496 894 012 211 717 786 828 8 × 2 = 1 + 0.669 245 888 479 053 974 151 611 328 124 993 788 024 423 435 573 657 6;
  • 61) 0.669 245 888 479 053 974 151 611 328 124 993 788 024 423 435 573 657 6 × 2 = 1 + 0.338 491 776 958 107 948 303 222 656 249 987 576 048 846 871 147 315 2;
  • 62) 0.338 491 776 958 107 948 303 222 656 249 987 576 048 846 871 147 315 2 × 2 = 0 + 0.676 983 553 916 215 896 606 445 312 499 975 152 097 693 742 294 630 4;
  • 63) 0.676 983 553 916 215 896 606 445 312 499 975 152 097 693 742 294 630 4 × 2 = 1 + 0.353 967 107 832 431 793 212 890 624 999 950 304 195 387 484 589 260 8;
  • 64) 0.353 967 107 832 431 793 212 890 624 999 950 304 195 387 484 589 260 8 × 2 = 0 + 0.707 934 215 664 863 586 425 781 249 999 900 608 390 774 969 178 521 6;
  • 65) 0.707 934 215 664 863 586 425 781 249 999 900 608 390 774 969 178 521 6 × 2 = 1 + 0.415 868 431 329 727 172 851 562 499 999 801 216 781 549 938 357 043 2;
  • 66) 0.415 868 431 329 727 172 851 562 499 999 801 216 781 549 938 357 043 2 × 2 = 0 + 0.831 736 862 659 454 345 703 124 999 999 602 433 563 099 876 714 086 4;
  • 67) 0.831 736 862 659 454 345 703 124 999 999 602 433 563 099 876 714 086 4 × 2 = 1 + 0.663 473 725 318 908 691 406 249 999 999 204 867 126 199 753 428 172 8;
  • 68) 0.663 473 725 318 908 691 406 249 999 999 204 867 126 199 753 428 172 8 × 2 = 1 + 0.326 947 450 637 817 382 812 499 999 998 409 734 252 399 506 856 345 6;
  • 69) 0.326 947 450 637 817 382 812 499 999 998 409 734 252 399 506 856 345 6 × 2 = 0 + 0.653 894 901 275 634 765 624 999 999 996 819 468 504 799 013 712 691 2;
  • 70) 0.653 894 901 275 634 765 624 999 999 996 819 468 504 799 013 712 691 2 × 2 = 1 + 0.307 789 802 551 269 531 249 999 999 993 638 937 009 598 027 425 382 4;
  • 71) 0.307 789 802 551 269 531 249 999 999 993 638 937 009 598 027 425 382 4 × 2 = 0 + 0.615 579 605 102 539 062 499 999 999 987 277 874 019 196 054 850 764 8;
  • 72) 0.615 579 605 102 539 062 499 999 999 987 277 874 019 196 054 850 764 8 × 2 = 1 + 0.231 159 210 205 078 124 999 999 999 974 555 748 038 392 109 701 529 6;
  • 73) 0.231 159 210 205 078 124 999 999 999 974 555 748 038 392 109 701 529 6 × 2 = 0 + 0.462 318 420 410 156 249 999 999 999 949 111 496 076 784 219 403 059 2;
  • 74) 0.462 318 420 410 156 249 999 999 999 949 111 496 076 784 219 403 059 2 × 2 = 0 + 0.924 636 840 820 312 499 999 999 999 898 222 992 153 568 438 806 118 4;
  • 75) 0.924 636 840 820 312 499 999 999 999 898 222 992 153 568 438 806 118 4 × 2 = 1 + 0.849 273 681 640 624 999 999 999 999 796 445 984 307 136 877 612 236 8;
  • 76) 0.849 273 681 640 624 999 999 999 999 796 445 984 307 136 877 612 236 8 × 2 = 1 + 0.698 547 363 281 249 999 999 999 999 592 891 968 614 273 755 224 473 6;
  • 77) 0.698 547 363 281 249 999 999 999 999 592 891 968 614 273 755 224 473 6 × 2 = 1 + 0.397 094 726 562 499 999 999 999 999 185 783 937 228 547 510 448 947 2;
  • 78) 0.397 094 726 562 499 999 999 999 999 185 783 937 228 547 510 448 947 2 × 2 = 0 + 0.794 189 453 124 999 999 999 999 998 371 567 874 457 095 020 897 894 4;
  • 79) 0.794 189 453 124 999 999 999 999 998 371 567 874 457 095 020 897 894 4 × 2 = 1 + 0.588 378 906 249 999 999 999 999 996 743 135 748 914 190 041 795 788 8;
  • 80) 0.588 378 906 249 999 999 999 999 996 743 135 748 914 190 041 795 788 8 × 2 = 1 + 0.176 757 812 499 999 999 999 999 993 486 271 497 828 380 083 591 577 6;
  • 81) 0.176 757 812 499 999 999 999 999 993 486 271 497 828 380 083 591 577 6 × 2 = 0 + 0.353 515 624 999 999 999 999 999 986 972 542 995 656 760 167 183 155 2;
  • 82) 0.353 515 624 999 999 999 999 999 986 972 542 995 656 760 167 183 155 2 × 2 = 0 + 0.707 031 249 999 999 999 999 999 973 945 085 991 313 520 334 366 310 4;
  • 83) 0.707 031 249 999 999 999 999 999 973 945 085 991 313 520 334 366 310 4 × 2 = 1 + 0.414 062 499 999 999 999 999 999 947 890 171 982 627 040 668 732 620 8;
  • 84) 0.414 062 499 999 999 999 999 999 947 890 171 982 627 040 668 732 620 8 × 2 = 0 + 0.828 124 999 999 999 999 999 999 895 780 343 965 254 081 337 465 241 6;
  • 85) 0.828 124 999 999 999 999 999 999 895 780 343 965 254 081 337 465 241 6 × 2 = 1 + 0.656 249 999 999 999 999 999 999 791 560 687 930 508 162 674 930 483 2;
  • 86) 0.656 249 999 999 999 999 999 999 791 560 687 930 508 162 674 930 483 2 × 2 = 1 + 0.312 499 999 999 999 999 999 999 583 121 375 861 016 325 349 860 966 4;
  • 87) 0.312 499 999 999 999 999 999 999 583 121 375 861 016 325 349 860 966 4 × 2 = 0 + 0.624 999 999 999 999 999 999 999 166 242 751 722 032 650 699 721 932 8;
  • 88) 0.624 999 999 999 999 999 999 999 166 242 751 722 032 650 699 721 932 8 × 2 = 1 + 0.249 999 999 999 999 999 999 998 332 485 503 444 065 301 399 443 865 6;
  • 89) 0.249 999 999 999 999 999 999 998 332 485 503 444 065 301 399 443 865 6 × 2 = 0 + 0.499 999 999 999 999 999 999 996 664 971 006 888 130 602 798 887 731 2;
  • 90) 0.499 999 999 999 999 999 999 996 664 971 006 888 130 602 798 887 731 2 × 2 = 0 + 0.999 999 999 999 999 999 999 993 329 942 013 776 261 205 597 775 462 4;
  • 91) 0.999 999 999 999 999 999 999 993 329 942 013 776 261 205 597 775 462 4 × 2 = 1 + 0.999 999 999 999 999 999 999 986 659 884 027 552 522 411 195 550 924 8;

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:


0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 147 6(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:

0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 147 6(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:


0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 147 6(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 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 147 6 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

Share

» Convert decimal 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 137 6 to 64 bit double precision IEEE 754 binary floating point representation standard

» Calculations Performed by Our Visitors: Decimal Numbers Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard. Data organized on a Monthly Basis

» Month 07, 2026 [July]: Decimal Numbers Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard. Calculations performed during the month of: July - by our visitors

» Improve your social skills: Reputation: Bridge Between Company's Actions & Market Value. Gemini Notebook YouTube Video of 6.55 minutes

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