0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 995 7 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 995 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
0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 995 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: 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 995 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) 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 995 7 × 2 = 0 + 0.000 000 000 007 269 999 999 999 999 516 522 114 064 601 356 670 305 991 4;
  • 2) 0.000 000 000 007 269 999 999 999 999 516 522 114 064 601 356 670 305 991 4 × 2 = 0 + 0.000 000 000 014 539 999 999 999 999 033 044 228 129 202 713 340 611 982 8;
  • 3) 0.000 000 000 014 539 999 999 999 999 033 044 228 129 202 713 340 611 982 8 × 2 = 0 + 0.000 000 000 029 079 999 999 999 998 066 088 456 258 405 426 681 223 965 6;
  • 4) 0.000 000 000 029 079 999 999 999 998 066 088 456 258 405 426 681 223 965 6 × 2 = 0 + 0.000 000 000 058 159 999 999 999 996 132 176 912 516 810 853 362 447 931 2;
  • 5) 0.000 000 000 058 159 999 999 999 996 132 176 912 516 810 853 362 447 931 2 × 2 = 0 + 0.000 000 000 116 319 999 999 999 992 264 353 825 033 621 706 724 895 862 4;
  • 6) 0.000 000 000 116 319 999 999 999 992 264 353 825 033 621 706 724 895 862 4 × 2 = 0 + 0.000 000 000 232 639 999 999 999 984 528 707 650 067 243 413 449 791 724 8;
  • 7) 0.000 000 000 232 639 999 999 999 984 528 707 650 067 243 413 449 791 724 8 × 2 = 0 + 0.000 000 000 465 279 999 999 999 969 057 415 300 134 486 826 899 583 449 6;
  • 8) 0.000 000 000 465 279 999 999 999 969 057 415 300 134 486 826 899 583 449 6 × 2 = 0 + 0.000 000 000 930 559 999 999 999 938 114 830 600 268 973 653 799 166 899 2;
  • 9) 0.000 000 000 930 559 999 999 999 938 114 830 600 268 973 653 799 166 899 2 × 2 = 0 + 0.000 000 001 861 119 999 999 999 876 229 661 200 537 947 307 598 333 798 4;
  • 10) 0.000 000 001 861 119 999 999 999 876 229 661 200 537 947 307 598 333 798 4 × 2 = 0 + 0.000 000 003 722 239 999 999 999 752 459 322 401 075 894 615 196 667 596 8;
  • 11) 0.000 000 003 722 239 999 999 999 752 459 322 401 075 894 615 196 667 596 8 × 2 = 0 + 0.000 000 007 444 479 999 999 999 504 918 644 802 151 789 230 393 335 193 6;
  • 12) 0.000 000 007 444 479 999 999 999 504 918 644 802 151 789 230 393 335 193 6 × 2 = 0 + 0.000 000 014 888 959 999 999 999 009 837 289 604 303 578 460 786 670 387 2;
  • 13) 0.000 000 014 888 959 999 999 999 009 837 289 604 303 578 460 786 670 387 2 × 2 = 0 + 0.000 000 029 777 919 999 999 998 019 674 579 208 607 156 921 573 340 774 4;
  • 14) 0.000 000 029 777 919 999 999 998 019 674 579 208 607 156 921 573 340 774 4 × 2 = 0 + 0.000 000 059 555 839 999 999 996 039 349 158 417 214 313 843 146 681 548 8;
  • 15) 0.000 000 059 555 839 999 999 996 039 349 158 417 214 313 843 146 681 548 8 × 2 = 0 + 0.000 000 119 111 679 999 999 992 078 698 316 834 428 627 686 293 363 097 6;
  • 16) 0.000 000 119 111 679 999 999 992 078 698 316 834 428 627 686 293 363 097 6 × 2 = 0 + 0.000 000 238 223 359 999 999 984 157 396 633 668 857 255 372 586 726 195 2;
  • 17) 0.000 000 238 223 359 999 999 984 157 396 633 668 857 255 372 586 726 195 2 × 2 = 0 + 0.000 000 476 446 719 999 999 968 314 793 267 337 714 510 745 173 452 390 4;
  • 18) 0.000 000 476 446 719 999 999 968 314 793 267 337 714 510 745 173 452 390 4 × 2 = 0 + 0.000 000 952 893 439 999 999 936 629 586 534 675 429 021 490 346 904 780 8;
  • 19) 0.000 000 952 893 439 999 999 936 629 586 534 675 429 021 490 346 904 780 8 × 2 = 0 + 0.000 001 905 786 879 999 999 873 259 173 069 350 858 042 980 693 809 561 6;
  • 20) 0.000 001 905 786 879 999 999 873 259 173 069 350 858 042 980 693 809 561 6 × 2 = 0 + 0.000 003 811 573 759 999 999 746 518 346 138 701 716 085 961 387 619 123 2;
  • 21) 0.000 003 811 573 759 999 999 746 518 346 138 701 716 085 961 387 619 123 2 × 2 = 0 + 0.000 007 623 147 519 999 999 493 036 692 277 403 432 171 922 775 238 246 4;
  • 22) 0.000 007 623 147 519 999 999 493 036 692 277 403 432 171 922 775 238 246 4 × 2 = 0 + 0.000 015 246 295 039 999 998 986 073 384 554 806 864 343 845 550 476 492 8;
  • 23) 0.000 015 246 295 039 999 998 986 073 384 554 806 864 343 845 550 476 492 8 × 2 = 0 + 0.000 030 492 590 079 999 997 972 146 769 109 613 728 687 691 100 952 985 6;
  • 24) 0.000 030 492 590 079 999 997 972 146 769 109 613 728 687 691 100 952 985 6 × 2 = 0 + 0.000 060 985 180 159 999 995 944 293 538 219 227 457 375 382 201 905 971 2;
  • 25) 0.000 060 985 180 159 999 995 944 293 538 219 227 457 375 382 201 905 971 2 × 2 = 0 + 0.000 121 970 360 319 999 991 888 587 076 438 454 914 750 764 403 811 942 4;
  • 26) 0.000 121 970 360 319 999 991 888 587 076 438 454 914 750 764 403 811 942 4 × 2 = 0 + 0.000 243 940 720 639 999 983 777 174 152 876 909 829 501 528 807 623 884 8;
  • 27) 0.000 243 940 720 639 999 983 777 174 152 876 909 829 501 528 807 623 884 8 × 2 = 0 + 0.000 487 881 441 279 999 967 554 348 305 753 819 659 003 057 615 247 769 6;
  • 28) 0.000 487 881 441 279 999 967 554 348 305 753 819 659 003 057 615 247 769 6 × 2 = 0 + 0.000 975 762 882 559 999 935 108 696 611 507 639 318 006 115 230 495 539 2;
  • 29) 0.000 975 762 882 559 999 935 108 696 611 507 639 318 006 115 230 495 539 2 × 2 = 0 + 0.001 951 525 765 119 999 870 217 393 223 015 278 636 012 230 460 991 078 4;
  • 30) 0.001 951 525 765 119 999 870 217 393 223 015 278 636 012 230 460 991 078 4 × 2 = 0 + 0.003 903 051 530 239 999 740 434 786 446 030 557 272 024 460 921 982 156 8;
  • 31) 0.003 903 051 530 239 999 740 434 786 446 030 557 272 024 460 921 982 156 8 × 2 = 0 + 0.007 806 103 060 479 999 480 869 572 892 061 114 544 048 921 843 964 313 6;
  • 32) 0.007 806 103 060 479 999 480 869 572 892 061 114 544 048 921 843 964 313 6 × 2 = 0 + 0.015 612 206 120 959 998 961 739 145 784 122 229 088 097 843 687 928 627 2;
  • 33) 0.015 612 206 120 959 998 961 739 145 784 122 229 088 097 843 687 928 627 2 × 2 = 0 + 0.031 224 412 241 919 997 923 478 291 568 244 458 176 195 687 375 857 254 4;
  • 34) 0.031 224 412 241 919 997 923 478 291 568 244 458 176 195 687 375 857 254 4 × 2 = 0 + 0.062 448 824 483 839 995 846 956 583 136 488 916 352 391 374 751 714 508 8;
  • 35) 0.062 448 824 483 839 995 846 956 583 136 488 916 352 391 374 751 714 508 8 × 2 = 0 + 0.124 897 648 967 679 991 693 913 166 272 977 832 704 782 749 503 429 017 6;
  • 36) 0.124 897 648 967 679 991 693 913 166 272 977 832 704 782 749 503 429 017 6 × 2 = 0 + 0.249 795 297 935 359 983 387 826 332 545 955 665 409 565 499 006 858 035 2;
  • 37) 0.249 795 297 935 359 983 387 826 332 545 955 665 409 565 499 006 858 035 2 × 2 = 0 + 0.499 590 595 870 719 966 775 652 665 091 911 330 819 130 998 013 716 070 4;
  • 38) 0.499 590 595 870 719 966 775 652 665 091 911 330 819 130 998 013 716 070 4 × 2 = 0 + 0.999 181 191 741 439 933 551 305 330 183 822 661 638 261 996 027 432 140 8;
  • 39) 0.999 181 191 741 439 933 551 305 330 183 822 661 638 261 996 027 432 140 8 × 2 = 1 + 0.998 362 383 482 879 867 102 610 660 367 645 323 276 523 992 054 864 281 6;
  • 40) 0.998 362 383 482 879 867 102 610 660 367 645 323 276 523 992 054 864 281 6 × 2 = 1 + 0.996 724 766 965 759 734 205 221 320 735 290 646 553 047 984 109 728 563 2;
  • 41) 0.996 724 766 965 759 734 205 221 320 735 290 646 553 047 984 109 728 563 2 × 2 = 1 + 0.993 449 533 931 519 468 410 442 641 470 581 293 106 095 968 219 457 126 4;
  • 42) 0.993 449 533 931 519 468 410 442 641 470 581 293 106 095 968 219 457 126 4 × 2 = 1 + 0.986 899 067 863 038 936 820 885 282 941 162 586 212 191 936 438 914 252 8;
  • 43) 0.986 899 067 863 038 936 820 885 282 941 162 586 212 191 936 438 914 252 8 × 2 = 1 + 0.973 798 135 726 077 873 641 770 565 882 325 172 424 383 872 877 828 505 6;
  • 44) 0.973 798 135 726 077 873 641 770 565 882 325 172 424 383 872 877 828 505 6 × 2 = 1 + 0.947 596 271 452 155 747 283 541 131 764 650 344 848 767 745 755 657 011 2;
  • 45) 0.947 596 271 452 155 747 283 541 131 764 650 344 848 767 745 755 657 011 2 × 2 = 1 + 0.895 192 542 904 311 494 567 082 263 529 300 689 697 535 491 511 314 022 4;
  • 46) 0.895 192 542 904 311 494 567 082 263 529 300 689 697 535 491 511 314 022 4 × 2 = 1 + 0.790 385 085 808 622 989 134 164 527 058 601 379 395 070 983 022 628 044 8;
  • 47) 0.790 385 085 808 622 989 134 164 527 058 601 379 395 070 983 022 628 044 8 × 2 = 1 + 0.580 770 171 617 245 978 268 329 054 117 202 758 790 141 966 045 256 089 6;
  • 48) 0.580 770 171 617 245 978 268 329 054 117 202 758 790 141 966 045 256 089 6 × 2 = 1 + 0.161 540 343 234 491 956 536 658 108 234 405 517 580 283 932 090 512 179 2;
  • 49) 0.161 540 343 234 491 956 536 658 108 234 405 517 580 283 932 090 512 179 2 × 2 = 0 + 0.323 080 686 468 983 913 073 316 216 468 811 035 160 567 864 181 024 358 4;
  • 50) 0.323 080 686 468 983 913 073 316 216 468 811 035 160 567 864 181 024 358 4 × 2 = 0 + 0.646 161 372 937 967 826 146 632 432 937 622 070 321 135 728 362 048 716 8;
  • 51) 0.646 161 372 937 967 826 146 632 432 937 622 070 321 135 728 362 048 716 8 × 2 = 1 + 0.292 322 745 875 935 652 293 264 865 875 244 140 642 271 456 724 097 433 6;
  • 52) 0.292 322 745 875 935 652 293 264 865 875 244 140 642 271 456 724 097 433 6 × 2 = 0 + 0.584 645 491 751 871 304 586 529 731 750 488 281 284 542 913 448 194 867 2;
  • 53) 0.584 645 491 751 871 304 586 529 731 750 488 281 284 542 913 448 194 867 2 × 2 = 1 + 0.169 290 983 503 742 609 173 059 463 500 976 562 569 085 826 896 389 734 4;
  • 54) 0.169 290 983 503 742 609 173 059 463 500 976 562 569 085 826 896 389 734 4 × 2 = 0 + 0.338 581 967 007 485 218 346 118 927 001 953 125 138 171 653 792 779 468 8;
  • 55) 0.338 581 967 007 485 218 346 118 927 001 953 125 138 171 653 792 779 468 8 × 2 = 0 + 0.677 163 934 014 970 436 692 237 854 003 906 250 276 343 307 585 558 937 6;
  • 56) 0.677 163 934 014 970 436 692 237 854 003 906 250 276 343 307 585 558 937 6 × 2 = 1 + 0.354 327 868 029 940 873 384 475 708 007 812 500 552 686 615 171 117 875 2;
  • 57) 0.354 327 868 029 940 873 384 475 708 007 812 500 552 686 615 171 117 875 2 × 2 = 0 + 0.708 655 736 059 881 746 768 951 416 015 625 001 105 373 230 342 235 750 4;
  • 58) 0.708 655 736 059 881 746 768 951 416 015 625 001 105 373 230 342 235 750 4 × 2 = 1 + 0.417 311 472 119 763 493 537 902 832 031 250 002 210 746 460 684 471 500 8;
  • 59) 0.417 311 472 119 763 493 537 902 832 031 250 002 210 746 460 684 471 500 8 × 2 = 0 + 0.834 622 944 239 526 987 075 805 664 062 500 004 421 492 921 368 943 001 6;
  • 60) 0.834 622 944 239 526 987 075 805 664 062 500 004 421 492 921 368 943 001 6 × 2 = 1 + 0.669 245 888 479 053 974 151 611 328 125 000 008 842 985 842 737 886 003 2;
  • 61) 0.669 245 888 479 053 974 151 611 328 125 000 008 842 985 842 737 886 003 2 × 2 = 1 + 0.338 491 776 958 107 948 303 222 656 250 000 017 685 971 685 475 772 006 4;
  • 62) 0.338 491 776 958 107 948 303 222 656 250 000 017 685 971 685 475 772 006 4 × 2 = 0 + 0.676 983 553 916 215 896 606 445 312 500 000 035 371 943 370 951 544 012 8;
  • 63) 0.676 983 553 916 215 896 606 445 312 500 000 035 371 943 370 951 544 012 8 × 2 = 1 + 0.353 967 107 832 431 793 212 890 625 000 000 070 743 886 741 903 088 025 6;
  • 64) 0.353 967 107 832 431 793 212 890 625 000 000 070 743 886 741 903 088 025 6 × 2 = 0 + 0.707 934 215 664 863 586 425 781 250 000 000 141 487 773 483 806 176 051 2;
  • 65) 0.707 934 215 664 863 586 425 781 250 000 000 141 487 773 483 806 176 051 2 × 2 = 1 + 0.415 868 431 329 727 172 851 562 500 000 000 282 975 546 967 612 352 102 4;
  • 66) 0.415 868 431 329 727 172 851 562 500 000 000 282 975 546 967 612 352 102 4 × 2 = 0 + 0.831 736 862 659 454 345 703 125 000 000 000 565 951 093 935 224 704 204 8;
  • 67) 0.831 736 862 659 454 345 703 125 000 000 000 565 951 093 935 224 704 204 8 × 2 = 1 + 0.663 473 725 318 908 691 406 250 000 000 001 131 902 187 870 449 408 409 6;
  • 68) 0.663 473 725 318 908 691 406 250 000 000 001 131 902 187 870 449 408 409 6 × 2 = 1 + 0.326 947 450 637 817 382 812 500 000 000 002 263 804 375 740 898 816 819 2;
  • 69) 0.326 947 450 637 817 382 812 500 000 000 002 263 804 375 740 898 816 819 2 × 2 = 0 + 0.653 894 901 275 634 765 625 000 000 000 004 527 608 751 481 797 633 638 4;
  • 70) 0.653 894 901 275 634 765 625 000 000 000 004 527 608 751 481 797 633 638 4 × 2 = 1 + 0.307 789 802 551 269 531 250 000 000 000 009 055 217 502 963 595 267 276 8;
  • 71) 0.307 789 802 551 269 531 250 000 000 000 009 055 217 502 963 595 267 276 8 × 2 = 0 + 0.615 579 605 102 539 062 500 000 000 000 018 110 435 005 927 190 534 553 6;
  • 72) 0.615 579 605 102 539 062 500 000 000 000 018 110 435 005 927 190 534 553 6 × 2 = 1 + 0.231 159 210 205 078 125 000 000 000 000 036 220 870 011 854 381 069 107 2;
  • 73) 0.231 159 210 205 078 125 000 000 000 000 036 220 870 011 854 381 069 107 2 × 2 = 0 + 0.462 318 420 410 156 250 000 000 000 000 072 441 740 023 708 762 138 214 4;
  • 74) 0.462 318 420 410 156 250 000 000 000 000 072 441 740 023 708 762 138 214 4 × 2 = 0 + 0.924 636 840 820 312 500 000 000 000 000 144 883 480 047 417 524 276 428 8;
  • 75) 0.924 636 840 820 312 500 000 000 000 000 144 883 480 047 417 524 276 428 8 × 2 = 1 + 0.849 273 681 640 625 000 000 000 000 000 289 766 960 094 835 048 552 857 6;
  • 76) 0.849 273 681 640 625 000 000 000 000 000 289 766 960 094 835 048 552 857 6 × 2 = 1 + 0.698 547 363 281 250 000 000 000 000 000 579 533 920 189 670 097 105 715 2;
  • 77) 0.698 547 363 281 250 000 000 000 000 000 579 533 920 189 670 097 105 715 2 × 2 = 1 + 0.397 094 726 562 500 000 000 000 000 001 159 067 840 379 340 194 211 430 4;
  • 78) 0.397 094 726 562 500 000 000 000 000 001 159 067 840 379 340 194 211 430 4 × 2 = 0 + 0.794 189 453 125 000 000 000 000 000 002 318 135 680 758 680 388 422 860 8;
  • 79) 0.794 189 453 125 000 000 000 000 000 002 318 135 680 758 680 388 422 860 8 × 2 = 1 + 0.588 378 906 250 000 000 000 000 000 004 636 271 361 517 360 776 845 721 6;
  • 80) 0.588 378 906 250 000 000 000 000 000 004 636 271 361 517 360 776 845 721 6 × 2 = 1 + 0.176 757 812 500 000 000 000 000 000 009 272 542 723 034 721 553 691 443 2;
  • 81) 0.176 757 812 500 000 000 000 000 000 009 272 542 723 034 721 553 691 443 2 × 2 = 0 + 0.353 515 625 000 000 000 000 000 000 018 545 085 446 069 443 107 382 886 4;
  • 82) 0.353 515 625 000 000 000 000 000 000 018 545 085 446 069 443 107 382 886 4 × 2 = 0 + 0.707 031 250 000 000 000 000 000 000 037 090 170 892 138 886 214 765 772 8;
  • 83) 0.707 031 250 000 000 000 000 000 000 037 090 170 892 138 886 214 765 772 8 × 2 = 1 + 0.414 062 500 000 000 000 000 000 000 074 180 341 784 277 772 429 531 545 6;
  • 84) 0.414 062 500 000 000 000 000 000 000 074 180 341 784 277 772 429 531 545 6 × 2 = 0 + 0.828 125 000 000 000 000 000 000 000 148 360 683 568 555 544 859 063 091 2;
  • 85) 0.828 125 000 000 000 000 000 000 000 148 360 683 568 555 544 859 063 091 2 × 2 = 1 + 0.656 250 000 000 000 000 000 000 000 296 721 367 137 111 089 718 126 182 4;
  • 86) 0.656 250 000 000 000 000 000 000 000 296 721 367 137 111 089 718 126 182 4 × 2 = 1 + 0.312 500 000 000 000 000 000 000 000 593 442 734 274 222 179 436 252 364 8;
  • 87) 0.312 500 000 000 000 000 000 000 000 593 442 734 274 222 179 436 252 364 8 × 2 = 0 + 0.625 000 000 000 000 000 000 000 001 186 885 468 548 444 358 872 504 729 6;
  • 88) 0.625 000 000 000 000 000 000 000 001 186 885 468 548 444 358 872 504 729 6 × 2 = 1 + 0.250 000 000 000 000 000 000 000 002 373 770 937 096 888 717 745 009 459 2;
  • 89) 0.250 000 000 000 000 000 000 000 002 373 770 937 096 888 717 745 009 459 2 × 2 = 0 + 0.500 000 000 000 000 000 000 000 004 747 541 874 193 777 435 490 018 918 4;
  • 90) 0.500 000 000 000 000 000 000 000 004 747 541 874 193 777 435 490 018 918 4 × 2 = 1 + 0.000 000 000 000 000 000 000 000 009 495 083 748 387 554 870 980 037 836 8;
  • 91) 0.000 000 000 000 000 000 000 000 009 495 083 748 387 554 870 980 037 836 8 × 2 = 0 + 0.000 000 000 000 000 000 000 000 018 990 167 496 775 109 741 960 075 673 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:


0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 995 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:

0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 995 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:


0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 995 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 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 995 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

Share

» Convert decimal 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 991 2 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: The Hidden Requirement in Every Single Job Description. NotebookLM YouTube Video of 8.15 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