0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 144 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 144(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
0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 144(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: 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 144.

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) 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 144 × 2 = 0 + 0.000 000 000 007 269 999 999 999 999 516 522 114 064 601 356 670 305 976 288;
  • 2) 0.000 000 000 007 269 999 999 999 999 516 522 114 064 601 356 670 305 976 288 × 2 = 0 + 0.000 000 000 014 539 999 999 999 999 033 044 228 129 202 713 340 611 952 576;
  • 3) 0.000 000 000 014 539 999 999 999 999 033 044 228 129 202 713 340 611 952 576 × 2 = 0 + 0.000 000 000 029 079 999 999 999 998 066 088 456 258 405 426 681 223 905 152;
  • 4) 0.000 000 000 029 079 999 999 999 998 066 088 456 258 405 426 681 223 905 152 × 2 = 0 + 0.000 000 000 058 159 999 999 999 996 132 176 912 516 810 853 362 447 810 304;
  • 5) 0.000 000 000 058 159 999 999 999 996 132 176 912 516 810 853 362 447 810 304 × 2 = 0 + 0.000 000 000 116 319 999 999 999 992 264 353 825 033 621 706 724 895 620 608;
  • 6) 0.000 000 000 116 319 999 999 999 992 264 353 825 033 621 706 724 895 620 608 × 2 = 0 + 0.000 000 000 232 639 999 999 999 984 528 707 650 067 243 413 449 791 241 216;
  • 7) 0.000 000 000 232 639 999 999 999 984 528 707 650 067 243 413 449 791 241 216 × 2 = 0 + 0.000 000 000 465 279 999 999 999 969 057 415 300 134 486 826 899 582 482 432;
  • 8) 0.000 000 000 465 279 999 999 999 969 057 415 300 134 486 826 899 582 482 432 × 2 = 0 + 0.000 000 000 930 559 999 999 999 938 114 830 600 268 973 653 799 164 964 864;
  • 9) 0.000 000 000 930 559 999 999 999 938 114 830 600 268 973 653 799 164 964 864 × 2 = 0 + 0.000 000 001 861 119 999 999 999 876 229 661 200 537 947 307 598 329 929 728;
  • 10) 0.000 000 001 861 119 999 999 999 876 229 661 200 537 947 307 598 329 929 728 × 2 = 0 + 0.000 000 003 722 239 999 999 999 752 459 322 401 075 894 615 196 659 859 456;
  • 11) 0.000 000 003 722 239 999 999 999 752 459 322 401 075 894 615 196 659 859 456 × 2 = 0 + 0.000 000 007 444 479 999 999 999 504 918 644 802 151 789 230 393 319 718 912;
  • 12) 0.000 000 007 444 479 999 999 999 504 918 644 802 151 789 230 393 319 718 912 × 2 = 0 + 0.000 000 014 888 959 999 999 999 009 837 289 604 303 578 460 786 639 437 824;
  • 13) 0.000 000 014 888 959 999 999 999 009 837 289 604 303 578 460 786 639 437 824 × 2 = 0 + 0.000 000 029 777 919 999 999 998 019 674 579 208 607 156 921 573 278 875 648;
  • 14) 0.000 000 029 777 919 999 999 998 019 674 579 208 607 156 921 573 278 875 648 × 2 = 0 + 0.000 000 059 555 839 999 999 996 039 349 158 417 214 313 843 146 557 751 296;
  • 15) 0.000 000 059 555 839 999 999 996 039 349 158 417 214 313 843 146 557 751 296 × 2 = 0 + 0.000 000 119 111 679 999 999 992 078 698 316 834 428 627 686 293 115 502 592;
  • 16) 0.000 000 119 111 679 999 999 992 078 698 316 834 428 627 686 293 115 502 592 × 2 = 0 + 0.000 000 238 223 359 999 999 984 157 396 633 668 857 255 372 586 231 005 184;
  • 17) 0.000 000 238 223 359 999 999 984 157 396 633 668 857 255 372 586 231 005 184 × 2 = 0 + 0.000 000 476 446 719 999 999 968 314 793 267 337 714 510 745 172 462 010 368;
  • 18) 0.000 000 476 446 719 999 999 968 314 793 267 337 714 510 745 172 462 010 368 × 2 = 0 + 0.000 000 952 893 439 999 999 936 629 586 534 675 429 021 490 344 924 020 736;
  • 19) 0.000 000 952 893 439 999 999 936 629 586 534 675 429 021 490 344 924 020 736 × 2 = 0 + 0.000 001 905 786 879 999 999 873 259 173 069 350 858 042 980 689 848 041 472;
  • 20) 0.000 001 905 786 879 999 999 873 259 173 069 350 858 042 980 689 848 041 472 × 2 = 0 + 0.000 003 811 573 759 999 999 746 518 346 138 701 716 085 961 379 696 082 944;
  • 21) 0.000 003 811 573 759 999 999 746 518 346 138 701 716 085 961 379 696 082 944 × 2 = 0 + 0.000 007 623 147 519 999 999 493 036 692 277 403 432 171 922 759 392 165 888;
  • 22) 0.000 007 623 147 519 999 999 493 036 692 277 403 432 171 922 759 392 165 888 × 2 = 0 + 0.000 015 246 295 039 999 998 986 073 384 554 806 864 343 845 518 784 331 776;
  • 23) 0.000 015 246 295 039 999 998 986 073 384 554 806 864 343 845 518 784 331 776 × 2 = 0 + 0.000 030 492 590 079 999 997 972 146 769 109 613 728 687 691 037 568 663 552;
  • 24) 0.000 030 492 590 079 999 997 972 146 769 109 613 728 687 691 037 568 663 552 × 2 = 0 + 0.000 060 985 180 159 999 995 944 293 538 219 227 457 375 382 075 137 327 104;
  • 25) 0.000 060 985 180 159 999 995 944 293 538 219 227 457 375 382 075 137 327 104 × 2 = 0 + 0.000 121 970 360 319 999 991 888 587 076 438 454 914 750 764 150 274 654 208;
  • 26) 0.000 121 970 360 319 999 991 888 587 076 438 454 914 750 764 150 274 654 208 × 2 = 0 + 0.000 243 940 720 639 999 983 777 174 152 876 909 829 501 528 300 549 308 416;
  • 27) 0.000 243 940 720 639 999 983 777 174 152 876 909 829 501 528 300 549 308 416 × 2 = 0 + 0.000 487 881 441 279 999 967 554 348 305 753 819 659 003 056 601 098 616 832;
  • 28) 0.000 487 881 441 279 999 967 554 348 305 753 819 659 003 056 601 098 616 832 × 2 = 0 + 0.000 975 762 882 559 999 935 108 696 611 507 639 318 006 113 202 197 233 664;
  • 29) 0.000 975 762 882 559 999 935 108 696 611 507 639 318 006 113 202 197 233 664 × 2 = 0 + 0.001 951 525 765 119 999 870 217 393 223 015 278 636 012 226 404 394 467 328;
  • 30) 0.001 951 525 765 119 999 870 217 393 223 015 278 636 012 226 404 394 467 328 × 2 = 0 + 0.003 903 051 530 239 999 740 434 786 446 030 557 272 024 452 808 788 934 656;
  • 31) 0.003 903 051 530 239 999 740 434 786 446 030 557 272 024 452 808 788 934 656 × 2 = 0 + 0.007 806 103 060 479 999 480 869 572 892 061 114 544 048 905 617 577 869 312;
  • 32) 0.007 806 103 060 479 999 480 869 572 892 061 114 544 048 905 617 577 869 312 × 2 = 0 + 0.015 612 206 120 959 998 961 739 145 784 122 229 088 097 811 235 155 738 624;
  • 33) 0.015 612 206 120 959 998 961 739 145 784 122 229 088 097 811 235 155 738 624 × 2 = 0 + 0.031 224 412 241 919 997 923 478 291 568 244 458 176 195 622 470 311 477 248;
  • 34) 0.031 224 412 241 919 997 923 478 291 568 244 458 176 195 622 470 311 477 248 × 2 = 0 + 0.062 448 824 483 839 995 846 956 583 136 488 916 352 391 244 940 622 954 496;
  • 35) 0.062 448 824 483 839 995 846 956 583 136 488 916 352 391 244 940 622 954 496 × 2 = 0 + 0.124 897 648 967 679 991 693 913 166 272 977 832 704 782 489 881 245 908 992;
  • 36) 0.124 897 648 967 679 991 693 913 166 272 977 832 704 782 489 881 245 908 992 × 2 = 0 + 0.249 795 297 935 359 983 387 826 332 545 955 665 409 564 979 762 491 817 984;
  • 37) 0.249 795 297 935 359 983 387 826 332 545 955 665 409 564 979 762 491 817 984 × 2 = 0 + 0.499 590 595 870 719 966 775 652 665 091 911 330 819 129 959 524 983 635 968;
  • 38) 0.499 590 595 870 719 966 775 652 665 091 911 330 819 129 959 524 983 635 968 × 2 = 0 + 0.999 181 191 741 439 933 551 305 330 183 822 661 638 259 919 049 967 271 936;
  • 39) 0.999 181 191 741 439 933 551 305 330 183 822 661 638 259 919 049 967 271 936 × 2 = 1 + 0.998 362 383 482 879 867 102 610 660 367 645 323 276 519 838 099 934 543 872;
  • 40) 0.998 362 383 482 879 867 102 610 660 367 645 323 276 519 838 099 934 543 872 × 2 = 1 + 0.996 724 766 965 759 734 205 221 320 735 290 646 553 039 676 199 869 087 744;
  • 41) 0.996 724 766 965 759 734 205 221 320 735 290 646 553 039 676 199 869 087 744 × 2 = 1 + 0.993 449 533 931 519 468 410 442 641 470 581 293 106 079 352 399 738 175 488;
  • 42) 0.993 449 533 931 519 468 410 442 641 470 581 293 106 079 352 399 738 175 488 × 2 = 1 + 0.986 899 067 863 038 936 820 885 282 941 162 586 212 158 704 799 476 350 976;
  • 43) 0.986 899 067 863 038 936 820 885 282 941 162 586 212 158 704 799 476 350 976 × 2 = 1 + 0.973 798 135 726 077 873 641 770 565 882 325 172 424 317 409 598 952 701 952;
  • 44) 0.973 798 135 726 077 873 641 770 565 882 325 172 424 317 409 598 952 701 952 × 2 = 1 + 0.947 596 271 452 155 747 283 541 131 764 650 344 848 634 819 197 905 403 904;
  • 45) 0.947 596 271 452 155 747 283 541 131 764 650 344 848 634 819 197 905 403 904 × 2 = 1 + 0.895 192 542 904 311 494 567 082 263 529 300 689 697 269 638 395 810 807 808;
  • 46) 0.895 192 542 904 311 494 567 082 263 529 300 689 697 269 638 395 810 807 808 × 2 = 1 + 0.790 385 085 808 622 989 134 164 527 058 601 379 394 539 276 791 621 615 616;
  • 47) 0.790 385 085 808 622 989 134 164 527 058 601 379 394 539 276 791 621 615 616 × 2 = 1 + 0.580 770 171 617 245 978 268 329 054 117 202 758 789 078 553 583 243 231 232;
  • 48) 0.580 770 171 617 245 978 268 329 054 117 202 758 789 078 553 583 243 231 232 × 2 = 1 + 0.161 540 343 234 491 956 536 658 108 234 405 517 578 157 107 166 486 462 464;
  • 49) 0.161 540 343 234 491 956 536 658 108 234 405 517 578 157 107 166 486 462 464 × 2 = 0 + 0.323 080 686 468 983 913 073 316 216 468 811 035 156 314 214 332 972 924 928;
  • 50) 0.323 080 686 468 983 913 073 316 216 468 811 035 156 314 214 332 972 924 928 × 2 = 0 + 0.646 161 372 937 967 826 146 632 432 937 622 070 312 628 428 665 945 849 856;
  • 51) 0.646 161 372 937 967 826 146 632 432 937 622 070 312 628 428 665 945 849 856 × 2 = 1 + 0.292 322 745 875 935 652 293 264 865 875 244 140 625 256 857 331 891 699 712;
  • 52) 0.292 322 745 875 935 652 293 264 865 875 244 140 625 256 857 331 891 699 712 × 2 = 0 + 0.584 645 491 751 871 304 586 529 731 750 488 281 250 513 714 663 783 399 424;
  • 53) 0.584 645 491 751 871 304 586 529 731 750 488 281 250 513 714 663 783 399 424 × 2 = 1 + 0.169 290 983 503 742 609 173 059 463 500 976 562 501 027 429 327 566 798 848;
  • 54) 0.169 290 983 503 742 609 173 059 463 500 976 562 501 027 429 327 566 798 848 × 2 = 0 + 0.338 581 967 007 485 218 346 118 927 001 953 125 002 054 858 655 133 597 696;
  • 55) 0.338 581 967 007 485 218 346 118 927 001 953 125 002 054 858 655 133 597 696 × 2 = 0 + 0.677 163 934 014 970 436 692 237 854 003 906 250 004 109 717 310 267 195 392;
  • 56) 0.677 163 934 014 970 436 692 237 854 003 906 250 004 109 717 310 267 195 392 × 2 = 1 + 0.354 327 868 029 940 873 384 475 708 007 812 500 008 219 434 620 534 390 784;
  • 57) 0.354 327 868 029 940 873 384 475 708 007 812 500 008 219 434 620 534 390 784 × 2 = 0 + 0.708 655 736 059 881 746 768 951 416 015 625 000 016 438 869 241 068 781 568;
  • 58) 0.708 655 736 059 881 746 768 951 416 015 625 000 016 438 869 241 068 781 568 × 2 = 1 + 0.417 311 472 119 763 493 537 902 832 031 250 000 032 877 738 482 137 563 136;
  • 59) 0.417 311 472 119 763 493 537 902 832 031 250 000 032 877 738 482 137 563 136 × 2 = 0 + 0.834 622 944 239 526 987 075 805 664 062 500 000 065 755 476 964 275 126 272;
  • 60) 0.834 622 944 239 526 987 075 805 664 062 500 000 065 755 476 964 275 126 272 × 2 = 1 + 0.669 245 888 479 053 974 151 611 328 125 000 000 131 510 953 928 550 252 544;
  • 61) 0.669 245 888 479 053 974 151 611 328 125 000 000 131 510 953 928 550 252 544 × 2 = 1 + 0.338 491 776 958 107 948 303 222 656 250 000 000 263 021 907 857 100 505 088;
  • 62) 0.338 491 776 958 107 948 303 222 656 250 000 000 263 021 907 857 100 505 088 × 2 = 0 + 0.676 983 553 916 215 896 606 445 312 500 000 000 526 043 815 714 201 010 176;
  • 63) 0.676 983 553 916 215 896 606 445 312 500 000 000 526 043 815 714 201 010 176 × 2 = 1 + 0.353 967 107 832 431 793 212 890 625 000 000 001 052 087 631 428 402 020 352;
  • 64) 0.353 967 107 832 431 793 212 890 625 000 000 001 052 087 631 428 402 020 352 × 2 = 0 + 0.707 934 215 664 863 586 425 781 250 000 000 002 104 175 262 856 804 040 704;
  • 65) 0.707 934 215 664 863 586 425 781 250 000 000 002 104 175 262 856 804 040 704 × 2 = 1 + 0.415 868 431 329 727 172 851 562 500 000 000 004 208 350 525 713 608 081 408;
  • 66) 0.415 868 431 329 727 172 851 562 500 000 000 004 208 350 525 713 608 081 408 × 2 = 0 + 0.831 736 862 659 454 345 703 125 000 000 000 008 416 701 051 427 216 162 816;
  • 67) 0.831 736 862 659 454 345 703 125 000 000 000 008 416 701 051 427 216 162 816 × 2 = 1 + 0.663 473 725 318 908 691 406 250 000 000 000 016 833 402 102 854 432 325 632;
  • 68) 0.663 473 725 318 908 691 406 250 000 000 000 016 833 402 102 854 432 325 632 × 2 = 1 + 0.326 947 450 637 817 382 812 500 000 000 000 033 666 804 205 708 864 651 264;
  • 69) 0.326 947 450 637 817 382 812 500 000 000 000 033 666 804 205 708 864 651 264 × 2 = 0 + 0.653 894 901 275 634 765 625 000 000 000 000 067 333 608 411 417 729 302 528;
  • 70) 0.653 894 901 275 634 765 625 000 000 000 000 067 333 608 411 417 729 302 528 × 2 = 1 + 0.307 789 802 551 269 531 250 000 000 000 000 134 667 216 822 835 458 605 056;
  • 71) 0.307 789 802 551 269 531 250 000 000 000 000 134 667 216 822 835 458 605 056 × 2 = 0 + 0.615 579 605 102 539 062 500 000 000 000 000 269 334 433 645 670 917 210 112;
  • 72) 0.615 579 605 102 539 062 500 000 000 000 000 269 334 433 645 670 917 210 112 × 2 = 1 + 0.231 159 210 205 078 125 000 000 000 000 000 538 668 867 291 341 834 420 224;
  • 73) 0.231 159 210 205 078 125 000 000 000 000 000 538 668 867 291 341 834 420 224 × 2 = 0 + 0.462 318 420 410 156 250 000 000 000 000 001 077 337 734 582 683 668 840 448;
  • 74) 0.462 318 420 410 156 250 000 000 000 000 001 077 337 734 582 683 668 840 448 × 2 = 0 + 0.924 636 840 820 312 500 000 000 000 000 002 154 675 469 165 367 337 680 896;
  • 75) 0.924 636 840 820 312 500 000 000 000 000 002 154 675 469 165 367 337 680 896 × 2 = 1 + 0.849 273 681 640 625 000 000 000 000 000 004 309 350 938 330 734 675 361 792;
  • 76) 0.849 273 681 640 625 000 000 000 000 000 004 309 350 938 330 734 675 361 792 × 2 = 1 + 0.698 547 363 281 250 000 000 000 000 000 008 618 701 876 661 469 350 723 584;
  • 77) 0.698 547 363 281 250 000 000 000 000 000 008 618 701 876 661 469 350 723 584 × 2 = 1 + 0.397 094 726 562 500 000 000 000 000 000 017 237 403 753 322 938 701 447 168;
  • 78) 0.397 094 726 562 500 000 000 000 000 000 017 237 403 753 322 938 701 447 168 × 2 = 0 + 0.794 189 453 125 000 000 000 000 000 000 034 474 807 506 645 877 402 894 336;
  • 79) 0.794 189 453 125 000 000 000 000 000 000 034 474 807 506 645 877 402 894 336 × 2 = 1 + 0.588 378 906 250 000 000 000 000 000 000 068 949 615 013 291 754 805 788 672;
  • 80) 0.588 378 906 250 000 000 000 000 000 000 068 949 615 013 291 754 805 788 672 × 2 = 1 + 0.176 757 812 500 000 000 000 000 000 000 137 899 230 026 583 509 611 577 344;
  • 81) 0.176 757 812 500 000 000 000 000 000 000 137 899 230 026 583 509 611 577 344 × 2 = 0 + 0.353 515 625 000 000 000 000 000 000 000 275 798 460 053 167 019 223 154 688;
  • 82) 0.353 515 625 000 000 000 000 000 000 000 275 798 460 053 167 019 223 154 688 × 2 = 0 + 0.707 031 250 000 000 000 000 000 000 000 551 596 920 106 334 038 446 309 376;
  • 83) 0.707 031 250 000 000 000 000 000 000 000 551 596 920 106 334 038 446 309 376 × 2 = 1 + 0.414 062 500 000 000 000 000 000 000 001 103 193 840 212 668 076 892 618 752;
  • 84) 0.414 062 500 000 000 000 000 000 000 001 103 193 840 212 668 076 892 618 752 × 2 = 0 + 0.828 125 000 000 000 000 000 000 000 002 206 387 680 425 336 153 785 237 504;
  • 85) 0.828 125 000 000 000 000 000 000 000 002 206 387 680 425 336 153 785 237 504 × 2 = 1 + 0.656 250 000 000 000 000 000 000 000 004 412 775 360 850 672 307 570 475 008;
  • 86) 0.656 250 000 000 000 000 000 000 000 004 412 775 360 850 672 307 570 475 008 × 2 = 1 + 0.312 500 000 000 000 000 000 000 000 008 825 550 721 701 344 615 140 950 016;
  • 87) 0.312 500 000 000 000 000 000 000 000 008 825 550 721 701 344 615 140 950 016 × 2 = 0 + 0.625 000 000 000 000 000 000 000 000 017 651 101 443 402 689 230 281 900 032;
  • 88) 0.625 000 000 000 000 000 000 000 000 017 651 101 443 402 689 230 281 900 032 × 2 = 1 + 0.250 000 000 000 000 000 000 000 000 035 302 202 886 805 378 460 563 800 064;
  • 89) 0.250 000 000 000 000 000 000 000 000 035 302 202 886 805 378 460 563 800 064 × 2 = 0 + 0.500 000 000 000 000 000 000 000 000 070 604 405 773 610 756 921 127 600 128;
  • 90) 0.500 000 000 000 000 000 000 000 000 070 604 405 773 610 756 921 127 600 128 × 2 = 1 + 0.000 000 000 000 000 000 000 000 000 141 208 811 547 221 513 842 255 200 256;
  • 91) 0.000 000 000 000 000 000 000 000 000 141 208 811 547 221 513 842 255 200 256 × 2 = 0 + 0.000 000 000 000 000 000 000 000 000 282 417 623 094 443 027 684 510 400 512;

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:


0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 144(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:

0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 144(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:


0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 144(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 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 144 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 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 192 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. NotebookLM 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