0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 159 2 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 159 2(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
0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 159 2(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: 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 159 2.

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) 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 159 2 × 2 = 0 + 0.000 000 000 007 269 999 999 999 999 516 522 114 064 601 356 670 318 4;
  • 2) 0.000 000 000 007 269 999 999 999 999 516 522 114 064 601 356 670 318 4 × 2 = 0 + 0.000 000 000 014 539 999 999 999 999 033 044 228 129 202 713 340 636 8;
  • 3) 0.000 000 000 014 539 999 999 999 999 033 044 228 129 202 713 340 636 8 × 2 = 0 + 0.000 000 000 029 079 999 999 999 998 066 088 456 258 405 426 681 273 6;
  • 4) 0.000 000 000 029 079 999 999 999 998 066 088 456 258 405 426 681 273 6 × 2 = 0 + 0.000 000 000 058 159 999 999 999 996 132 176 912 516 810 853 362 547 2;
  • 5) 0.000 000 000 058 159 999 999 999 996 132 176 912 516 810 853 362 547 2 × 2 = 0 + 0.000 000 000 116 319 999 999 999 992 264 353 825 033 621 706 725 094 4;
  • 6) 0.000 000 000 116 319 999 999 999 992 264 353 825 033 621 706 725 094 4 × 2 = 0 + 0.000 000 000 232 639 999 999 999 984 528 707 650 067 243 413 450 188 8;
  • 7) 0.000 000 000 232 639 999 999 999 984 528 707 650 067 243 413 450 188 8 × 2 = 0 + 0.000 000 000 465 279 999 999 999 969 057 415 300 134 486 826 900 377 6;
  • 8) 0.000 000 000 465 279 999 999 999 969 057 415 300 134 486 826 900 377 6 × 2 = 0 + 0.000 000 000 930 559 999 999 999 938 114 830 600 268 973 653 800 755 2;
  • 9) 0.000 000 000 930 559 999 999 999 938 114 830 600 268 973 653 800 755 2 × 2 = 0 + 0.000 000 001 861 119 999 999 999 876 229 661 200 537 947 307 601 510 4;
  • 10) 0.000 000 001 861 119 999 999 999 876 229 661 200 537 947 307 601 510 4 × 2 = 0 + 0.000 000 003 722 239 999 999 999 752 459 322 401 075 894 615 203 020 8;
  • 11) 0.000 000 003 722 239 999 999 999 752 459 322 401 075 894 615 203 020 8 × 2 = 0 + 0.000 000 007 444 479 999 999 999 504 918 644 802 151 789 230 406 041 6;
  • 12) 0.000 000 007 444 479 999 999 999 504 918 644 802 151 789 230 406 041 6 × 2 = 0 + 0.000 000 014 888 959 999 999 999 009 837 289 604 303 578 460 812 083 2;
  • 13) 0.000 000 014 888 959 999 999 999 009 837 289 604 303 578 460 812 083 2 × 2 = 0 + 0.000 000 029 777 919 999 999 998 019 674 579 208 607 156 921 624 166 4;
  • 14) 0.000 000 029 777 919 999 999 998 019 674 579 208 607 156 921 624 166 4 × 2 = 0 + 0.000 000 059 555 839 999 999 996 039 349 158 417 214 313 843 248 332 8;
  • 15) 0.000 000 059 555 839 999 999 996 039 349 158 417 214 313 843 248 332 8 × 2 = 0 + 0.000 000 119 111 679 999 999 992 078 698 316 834 428 627 686 496 665 6;
  • 16) 0.000 000 119 111 679 999 999 992 078 698 316 834 428 627 686 496 665 6 × 2 = 0 + 0.000 000 238 223 359 999 999 984 157 396 633 668 857 255 372 993 331 2;
  • 17) 0.000 000 238 223 359 999 999 984 157 396 633 668 857 255 372 993 331 2 × 2 = 0 + 0.000 000 476 446 719 999 999 968 314 793 267 337 714 510 745 986 662 4;
  • 18) 0.000 000 476 446 719 999 999 968 314 793 267 337 714 510 745 986 662 4 × 2 = 0 + 0.000 000 952 893 439 999 999 936 629 586 534 675 429 021 491 973 324 8;
  • 19) 0.000 000 952 893 439 999 999 936 629 586 534 675 429 021 491 973 324 8 × 2 = 0 + 0.000 001 905 786 879 999 999 873 259 173 069 350 858 042 983 946 649 6;
  • 20) 0.000 001 905 786 879 999 999 873 259 173 069 350 858 042 983 946 649 6 × 2 = 0 + 0.000 003 811 573 759 999 999 746 518 346 138 701 716 085 967 893 299 2;
  • 21) 0.000 003 811 573 759 999 999 746 518 346 138 701 716 085 967 893 299 2 × 2 = 0 + 0.000 007 623 147 519 999 999 493 036 692 277 403 432 171 935 786 598 4;
  • 22) 0.000 007 623 147 519 999 999 493 036 692 277 403 432 171 935 786 598 4 × 2 = 0 + 0.000 015 246 295 039 999 998 986 073 384 554 806 864 343 871 573 196 8;
  • 23) 0.000 015 246 295 039 999 998 986 073 384 554 806 864 343 871 573 196 8 × 2 = 0 + 0.000 030 492 590 079 999 997 972 146 769 109 613 728 687 743 146 393 6;
  • 24) 0.000 030 492 590 079 999 997 972 146 769 109 613 728 687 743 146 393 6 × 2 = 0 + 0.000 060 985 180 159 999 995 944 293 538 219 227 457 375 486 292 787 2;
  • 25) 0.000 060 985 180 159 999 995 944 293 538 219 227 457 375 486 292 787 2 × 2 = 0 + 0.000 121 970 360 319 999 991 888 587 076 438 454 914 750 972 585 574 4;
  • 26) 0.000 121 970 360 319 999 991 888 587 076 438 454 914 750 972 585 574 4 × 2 = 0 + 0.000 243 940 720 639 999 983 777 174 152 876 909 829 501 945 171 148 8;
  • 27) 0.000 243 940 720 639 999 983 777 174 152 876 909 829 501 945 171 148 8 × 2 = 0 + 0.000 487 881 441 279 999 967 554 348 305 753 819 659 003 890 342 297 6;
  • 28) 0.000 487 881 441 279 999 967 554 348 305 753 819 659 003 890 342 297 6 × 2 = 0 + 0.000 975 762 882 559 999 935 108 696 611 507 639 318 007 780 684 595 2;
  • 29) 0.000 975 762 882 559 999 935 108 696 611 507 639 318 007 780 684 595 2 × 2 = 0 + 0.001 951 525 765 119 999 870 217 393 223 015 278 636 015 561 369 190 4;
  • 30) 0.001 951 525 765 119 999 870 217 393 223 015 278 636 015 561 369 190 4 × 2 = 0 + 0.003 903 051 530 239 999 740 434 786 446 030 557 272 031 122 738 380 8;
  • 31) 0.003 903 051 530 239 999 740 434 786 446 030 557 272 031 122 738 380 8 × 2 = 0 + 0.007 806 103 060 479 999 480 869 572 892 061 114 544 062 245 476 761 6;
  • 32) 0.007 806 103 060 479 999 480 869 572 892 061 114 544 062 245 476 761 6 × 2 = 0 + 0.015 612 206 120 959 998 961 739 145 784 122 229 088 124 490 953 523 2;
  • 33) 0.015 612 206 120 959 998 961 739 145 784 122 229 088 124 490 953 523 2 × 2 = 0 + 0.031 224 412 241 919 997 923 478 291 568 244 458 176 248 981 907 046 4;
  • 34) 0.031 224 412 241 919 997 923 478 291 568 244 458 176 248 981 907 046 4 × 2 = 0 + 0.062 448 824 483 839 995 846 956 583 136 488 916 352 497 963 814 092 8;
  • 35) 0.062 448 824 483 839 995 846 956 583 136 488 916 352 497 963 814 092 8 × 2 = 0 + 0.124 897 648 967 679 991 693 913 166 272 977 832 704 995 927 628 185 6;
  • 36) 0.124 897 648 967 679 991 693 913 166 272 977 832 704 995 927 628 185 6 × 2 = 0 + 0.249 795 297 935 359 983 387 826 332 545 955 665 409 991 855 256 371 2;
  • 37) 0.249 795 297 935 359 983 387 826 332 545 955 665 409 991 855 256 371 2 × 2 = 0 + 0.499 590 595 870 719 966 775 652 665 091 911 330 819 983 710 512 742 4;
  • 38) 0.499 590 595 870 719 966 775 652 665 091 911 330 819 983 710 512 742 4 × 2 = 0 + 0.999 181 191 741 439 933 551 305 330 183 822 661 639 967 421 025 484 8;
  • 39) 0.999 181 191 741 439 933 551 305 330 183 822 661 639 967 421 025 484 8 × 2 = 1 + 0.998 362 383 482 879 867 102 610 660 367 645 323 279 934 842 050 969 6;
  • 40) 0.998 362 383 482 879 867 102 610 660 367 645 323 279 934 842 050 969 6 × 2 = 1 + 0.996 724 766 965 759 734 205 221 320 735 290 646 559 869 684 101 939 2;
  • 41) 0.996 724 766 965 759 734 205 221 320 735 290 646 559 869 684 101 939 2 × 2 = 1 + 0.993 449 533 931 519 468 410 442 641 470 581 293 119 739 368 203 878 4;
  • 42) 0.993 449 533 931 519 468 410 442 641 470 581 293 119 739 368 203 878 4 × 2 = 1 + 0.986 899 067 863 038 936 820 885 282 941 162 586 239 478 736 407 756 8;
  • 43) 0.986 899 067 863 038 936 820 885 282 941 162 586 239 478 736 407 756 8 × 2 = 1 + 0.973 798 135 726 077 873 641 770 565 882 325 172 478 957 472 815 513 6;
  • 44) 0.973 798 135 726 077 873 641 770 565 882 325 172 478 957 472 815 513 6 × 2 = 1 + 0.947 596 271 452 155 747 283 541 131 764 650 344 957 914 945 631 027 2;
  • 45) 0.947 596 271 452 155 747 283 541 131 764 650 344 957 914 945 631 027 2 × 2 = 1 + 0.895 192 542 904 311 494 567 082 263 529 300 689 915 829 891 262 054 4;
  • 46) 0.895 192 542 904 311 494 567 082 263 529 300 689 915 829 891 262 054 4 × 2 = 1 + 0.790 385 085 808 622 989 134 164 527 058 601 379 831 659 782 524 108 8;
  • 47) 0.790 385 085 808 622 989 134 164 527 058 601 379 831 659 782 524 108 8 × 2 = 1 + 0.580 770 171 617 245 978 268 329 054 117 202 759 663 319 565 048 217 6;
  • 48) 0.580 770 171 617 245 978 268 329 054 117 202 759 663 319 565 048 217 6 × 2 = 1 + 0.161 540 343 234 491 956 536 658 108 234 405 519 326 639 130 096 435 2;
  • 49) 0.161 540 343 234 491 956 536 658 108 234 405 519 326 639 130 096 435 2 × 2 = 0 + 0.323 080 686 468 983 913 073 316 216 468 811 038 653 278 260 192 870 4;
  • 50) 0.323 080 686 468 983 913 073 316 216 468 811 038 653 278 260 192 870 4 × 2 = 0 + 0.646 161 372 937 967 826 146 632 432 937 622 077 306 556 520 385 740 8;
  • 51) 0.646 161 372 937 967 826 146 632 432 937 622 077 306 556 520 385 740 8 × 2 = 1 + 0.292 322 745 875 935 652 293 264 865 875 244 154 613 113 040 771 481 6;
  • 52) 0.292 322 745 875 935 652 293 264 865 875 244 154 613 113 040 771 481 6 × 2 = 0 + 0.584 645 491 751 871 304 586 529 731 750 488 309 226 226 081 542 963 2;
  • 53) 0.584 645 491 751 871 304 586 529 731 750 488 309 226 226 081 542 963 2 × 2 = 1 + 0.169 290 983 503 742 609 173 059 463 500 976 618 452 452 163 085 926 4;
  • 54) 0.169 290 983 503 742 609 173 059 463 500 976 618 452 452 163 085 926 4 × 2 = 0 + 0.338 581 967 007 485 218 346 118 927 001 953 236 904 904 326 171 852 8;
  • 55) 0.338 581 967 007 485 218 346 118 927 001 953 236 904 904 326 171 852 8 × 2 = 0 + 0.677 163 934 014 970 436 692 237 854 003 906 473 809 808 652 343 705 6;
  • 56) 0.677 163 934 014 970 436 692 237 854 003 906 473 809 808 652 343 705 6 × 2 = 1 + 0.354 327 868 029 940 873 384 475 708 007 812 947 619 617 304 687 411 2;
  • 57) 0.354 327 868 029 940 873 384 475 708 007 812 947 619 617 304 687 411 2 × 2 = 0 + 0.708 655 736 059 881 746 768 951 416 015 625 895 239 234 609 374 822 4;
  • 58) 0.708 655 736 059 881 746 768 951 416 015 625 895 239 234 609 374 822 4 × 2 = 1 + 0.417 311 472 119 763 493 537 902 832 031 251 790 478 469 218 749 644 8;
  • 59) 0.417 311 472 119 763 493 537 902 832 031 251 790 478 469 218 749 644 8 × 2 = 0 + 0.834 622 944 239 526 987 075 805 664 062 503 580 956 938 437 499 289 6;
  • 60) 0.834 622 944 239 526 987 075 805 664 062 503 580 956 938 437 499 289 6 × 2 = 1 + 0.669 245 888 479 053 974 151 611 328 125 007 161 913 876 874 998 579 2;
  • 61) 0.669 245 888 479 053 974 151 611 328 125 007 161 913 876 874 998 579 2 × 2 = 1 + 0.338 491 776 958 107 948 303 222 656 250 014 323 827 753 749 997 158 4;
  • 62) 0.338 491 776 958 107 948 303 222 656 250 014 323 827 753 749 997 158 4 × 2 = 0 + 0.676 983 553 916 215 896 606 445 312 500 028 647 655 507 499 994 316 8;
  • 63) 0.676 983 553 916 215 896 606 445 312 500 028 647 655 507 499 994 316 8 × 2 = 1 + 0.353 967 107 832 431 793 212 890 625 000 057 295 311 014 999 988 633 6;
  • 64) 0.353 967 107 832 431 793 212 890 625 000 057 295 311 014 999 988 633 6 × 2 = 0 + 0.707 934 215 664 863 586 425 781 250 000 114 590 622 029 999 977 267 2;
  • 65) 0.707 934 215 664 863 586 425 781 250 000 114 590 622 029 999 977 267 2 × 2 = 1 + 0.415 868 431 329 727 172 851 562 500 000 229 181 244 059 999 954 534 4;
  • 66) 0.415 868 431 329 727 172 851 562 500 000 229 181 244 059 999 954 534 4 × 2 = 0 + 0.831 736 862 659 454 345 703 125 000 000 458 362 488 119 999 909 068 8;
  • 67) 0.831 736 862 659 454 345 703 125 000 000 458 362 488 119 999 909 068 8 × 2 = 1 + 0.663 473 725 318 908 691 406 250 000 000 916 724 976 239 999 818 137 6;
  • 68) 0.663 473 725 318 908 691 406 250 000 000 916 724 976 239 999 818 137 6 × 2 = 1 + 0.326 947 450 637 817 382 812 500 000 001 833 449 952 479 999 636 275 2;
  • 69) 0.326 947 450 637 817 382 812 500 000 001 833 449 952 479 999 636 275 2 × 2 = 0 + 0.653 894 901 275 634 765 625 000 000 003 666 899 904 959 999 272 550 4;
  • 70) 0.653 894 901 275 634 765 625 000 000 003 666 899 904 959 999 272 550 4 × 2 = 1 + 0.307 789 802 551 269 531 250 000 000 007 333 799 809 919 998 545 100 8;
  • 71) 0.307 789 802 551 269 531 250 000 000 007 333 799 809 919 998 545 100 8 × 2 = 0 + 0.615 579 605 102 539 062 500 000 000 014 667 599 619 839 997 090 201 6;
  • 72) 0.615 579 605 102 539 062 500 000 000 014 667 599 619 839 997 090 201 6 × 2 = 1 + 0.231 159 210 205 078 125 000 000 000 029 335 199 239 679 994 180 403 2;
  • 73) 0.231 159 210 205 078 125 000 000 000 029 335 199 239 679 994 180 403 2 × 2 = 0 + 0.462 318 420 410 156 250 000 000 000 058 670 398 479 359 988 360 806 4;
  • 74) 0.462 318 420 410 156 250 000 000 000 058 670 398 479 359 988 360 806 4 × 2 = 0 + 0.924 636 840 820 312 500 000 000 000 117 340 796 958 719 976 721 612 8;
  • 75) 0.924 636 840 820 312 500 000 000 000 117 340 796 958 719 976 721 612 8 × 2 = 1 + 0.849 273 681 640 625 000 000 000 000 234 681 593 917 439 953 443 225 6;
  • 76) 0.849 273 681 640 625 000 000 000 000 234 681 593 917 439 953 443 225 6 × 2 = 1 + 0.698 547 363 281 250 000 000 000 000 469 363 187 834 879 906 886 451 2;
  • 77) 0.698 547 363 281 250 000 000 000 000 469 363 187 834 879 906 886 451 2 × 2 = 1 + 0.397 094 726 562 500 000 000 000 000 938 726 375 669 759 813 772 902 4;
  • 78) 0.397 094 726 562 500 000 000 000 000 938 726 375 669 759 813 772 902 4 × 2 = 0 + 0.794 189 453 125 000 000 000 000 001 877 452 751 339 519 627 545 804 8;
  • 79) 0.794 189 453 125 000 000 000 000 001 877 452 751 339 519 627 545 804 8 × 2 = 1 + 0.588 378 906 250 000 000 000 000 003 754 905 502 679 039 255 091 609 6;
  • 80) 0.588 378 906 250 000 000 000 000 003 754 905 502 679 039 255 091 609 6 × 2 = 1 + 0.176 757 812 500 000 000 000 000 007 509 811 005 358 078 510 183 219 2;
  • 81) 0.176 757 812 500 000 000 000 000 007 509 811 005 358 078 510 183 219 2 × 2 = 0 + 0.353 515 625 000 000 000 000 000 015 019 622 010 716 157 020 366 438 4;
  • 82) 0.353 515 625 000 000 000 000 000 015 019 622 010 716 157 020 366 438 4 × 2 = 0 + 0.707 031 250 000 000 000 000 000 030 039 244 021 432 314 040 732 876 8;
  • 83) 0.707 031 250 000 000 000 000 000 030 039 244 021 432 314 040 732 876 8 × 2 = 1 + 0.414 062 500 000 000 000 000 000 060 078 488 042 864 628 081 465 753 6;
  • 84) 0.414 062 500 000 000 000 000 000 060 078 488 042 864 628 081 465 753 6 × 2 = 0 + 0.828 125 000 000 000 000 000 000 120 156 976 085 729 256 162 931 507 2;
  • 85) 0.828 125 000 000 000 000 000 000 120 156 976 085 729 256 162 931 507 2 × 2 = 1 + 0.656 250 000 000 000 000 000 000 240 313 952 171 458 512 325 863 014 4;
  • 86) 0.656 250 000 000 000 000 000 000 240 313 952 171 458 512 325 863 014 4 × 2 = 1 + 0.312 500 000 000 000 000 000 000 480 627 904 342 917 024 651 726 028 8;
  • 87) 0.312 500 000 000 000 000 000 000 480 627 904 342 917 024 651 726 028 8 × 2 = 0 + 0.625 000 000 000 000 000 000 000 961 255 808 685 834 049 303 452 057 6;
  • 88) 0.625 000 000 000 000 000 000 000 961 255 808 685 834 049 303 452 057 6 × 2 = 1 + 0.250 000 000 000 000 000 000 001 922 511 617 371 668 098 606 904 115 2;
  • 89) 0.250 000 000 000 000 000 000 001 922 511 617 371 668 098 606 904 115 2 × 2 = 0 + 0.500 000 000 000 000 000 000 003 845 023 234 743 336 197 213 808 230 4;
  • 90) 0.500 000 000 000 000 000 000 003 845 023 234 743 336 197 213 808 230 4 × 2 = 1 + 0.000 000 000 000 000 000 000 007 690 046 469 486 672 394 427 616 460 8;
  • 91) 0.000 000 000 000 000 000 000 007 690 046 469 486 672 394 427 616 460 8 × 2 = 0 + 0.000 000 000 000 000 000 000 015 380 092 938 973 344 788 855 232 921 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:


0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 159 2(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:

0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 159 2(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:


0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 159 2(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 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 159 2 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 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 156 3 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: My Manager Only Says “We” When Referring to “My Work”. Gemini Notebook YouTube Video of 11.13 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