0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 032 23 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 032 23(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
0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 032 23(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: 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 032 23.

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) 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 032 23 × 2 = 0 + 0.000 000 000 007 269 999 999 999 999 516 522 114 064 601 356 670 305 976 064 46;
  • 2) 0.000 000 000 007 269 999 999 999 999 516 522 114 064 601 356 670 305 976 064 46 × 2 = 0 + 0.000 000 000 014 539 999 999 999 999 033 044 228 129 202 713 340 611 952 128 92;
  • 3) 0.000 000 000 014 539 999 999 999 999 033 044 228 129 202 713 340 611 952 128 92 × 2 = 0 + 0.000 000 000 029 079 999 999 999 998 066 088 456 258 405 426 681 223 904 257 84;
  • 4) 0.000 000 000 029 079 999 999 999 998 066 088 456 258 405 426 681 223 904 257 84 × 2 = 0 + 0.000 000 000 058 159 999 999 999 996 132 176 912 516 810 853 362 447 808 515 68;
  • 5) 0.000 000 000 058 159 999 999 999 996 132 176 912 516 810 853 362 447 808 515 68 × 2 = 0 + 0.000 000 000 116 319 999 999 999 992 264 353 825 033 621 706 724 895 617 031 36;
  • 6) 0.000 000 000 116 319 999 999 999 992 264 353 825 033 621 706 724 895 617 031 36 × 2 = 0 + 0.000 000 000 232 639 999 999 999 984 528 707 650 067 243 413 449 791 234 062 72;
  • 7) 0.000 000 000 232 639 999 999 999 984 528 707 650 067 243 413 449 791 234 062 72 × 2 = 0 + 0.000 000 000 465 279 999 999 999 969 057 415 300 134 486 826 899 582 468 125 44;
  • 8) 0.000 000 000 465 279 999 999 999 969 057 415 300 134 486 826 899 582 468 125 44 × 2 = 0 + 0.000 000 000 930 559 999 999 999 938 114 830 600 268 973 653 799 164 936 250 88;
  • 9) 0.000 000 000 930 559 999 999 999 938 114 830 600 268 973 653 799 164 936 250 88 × 2 = 0 + 0.000 000 001 861 119 999 999 999 876 229 661 200 537 947 307 598 329 872 501 76;
  • 10) 0.000 000 001 861 119 999 999 999 876 229 661 200 537 947 307 598 329 872 501 76 × 2 = 0 + 0.000 000 003 722 239 999 999 999 752 459 322 401 075 894 615 196 659 745 003 52;
  • 11) 0.000 000 003 722 239 999 999 999 752 459 322 401 075 894 615 196 659 745 003 52 × 2 = 0 + 0.000 000 007 444 479 999 999 999 504 918 644 802 151 789 230 393 319 490 007 04;
  • 12) 0.000 000 007 444 479 999 999 999 504 918 644 802 151 789 230 393 319 490 007 04 × 2 = 0 + 0.000 000 014 888 959 999 999 999 009 837 289 604 303 578 460 786 638 980 014 08;
  • 13) 0.000 000 014 888 959 999 999 999 009 837 289 604 303 578 460 786 638 980 014 08 × 2 = 0 + 0.000 000 029 777 919 999 999 998 019 674 579 208 607 156 921 573 277 960 028 16;
  • 14) 0.000 000 029 777 919 999 999 998 019 674 579 208 607 156 921 573 277 960 028 16 × 2 = 0 + 0.000 000 059 555 839 999 999 996 039 349 158 417 214 313 843 146 555 920 056 32;
  • 15) 0.000 000 059 555 839 999 999 996 039 349 158 417 214 313 843 146 555 920 056 32 × 2 = 0 + 0.000 000 119 111 679 999 999 992 078 698 316 834 428 627 686 293 111 840 112 64;
  • 16) 0.000 000 119 111 679 999 999 992 078 698 316 834 428 627 686 293 111 840 112 64 × 2 = 0 + 0.000 000 238 223 359 999 999 984 157 396 633 668 857 255 372 586 223 680 225 28;
  • 17) 0.000 000 238 223 359 999 999 984 157 396 633 668 857 255 372 586 223 680 225 28 × 2 = 0 + 0.000 000 476 446 719 999 999 968 314 793 267 337 714 510 745 172 447 360 450 56;
  • 18) 0.000 000 476 446 719 999 999 968 314 793 267 337 714 510 745 172 447 360 450 56 × 2 = 0 + 0.000 000 952 893 439 999 999 936 629 586 534 675 429 021 490 344 894 720 901 12;
  • 19) 0.000 000 952 893 439 999 999 936 629 586 534 675 429 021 490 344 894 720 901 12 × 2 = 0 + 0.000 001 905 786 879 999 999 873 259 173 069 350 858 042 980 689 789 441 802 24;
  • 20) 0.000 001 905 786 879 999 999 873 259 173 069 350 858 042 980 689 789 441 802 24 × 2 = 0 + 0.000 003 811 573 759 999 999 746 518 346 138 701 716 085 961 379 578 883 604 48;
  • 21) 0.000 003 811 573 759 999 999 746 518 346 138 701 716 085 961 379 578 883 604 48 × 2 = 0 + 0.000 007 623 147 519 999 999 493 036 692 277 403 432 171 922 759 157 767 208 96;
  • 22) 0.000 007 623 147 519 999 999 493 036 692 277 403 432 171 922 759 157 767 208 96 × 2 = 0 + 0.000 015 246 295 039 999 998 986 073 384 554 806 864 343 845 518 315 534 417 92;
  • 23) 0.000 015 246 295 039 999 998 986 073 384 554 806 864 343 845 518 315 534 417 92 × 2 = 0 + 0.000 030 492 590 079 999 997 972 146 769 109 613 728 687 691 036 631 068 835 84;
  • 24) 0.000 030 492 590 079 999 997 972 146 769 109 613 728 687 691 036 631 068 835 84 × 2 = 0 + 0.000 060 985 180 159 999 995 944 293 538 219 227 457 375 382 073 262 137 671 68;
  • 25) 0.000 060 985 180 159 999 995 944 293 538 219 227 457 375 382 073 262 137 671 68 × 2 = 0 + 0.000 121 970 360 319 999 991 888 587 076 438 454 914 750 764 146 524 275 343 36;
  • 26) 0.000 121 970 360 319 999 991 888 587 076 438 454 914 750 764 146 524 275 343 36 × 2 = 0 + 0.000 243 940 720 639 999 983 777 174 152 876 909 829 501 528 293 048 550 686 72;
  • 27) 0.000 243 940 720 639 999 983 777 174 152 876 909 829 501 528 293 048 550 686 72 × 2 = 0 + 0.000 487 881 441 279 999 967 554 348 305 753 819 659 003 056 586 097 101 373 44;
  • 28) 0.000 487 881 441 279 999 967 554 348 305 753 819 659 003 056 586 097 101 373 44 × 2 = 0 + 0.000 975 762 882 559 999 935 108 696 611 507 639 318 006 113 172 194 202 746 88;
  • 29) 0.000 975 762 882 559 999 935 108 696 611 507 639 318 006 113 172 194 202 746 88 × 2 = 0 + 0.001 951 525 765 119 999 870 217 393 223 015 278 636 012 226 344 388 405 493 76;
  • 30) 0.001 951 525 765 119 999 870 217 393 223 015 278 636 012 226 344 388 405 493 76 × 2 = 0 + 0.003 903 051 530 239 999 740 434 786 446 030 557 272 024 452 688 776 810 987 52;
  • 31) 0.003 903 051 530 239 999 740 434 786 446 030 557 272 024 452 688 776 810 987 52 × 2 = 0 + 0.007 806 103 060 479 999 480 869 572 892 061 114 544 048 905 377 553 621 975 04;
  • 32) 0.007 806 103 060 479 999 480 869 572 892 061 114 544 048 905 377 553 621 975 04 × 2 = 0 + 0.015 612 206 120 959 998 961 739 145 784 122 229 088 097 810 755 107 243 950 08;
  • 33) 0.015 612 206 120 959 998 961 739 145 784 122 229 088 097 810 755 107 243 950 08 × 2 = 0 + 0.031 224 412 241 919 997 923 478 291 568 244 458 176 195 621 510 214 487 900 16;
  • 34) 0.031 224 412 241 919 997 923 478 291 568 244 458 176 195 621 510 214 487 900 16 × 2 = 0 + 0.062 448 824 483 839 995 846 956 583 136 488 916 352 391 243 020 428 975 800 32;
  • 35) 0.062 448 824 483 839 995 846 956 583 136 488 916 352 391 243 020 428 975 800 32 × 2 = 0 + 0.124 897 648 967 679 991 693 913 166 272 977 832 704 782 486 040 857 951 600 64;
  • 36) 0.124 897 648 967 679 991 693 913 166 272 977 832 704 782 486 040 857 951 600 64 × 2 = 0 + 0.249 795 297 935 359 983 387 826 332 545 955 665 409 564 972 081 715 903 201 28;
  • 37) 0.249 795 297 935 359 983 387 826 332 545 955 665 409 564 972 081 715 903 201 28 × 2 = 0 + 0.499 590 595 870 719 966 775 652 665 091 911 330 819 129 944 163 431 806 402 56;
  • 38) 0.499 590 595 870 719 966 775 652 665 091 911 330 819 129 944 163 431 806 402 56 × 2 = 0 + 0.999 181 191 741 439 933 551 305 330 183 822 661 638 259 888 326 863 612 805 12;
  • 39) 0.999 181 191 741 439 933 551 305 330 183 822 661 638 259 888 326 863 612 805 12 × 2 = 1 + 0.998 362 383 482 879 867 102 610 660 367 645 323 276 519 776 653 727 225 610 24;
  • 40) 0.998 362 383 482 879 867 102 610 660 367 645 323 276 519 776 653 727 225 610 24 × 2 = 1 + 0.996 724 766 965 759 734 205 221 320 735 290 646 553 039 553 307 454 451 220 48;
  • 41) 0.996 724 766 965 759 734 205 221 320 735 290 646 553 039 553 307 454 451 220 48 × 2 = 1 + 0.993 449 533 931 519 468 410 442 641 470 581 293 106 079 106 614 908 902 440 96;
  • 42) 0.993 449 533 931 519 468 410 442 641 470 581 293 106 079 106 614 908 902 440 96 × 2 = 1 + 0.986 899 067 863 038 936 820 885 282 941 162 586 212 158 213 229 817 804 881 92;
  • 43) 0.986 899 067 863 038 936 820 885 282 941 162 586 212 158 213 229 817 804 881 92 × 2 = 1 + 0.973 798 135 726 077 873 641 770 565 882 325 172 424 316 426 459 635 609 763 84;
  • 44) 0.973 798 135 726 077 873 641 770 565 882 325 172 424 316 426 459 635 609 763 84 × 2 = 1 + 0.947 596 271 452 155 747 283 541 131 764 650 344 848 632 852 919 271 219 527 68;
  • 45) 0.947 596 271 452 155 747 283 541 131 764 650 344 848 632 852 919 271 219 527 68 × 2 = 1 + 0.895 192 542 904 311 494 567 082 263 529 300 689 697 265 705 838 542 439 055 36;
  • 46) 0.895 192 542 904 311 494 567 082 263 529 300 689 697 265 705 838 542 439 055 36 × 2 = 1 + 0.790 385 085 808 622 989 134 164 527 058 601 379 394 531 411 677 084 878 110 72;
  • 47) 0.790 385 085 808 622 989 134 164 527 058 601 379 394 531 411 677 084 878 110 72 × 2 = 1 + 0.580 770 171 617 245 978 268 329 054 117 202 758 789 062 823 354 169 756 221 44;
  • 48) 0.580 770 171 617 245 978 268 329 054 117 202 758 789 062 823 354 169 756 221 44 × 2 = 1 + 0.161 540 343 234 491 956 536 658 108 234 405 517 578 125 646 708 339 512 442 88;
  • 49) 0.161 540 343 234 491 956 536 658 108 234 405 517 578 125 646 708 339 512 442 88 × 2 = 0 + 0.323 080 686 468 983 913 073 316 216 468 811 035 156 251 293 416 679 024 885 76;
  • 50) 0.323 080 686 468 983 913 073 316 216 468 811 035 156 251 293 416 679 024 885 76 × 2 = 0 + 0.646 161 372 937 967 826 146 632 432 937 622 070 312 502 586 833 358 049 771 52;
  • 51) 0.646 161 372 937 967 826 146 632 432 937 622 070 312 502 586 833 358 049 771 52 × 2 = 1 + 0.292 322 745 875 935 652 293 264 865 875 244 140 625 005 173 666 716 099 543 04;
  • 52) 0.292 322 745 875 935 652 293 264 865 875 244 140 625 005 173 666 716 099 543 04 × 2 = 0 + 0.584 645 491 751 871 304 586 529 731 750 488 281 250 010 347 333 432 199 086 08;
  • 53) 0.584 645 491 751 871 304 586 529 731 750 488 281 250 010 347 333 432 199 086 08 × 2 = 1 + 0.169 290 983 503 742 609 173 059 463 500 976 562 500 020 694 666 864 398 172 16;
  • 54) 0.169 290 983 503 742 609 173 059 463 500 976 562 500 020 694 666 864 398 172 16 × 2 = 0 + 0.338 581 967 007 485 218 346 118 927 001 953 125 000 041 389 333 728 796 344 32;
  • 55) 0.338 581 967 007 485 218 346 118 927 001 953 125 000 041 389 333 728 796 344 32 × 2 = 0 + 0.677 163 934 014 970 436 692 237 854 003 906 250 000 082 778 667 457 592 688 64;
  • 56) 0.677 163 934 014 970 436 692 237 854 003 906 250 000 082 778 667 457 592 688 64 × 2 = 1 + 0.354 327 868 029 940 873 384 475 708 007 812 500 000 165 557 334 915 185 377 28;
  • 57) 0.354 327 868 029 940 873 384 475 708 007 812 500 000 165 557 334 915 185 377 28 × 2 = 0 + 0.708 655 736 059 881 746 768 951 416 015 625 000 000 331 114 669 830 370 754 56;
  • 58) 0.708 655 736 059 881 746 768 951 416 015 625 000 000 331 114 669 830 370 754 56 × 2 = 1 + 0.417 311 472 119 763 493 537 902 832 031 250 000 000 662 229 339 660 741 509 12;
  • 59) 0.417 311 472 119 763 493 537 902 832 031 250 000 000 662 229 339 660 741 509 12 × 2 = 0 + 0.834 622 944 239 526 987 075 805 664 062 500 000 001 324 458 679 321 483 018 24;
  • 60) 0.834 622 944 239 526 987 075 805 664 062 500 000 001 324 458 679 321 483 018 24 × 2 = 1 + 0.669 245 888 479 053 974 151 611 328 125 000 000 002 648 917 358 642 966 036 48;
  • 61) 0.669 245 888 479 053 974 151 611 328 125 000 000 002 648 917 358 642 966 036 48 × 2 = 1 + 0.338 491 776 958 107 948 303 222 656 250 000 000 005 297 834 717 285 932 072 96;
  • 62) 0.338 491 776 958 107 948 303 222 656 250 000 000 005 297 834 717 285 932 072 96 × 2 = 0 + 0.676 983 553 916 215 896 606 445 312 500 000 000 010 595 669 434 571 864 145 92;
  • 63) 0.676 983 553 916 215 896 606 445 312 500 000 000 010 595 669 434 571 864 145 92 × 2 = 1 + 0.353 967 107 832 431 793 212 890 625 000 000 000 021 191 338 869 143 728 291 84;
  • 64) 0.353 967 107 832 431 793 212 890 625 000 000 000 021 191 338 869 143 728 291 84 × 2 = 0 + 0.707 934 215 664 863 586 425 781 250 000 000 000 042 382 677 738 287 456 583 68;
  • 65) 0.707 934 215 664 863 586 425 781 250 000 000 000 042 382 677 738 287 456 583 68 × 2 = 1 + 0.415 868 431 329 727 172 851 562 500 000 000 000 084 765 355 476 574 913 167 36;
  • 66) 0.415 868 431 329 727 172 851 562 500 000 000 000 084 765 355 476 574 913 167 36 × 2 = 0 + 0.831 736 862 659 454 345 703 125 000 000 000 000 169 530 710 953 149 826 334 72;
  • 67) 0.831 736 862 659 454 345 703 125 000 000 000 000 169 530 710 953 149 826 334 72 × 2 = 1 + 0.663 473 725 318 908 691 406 250 000 000 000 000 339 061 421 906 299 652 669 44;
  • 68) 0.663 473 725 318 908 691 406 250 000 000 000 000 339 061 421 906 299 652 669 44 × 2 = 1 + 0.326 947 450 637 817 382 812 500 000 000 000 000 678 122 843 812 599 305 338 88;
  • 69) 0.326 947 450 637 817 382 812 500 000 000 000 000 678 122 843 812 599 305 338 88 × 2 = 0 + 0.653 894 901 275 634 765 625 000 000 000 000 001 356 245 687 625 198 610 677 76;
  • 70) 0.653 894 901 275 634 765 625 000 000 000 000 001 356 245 687 625 198 610 677 76 × 2 = 1 + 0.307 789 802 551 269 531 250 000 000 000 000 002 712 491 375 250 397 221 355 52;
  • 71) 0.307 789 802 551 269 531 250 000 000 000 000 002 712 491 375 250 397 221 355 52 × 2 = 0 + 0.615 579 605 102 539 062 500 000 000 000 000 005 424 982 750 500 794 442 711 04;
  • 72) 0.615 579 605 102 539 062 500 000 000 000 000 005 424 982 750 500 794 442 711 04 × 2 = 1 + 0.231 159 210 205 078 125 000 000 000 000 000 010 849 965 501 001 588 885 422 08;
  • 73) 0.231 159 210 205 078 125 000 000 000 000 000 010 849 965 501 001 588 885 422 08 × 2 = 0 + 0.462 318 420 410 156 250 000 000 000 000 000 021 699 931 002 003 177 770 844 16;
  • 74) 0.462 318 420 410 156 250 000 000 000 000 000 021 699 931 002 003 177 770 844 16 × 2 = 0 + 0.924 636 840 820 312 500 000 000 000 000 000 043 399 862 004 006 355 541 688 32;
  • 75) 0.924 636 840 820 312 500 000 000 000 000 000 043 399 862 004 006 355 541 688 32 × 2 = 1 + 0.849 273 681 640 625 000 000 000 000 000 000 086 799 724 008 012 711 083 376 64;
  • 76) 0.849 273 681 640 625 000 000 000 000 000 000 086 799 724 008 012 711 083 376 64 × 2 = 1 + 0.698 547 363 281 250 000 000 000 000 000 000 173 599 448 016 025 422 166 753 28;
  • 77) 0.698 547 363 281 250 000 000 000 000 000 000 173 599 448 016 025 422 166 753 28 × 2 = 1 + 0.397 094 726 562 500 000 000 000 000 000 000 347 198 896 032 050 844 333 506 56;
  • 78) 0.397 094 726 562 500 000 000 000 000 000 000 347 198 896 032 050 844 333 506 56 × 2 = 0 + 0.794 189 453 125 000 000 000 000 000 000 000 694 397 792 064 101 688 667 013 12;
  • 79) 0.794 189 453 125 000 000 000 000 000 000 000 694 397 792 064 101 688 667 013 12 × 2 = 1 + 0.588 378 906 250 000 000 000 000 000 000 001 388 795 584 128 203 377 334 026 24;
  • 80) 0.588 378 906 250 000 000 000 000 000 000 001 388 795 584 128 203 377 334 026 24 × 2 = 1 + 0.176 757 812 500 000 000 000 000 000 000 002 777 591 168 256 406 754 668 052 48;
  • 81) 0.176 757 812 500 000 000 000 000 000 000 002 777 591 168 256 406 754 668 052 48 × 2 = 0 + 0.353 515 625 000 000 000 000 000 000 000 005 555 182 336 512 813 509 336 104 96;
  • 82) 0.353 515 625 000 000 000 000 000 000 000 005 555 182 336 512 813 509 336 104 96 × 2 = 0 + 0.707 031 250 000 000 000 000 000 000 000 011 110 364 673 025 627 018 672 209 92;
  • 83) 0.707 031 250 000 000 000 000 000 000 000 011 110 364 673 025 627 018 672 209 92 × 2 = 1 + 0.414 062 500 000 000 000 000 000 000 000 022 220 729 346 051 254 037 344 419 84;
  • 84) 0.414 062 500 000 000 000 000 000 000 000 022 220 729 346 051 254 037 344 419 84 × 2 = 0 + 0.828 125 000 000 000 000 000 000 000 000 044 441 458 692 102 508 074 688 839 68;
  • 85) 0.828 125 000 000 000 000 000 000 000 000 044 441 458 692 102 508 074 688 839 68 × 2 = 1 + 0.656 250 000 000 000 000 000 000 000 000 088 882 917 384 205 016 149 377 679 36;
  • 86) 0.656 250 000 000 000 000 000 000 000 000 088 882 917 384 205 016 149 377 679 36 × 2 = 1 + 0.312 500 000 000 000 000 000 000 000 000 177 765 834 768 410 032 298 755 358 72;
  • 87) 0.312 500 000 000 000 000 000 000 000 000 177 765 834 768 410 032 298 755 358 72 × 2 = 0 + 0.625 000 000 000 000 000 000 000 000 000 355 531 669 536 820 064 597 510 717 44;
  • 88) 0.625 000 000 000 000 000 000 000 000 000 355 531 669 536 820 064 597 510 717 44 × 2 = 1 + 0.250 000 000 000 000 000 000 000 000 000 711 063 339 073 640 129 195 021 434 88;
  • 89) 0.250 000 000 000 000 000 000 000 000 000 711 063 339 073 640 129 195 021 434 88 × 2 = 0 + 0.500 000 000 000 000 000 000 000 000 001 422 126 678 147 280 258 390 042 869 76;
  • 90) 0.500 000 000 000 000 000 000 000 000 001 422 126 678 147 280 258 390 042 869 76 × 2 = 1 + 0.000 000 000 000 000 000 000 000 000 002 844 253 356 294 560 516 780 085 739 52;
  • 91) 0.000 000 000 000 000 000 000 000 000 002 844 253 356 294 560 516 780 085 739 52 × 2 = 0 + 0.000 000 000 000 000 000 000 000 000 005 688 506 712 589 121 033 560 171 479 04;

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:


0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 032 23(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:

0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 032 23(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:


0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 032 23(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 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 032 23 converted to 64 bit double precision IEEE 754 binary floating point representation:

0 - 011 1101 1000 - 1111 1111 1001 0100 1010 1101 0101 1010 1001 1101 1001 0110 1010

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How to convert numbers from the decimal system (base ten) to 64 bit double precision IEEE 754 binary floating point standard

Follow the steps below to convert a base 10 decimal number to 64 bit double precision IEEE 754 binary floating point:

  • 1. If the number to be converted is negative, start with its the positive version.
  • 2. First convert the integer part. Divide repeatedly by 2 the positive representation of the integer number that is to be converted to binary, until we get a quotient that is equal to zero, keeping track of each remainder.
  • 3. Construct the base 2 representation of the positive integer part of the number, by taking all the remainders from the previous operations, starting from the bottom of the list constructed above. Thus, the last remainder of the divisions becomes the first symbol (the leftmost) of the base two number, while the first remainder becomes the last symbol (the rightmost).
  • 4. Then convert the fractional part. Multiply the number repeatedly by 2, until we get a fractional part that is equal to zero, keeping track of each integer part of the results.
  • 5. Construct the base 2 representation of the fractional part of the number, by taking all the integer parts of the multiplying operations, starting from the top of the list constructed above (they should appear in the binary representation, from left to right, in the order they have been calculated).
  • 6. Normalize the binary representation of the number, shifting the decimal mark (the decimal point) "n" positions either to the left, or to the right, so that only one non zero digit remains to the left of the decimal mark.
  • 7. Adjust the exponent in 11 bit excess/bias notation and then convert it from decimal (base 10) to 11 bit binary, by using the same technique of repeatedly dividing by 2, as shown above:
    Exponent (adjusted) = Exponent (unadjusted) + 2(11-1) - 1
  • 8. Normalize mantissa, remove the leading (leftmost) bit, since it's allways '1' (and the decimal mark, if the case) and adjust its length to 52 bits, either by removing the excess bits from the right (losing precision...) or by adding extra bits set on '0' to the right.
  • 9. Sign (it takes 1 bit) is either 1 for a negative or 0 for a positive number.

Example: convert the negative number -31.640 215 from the decimal system (base ten) to 64 bit double precision IEEE 754 binary floating point:

  • 1. Start with the positive version of the number:

    |-31.640 215| = 31.640 215

  • 2. First convert the integer part, 31. Divide it repeatedly by 2, keeping track of each remainder, until we get a quotient that is equal to zero:
    • division = quotient + remainder;
    • 31 ÷ 2 = 15 + 1;
    • 15 ÷ 2 = 7 + 1;
    • 7 ÷ 2 = 3 + 1;
    • 3 ÷ 2 = 1 + 1;
    • 1 ÷ 2 = 0 + 1;
    • We have encountered a quotient that is ZERO => FULL STOP
  • 3. Construct the base 2 representation of the integer part of the number by taking all the remainders of the previous dividing operations, starting from the bottom of the list constructed above:

    31(10) = 1 1111(2)

  • 4. Then, convert the fractional part, 0.640 215. Multiply repeatedly by 2, keeping track of each integer part of the results, until we get a fractional part that is equal to zero:
    • #) multiplying = integer + fractional part;
    • 1) 0.640 215 × 2 = 1 + 0.280 43;
    • 2) 0.280 43 × 2 = 0 + 0.560 86;
    • 3) 0.560 86 × 2 = 1 + 0.121 72;
    • 4) 0.121 72 × 2 = 0 + 0.243 44;
    • 5) 0.243 44 × 2 = 0 + 0.486 88;
    • 6) 0.486 88 × 2 = 0 + 0.973 76;
    • 7) 0.973 76 × 2 = 1 + 0.947 52;
    • 8) 0.947 52 × 2 = 1 + 0.895 04;
    • 9) 0.895 04 × 2 = 1 + 0.790 08;
    • 10) 0.790 08 × 2 = 1 + 0.580 16;
    • 11) 0.580 16 × 2 = 1 + 0.160 32;
    • 12) 0.160 32 × 2 = 0 + 0.320 64;
    • 13) 0.320 64 × 2 = 0 + 0.641 28;
    • 14) 0.641 28 × 2 = 1 + 0.282 56;
    • 15) 0.282 56 × 2 = 0 + 0.565 12;
    • 16) 0.565 12 × 2 = 1 + 0.130 24;
    • 17) 0.130 24 × 2 = 0 + 0.260 48;
    • 18) 0.260 48 × 2 = 0 + 0.520 96;
    • 19) 0.520 96 × 2 = 1 + 0.041 92;
    • 20) 0.041 92 × 2 = 0 + 0.083 84;
    • 21) 0.083 84 × 2 = 0 + 0.167 68;
    • 22) 0.167 68 × 2 = 0 + 0.335 36;
    • 23) 0.335 36 × 2 = 0 + 0.670 72;
    • 24) 0.670 72 × 2 = 1 + 0.341 44;
    • 25) 0.341 44 × 2 = 0 + 0.682 88;
    • 26) 0.682 88 × 2 = 1 + 0.365 76;
    • 27) 0.365 76 × 2 = 0 + 0.731 52;
    • 28) 0.731 52 × 2 = 1 + 0.463 04;
    • 29) 0.463 04 × 2 = 0 + 0.926 08;
    • 30) 0.926 08 × 2 = 1 + 0.852 16;
    • 31) 0.852 16 × 2 = 1 + 0.704 32;
    • 32) 0.704 32 × 2 = 1 + 0.408 64;
    • 33) 0.408 64 × 2 = 0 + 0.817 28;
    • 34) 0.817 28 × 2 = 1 + 0.634 56;
    • 35) 0.634 56 × 2 = 1 + 0.269 12;
    • 36) 0.269 12 × 2 = 0 + 0.538 24;
    • 37) 0.538 24 × 2 = 1 + 0.076 48;
    • 38) 0.076 48 × 2 = 0 + 0.152 96;
    • 39) 0.152 96 × 2 = 0 + 0.305 92;
    • 40) 0.305 92 × 2 = 0 + 0.611 84;
    • 41) 0.611 84 × 2 = 1 + 0.223 68;
    • 42) 0.223 68 × 2 = 0 + 0.447 36;
    • 43) 0.447 36 × 2 = 0 + 0.894 72;
    • 44) 0.894 72 × 2 = 1 + 0.789 44;
    • 45) 0.789 44 × 2 = 1 + 0.578 88;
    • 46) 0.578 88 × 2 = 1 + 0.157 76;
    • 47) 0.157 76 × 2 = 0 + 0.315 52;
    • 48) 0.315 52 × 2 = 0 + 0.631 04;
    • 49) 0.631 04 × 2 = 1 + 0.262 08;
    • 50) 0.262 08 × 2 = 0 + 0.524 16;
    • 51) 0.524 16 × 2 = 1 + 0.048 32;
    • 52) 0.048 32 × 2 = 0 + 0.096 64;
    • 53) 0.096 64 × 2 = 0 + 0.193 28;
    • We didn't get any fractional part that was equal to zero. But we had enough iterations (over Mantissa limit = 52) and at least one integer part that was different from zero => FULL STOP (losing precision...).
  • 5. Construct the base 2 representation of the fractional part of the number, by taking all the integer parts of the previous multiplying operations, starting from the top of the constructed list above:

    0.640 215(10) = 0.1010 0011 1110 0101 0010 0001 0101 0111 0110 1000 1001 1100 1010 0(2)

  • 6. Summarizing - the positive number before normalization:

    31.640 215(10) = 1 1111.1010 0011 1110 0101 0010 0001 0101 0111 0110 1000 1001 1100 1010 0(2)

  • 7. Normalize the binary representation of the number, shifting the decimal mark 4 positions to the left so that only one non-zero digit stays to the left of the decimal mark:

    31.640 215(10) =
    1 1111.1010 0011 1110 0101 0010 0001 0101 0111 0110 1000 1001 1100 1010 0(2) =
    1 1111.1010 0011 1110 0101 0010 0001 0101 0111 0110 1000 1001 1100 1010 0(2) × 20 =
    1.1111 1010 0011 1110 0101 0010 0001 0101 0111 0110 1000 1001 1100 1010 0(2) × 24

  • 8. Up to this moment, there are the following elements that would feed into the 64 bit double precision IEEE 754 binary floating point representation:

    Sign: 1 (a negative number)

    Exponent (unadjusted): 4

    Mantissa (not-normalized): 1.1111 1010 0011 1110 0101 0010 0001 0101 0111 0110 1000 1001 1100 1010 0

  • 9. Adjust the exponent in 11 bit excess/bias notation and then convert it from decimal (base 10) to 11 bit binary (base 2), by using the same technique of repeatedly dividing it by 2, as shown above:

    Exponent (adjusted) = Exponent (unadjusted) + 2(11-1) - 1 = (4 + 1023)(10) = 1027(10) =
    100 0000 0011(2)

  • 10. Normalize mantissa, remove the leading (leftmost) bit, since it's allways '1' (and the decimal sign) and adjust its length to 52 bits, by removing the excess bits, from the right (losing precision...):

    Mantissa (not-normalized): 1.1111 1010 0011 1110 0101 0010 0001 0101 0111 0110 1000 1001 1100 1010 0

    Mantissa (normalized): 1111 1010 0011 1110 0101 0010 0001 0101 0111 0110 1000 1001 1100

  • Conclusion:

    Sign (1 bit) = 1 (a negative number)

    Exponent (8 bits) = 100 0000 0011

    Mantissa (52 bits) = 1111 1010 0011 1110 0101 0010 0001 0101 0111 0110 1000 1001 1100

  • Number -31.640 215, converted from decimal system (base 10) to 64 bit double precision IEEE 754 binary floating point =
    1 - 100 0000 0011 - 1111 1010 0011 1110 0101 0010 0001 0101 0111 0110 1000 1001 1100