0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 029 940 5 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 029 940 5(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
0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 029 940 5(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: 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 029 940 5.

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) 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 029 940 5 × 2 = 0 + 0.000 000 000 007 269 999 999 999 999 516 522 114 064 601 356 670 305 976 059 881;
  • 2) 0.000 000 000 007 269 999 999 999 999 516 522 114 064 601 356 670 305 976 059 881 × 2 = 0 + 0.000 000 000 014 539 999 999 999 999 033 044 228 129 202 713 340 611 952 119 762;
  • 3) 0.000 000 000 014 539 999 999 999 999 033 044 228 129 202 713 340 611 952 119 762 × 2 = 0 + 0.000 000 000 029 079 999 999 999 998 066 088 456 258 405 426 681 223 904 239 524;
  • 4) 0.000 000 000 029 079 999 999 999 998 066 088 456 258 405 426 681 223 904 239 524 × 2 = 0 + 0.000 000 000 058 159 999 999 999 996 132 176 912 516 810 853 362 447 808 479 048;
  • 5) 0.000 000 000 058 159 999 999 999 996 132 176 912 516 810 853 362 447 808 479 048 × 2 = 0 + 0.000 000 000 116 319 999 999 999 992 264 353 825 033 621 706 724 895 616 958 096;
  • 6) 0.000 000 000 116 319 999 999 999 992 264 353 825 033 621 706 724 895 616 958 096 × 2 = 0 + 0.000 000 000 232 639 999 999 999 984 528 707 650 067 243 413 449 791 233 916 192;
  • 7) 0.000 000 000 232 639 999 999 999 984 528 707 650 067 243 413 449 791 233 916 192 × 2 = 0 + 0.000 000 000 465 279 999 999 999 969 057 415 300 134 486 826 899 582 467 832 384;
  • 8) 0.000 000 000 465 279 999 999 999 969 057 415 300 134 486 826 899 582 467 832 384 × 2 = 0 + 0.000 000 000 930 559 999 999 999 938 114 830 600 268 973 653 799 164 935 664 768;
  • 9) 0.000 000 000 930 559 999 999 999 938 114 830 600 268 973 653 799 164 935 664 768 × 2 = 0 + 0.000 000 001 861 119 999 999 999 876 229 661 200 537 947 307 598 329 871 329 536;
  • 10) 0.000 000 001 861 119 999 999 999 876 229 661 200 537 947 307 598 329 871 329 536 × 2 = 0 + 0.000 000 003 722 239 999 999 999 752 459 322 401 075 894 615 196 659 742 659 072;
  • 11) 0.000 000 003 722 239 999 999 999 752 459 322 401 075 894 615 196 659 742 659 072 × 2 = 0 + 0.000 000 007 444 479 999 999 999 504 918 644 802 151 789 230 393 319 485 318 144;
  • 12) 0.000 000 007 444 479 999 999 999 504 918 644 802 151 789 230 393 319 485 318 144 × 2 = 0 + 0.000 000 014 888 959 999 999 999 009 837 289 604 303 578 460 786 638 970 636 288;
  • 13) 0.000 000 014 888 959 999 999 999 009 837 289 604 303 578 460 786 638 970 636 288 × 2 = 0 + 0.000 000 029 777 919 999 999 998 019 674 579 208 607 156 921 573 277 941 272 576;
  • 14) 0.000 000 029 777 919 999 999 998 019 674 579 208 607 156 921 573 277 941 272 576 × 2 = 0 + 0.000 000 059 555 839 999 999 996 039 349 158 417 214 313 843 146 555 882 545 152;
  • 15) 0.000 000 059 555 839 999 999 996 039 349 158 417 214 313 843 146 555 882 545 152 × 2 = 0 + 0.000 000 119 111 679 999 999 992 078 698 316 834 428 627 686 293 111 765 090 304;
  • 16) 0.000 000 119 111 679 999 999 992 078 698 316 834 428 627 686 293 111 765 090 304 × 2 = 0 + 0.000 000 238 223 359 999 999 984 157 396 633 668 857 255 372 586 223 530 180 608;
  • 17) 0.000 000 238 223 359 999 999 984 157 396 633 668 857 255 372 586 223 530 180 608 × 2 = 0 + 0.000 000 476 446 719 999 999 968 314 793 267 337 714 510 745 172 447 060 361 216;
  • 18) 0.000 000 476 446 719 999 999 968 314 793 267 337 714 510 745 172 447 060 361 216 × 2 = 0 + 0.000 000 952 893 439 999 999 936 629 586 534 675 429 021 490 344 894 120 722 432;
  • 19) 0.000 000 952 893 439 999 999 936 629 586 534 675 429 021 490 344 894 120 722 432 × 2 = 0 + 0.000 001 905 786 879 999 999 873 259 173 069 350 858 042 980 689 788 241 444 864;
  • 20) 0.000 001 905 786 879 999 999 873 259 173 069 350 858 042 980 689 788 241 444 864 × 2 = 0 + 0.000 003 811 573 759 999 999 746 518 346 138 701 716 085 961 379 576 482 889 728;
  • 21) 0.000 003 811 573 759 999 999 746 518 346 138 701 716 085 961 379 576 482 889 728 × 2 = 0 + 0.000 007 623 147 519 999 999 493 036 692 277 403 432 171 922 759 152 965 779 456;
  • 22) 0.000 007 623 147 519 999 999 493 036 692 277 403 432 171 922 759 152 965 779 456 × 2 = 0 + 0.000 015 246 295 039 999 998 986 073 384 554 806 864 343 845 518 305 931 558 912;
  • 23) 0.000 015 246 295 039 999 998 986 073 384 554 806 864 343 845 518 305 931 558 912 × 2 = 0 + 0.000 030 492 590 079 999 997 972 146 769 109 613 728 687 691 036 611 863 117 824;
  • 24) 0.000 030 492 590 079 999 997 972 146 769 109 613 728 687 691 036 611 863 117 824 × 2 = 0 + 0.000 060 985 180 159 999 995 944 293 538 219 227 457 375 382 073 223 726 235 648;
  • 25) 0.000 060 985 180 159 999 995 944 293 538 219 227 457 375 382 073 223 726 235 648 × 2 = 0 + 0.000 121 970 360 319 999 991 888 587 076 438 454 914 750 764 146 447 452 471 296;
  • 26) 0.000 121 970 360 319 999 991 888 587 076 438 454 914 750 764 146 447 452 471 296 × 2 = 0 + 0.000 243 940 720 639 999 983 777 174 152 876 909 829 501 528 292 894 904 942 592;
  • 27) 0.000 243 940 720 639 999 983 777 174 152 876 909 829 501 528 292 894 904 942 592 × 2 = 0 + 0.000 487 881 441 279 999 967 554 348 305 753 819 659 003 056 585 789 809 885 184;
  • 28) 0.000 487 881 441 279 999 967 554 348 305 753 819 659 003 056 585 789 809 885 184 × 2 = 0 + 0.000 975 762 882 559 999 935 108 696 611 507 639 318 006 113 171 579 619 770 368;
  • 29) 0.000 975 762 882 559 999 935 108 696 611 507 639 318 006 113 171 579 619 770 368 × 2 = 0 + 0.001 951 525 765 119 999 870 217 393 223 015 278 636 012 226 343 159 239 540 736;
  • 30) 0.001 951 525 765 119 999 870 217 393 223 015 278 636 012 226 343 159 239 540 736 × 2 = 0 + 0.003 903 051 530 239 999 740 434 786 446 030 557 272 024 452 686 318 479 081 472;
  • 31) 0.003 903 051 530 239 999 740 434 786 446 030 557 272 024 452 686 318 479 081 472 × 2 = 0 + 0.007 806 103 060 479 999 480 869 572 892 061 114 544 048 905 372 636 958 162 944;
  • 32) 0.007 806 103 060 479 999 480 869 572 892 061 114 544 048 905 372 636 958 162 944 × 2 = 0 + 0.015 612 206 120 959 998 961 739 145 784 122 229 088 097 810 745 273 916 325 888;
  • 33) 0.015 612 206 120 959 998 961 739 145 784 122 229 088 097 810 745 273 916 325 888 × 2 = 0 + 0.031 224 412 241 919 997 923 478 291 568 244 458 176 195 621 490 547 832 651 776;
  • 34) 0.031 224 412 241 919 997 923 478 291 568 244 458 176 195 621 490 547 832 651 776 × 2 = 0 + 0.062 448 824 483 839 995 846 956 583 136 488 916 352 391 242 981 095 665 303 552;
  • 35) 0.062 448 824 483 839 995 846 956 583 136 488 916 352 391 242 981 095 665 303 552 × 2 = 0 + 0.124 897 648 967 679 991 693 913 166 272 977 832 704 782 485 962 191 330 607 104;
  • 36) 0.124 897 648 967 679 991 693 913 166 272 977 832 704 782 485 962 191 330 607 104 × 2 = 0 + 0.249 795 297 935 359 983 387 826 332 545 955 665 409 564 971 924 382 661 214 208;
  • 37) 0.249 795 297 935 359 983 387 826 332 545 955 665 409 564 971 924 382 661 214 208 × 2 = 0 + 0.499 590 595 870 719 966 775 652 665 091 911 330 819 129 943 848 765 322 428 416;
  • 38) 0.499 590 595 870 719 966 775 652 665 091 911 330 819 129 943 848 765 322 428 416 × 2 = 0 + 0.999 181 191 741 439 933 551 305 330 183 822 661 638 259 887 697 530 644 856 832;
  • 39) 0.999 181 191 741 439 933 551 305 330 183 822 661 638 259 887 697 530 644 856 832 × 2 = 1 + 0.998 362 383 482 879 867 102 610 660 367 645 323 276 519 775 395 061 289 713 664;
  • 40) 0.998 362 383 482 879 867 102 610 660 367 645 323 276 519 775 395 061 289 713 664 × 2 = 1 + 0.996 724 766 965 759 734 205 221 320 735 290 646 553 039 550 790 122 579 427 328;
  • 41) 0.996 724 766 965 759 734 205 221 320 735 290 646 553 039 550 790 122 579 427 328 × 2 = 1 + 0.993 449 533 931 519 468 410 442 641 470 581 293 106 079 101 580 245 158 854 656;
  • 42) 0.993 449 533 931 519 468 410 442 641 470 581 293 106 079 101 580 245 158 854 656 × 2 = 1 + 0.986 899 067 863 038 936 820 885 282 941 162 586 212 158 203 160 490 317 709 312;
  • 43) 0.986 899 067 863 038 936 820 885 282 941 162 586 212 158 203 160 490 317 709 312 × 2 = 1 + 0.973 798 135 726 077 873 641 770 565 882 325 172 424 316 406 320 980 635 418 624;
  • 44) 0.973 798 135 726 077 873 641 770 565 882 325 172 424 316 406 320 980 635 418 624 × 2 = 1 + 0.947 596 271 452 155 747 283 541 131 764 650 344 848 632 812 641 961 270 837 248;
  • 45) 0.947 596 271 452 155 747 283 541 131 764 650 344 848 632 812 641 961 270 837 248 × 2 = 1 + 0.895 192 542 904 311 494 567 082 263 529 300 689 697 265 625 283 922 541 674 496;
  • 46) 0.895 192 542 904 311 494 567 082 263 529 300 689 697 265 625 283 922 541 674 496 × 2 = 1 + 0.790 385 085 808 622 989 134 164 527 058 601 379 394 531 250 567 845 083 348 992;
  • 47) 0.790 385 085 808 622 989 134 164 527 058 601 379 394 531 250 567 845 083 348 992 × 2 = 1 + 0.580 770 171 617 245 978 268 329 054 117 202 758 789 062 501 135 690 166 697 984;
  • 48) 0.580 770 171 617 245 978 268 329 054 117 202 758 789 062 501 135 690 166 697 984 × 2 = 1 + 0.161 540 343 234 491 956 536 658 108 234 405 517 578 125 002 271 380 333 395 968;
  • 49) 0.161 540 343 234 491 956 536 658 108 234 405 517 578 125 002 271 380 333 395 968 × 2 = 0 + 0.323 080 686 468 983 913 073 316 216 468 811 035 156 250 004 542 760 666 791 936;
  • 50) 0.323 080 686 468 983 913 073 316 216 468 811 035 156 250 004 542 760 666 791 936 × 2 = 0 + 0.646 161 372 937 967 826 146 632 432 937 622 070 312 500 009 085 521 333 583 872;
  • 51) 0.646 161 372 937 967 826 146 632 432 937 622 070 312 500 009 085 521 333 583 872 × 2 = 1 + 0.292 322 745 875 935 652 293 264 865 875 244 140 625 000 018 171 042 667 167 744;
  • 52) 0.292 322 745 875 935 652 293 264 865 875 244 140 625 000 018 171 042 667 167 744 × 2 = 0 + 0.584 645 491 751 871 304 586 529 731 750 488 281 250 000 036 342 085 334 335 488;
  • 53) 0.584 645 491 751 871 304 586 529 731 750 488 281 250 000 036 342 085 334 335 488 × 2 = 1 + 0.169 290 983 503 742 609 173 059 463 500 976 562 500 000 072 684 170 668 670 976;
  • 54) 0.169 290 983 503 742 609 173 059 463 500 976 562 500 000 072 684 170 668 670 976 × 2 = 0 + 0.338 581 967 007 485 218 346 118 927 001 953 125 000 000 145 368 341 337 341 952;
  • 55) 0.338 581 967 007 485 218 346 118 927 001 953 125 000 000 145 368 341 337 341 952 × 2 = 0 + 0.677 163 934 014 970 436 692 237 854 003 906 250 000 000 290 736 682 674 683 904;
  • 56) 0.677 163 934 014 970 436 692 237 854 003 906 250 000 000 290 736 682 674 683 904 × 2 = 1 + 0.354 327 868 029 940 873 384 475 708 007 812 500 000 000 581 473 365 349 367 808;
  • 57) 0.354 327 868 029 940 873 384 475 708 007 812 500 000 000 581 473 365 349 367 808 × 2 = 0 + 0.708 655 736 059 881 746 768 951 416 015 625 000 000 001 162 946 730 698 735 616;
  • 58) 0.708 655 736 059 881 746 768 951 416 015 625 000 000 001 162 946 730 698 735 616 × 2 = 1 + 0.417 311 472 119 763 493 537 902 832 031 250 000 000 002 325 893 461 397 471 232;
  • 59) 0.417 311 472 119 763 493 537 902 832 031 250 000 000 002 325 893 461 397 471 232 × 2 = 0 + 0.834 622 944 239 526 987 075 805 664 062 500 000 000 004 651 786 922 794 942 464;
  • 60) 0.834 622 944 239 526 987 075 805 664 062 500 000 000 004 651 786 922 794 942 464 × 2 = 1 + 0.669 245 888 479 053 974 151 611 328 125 000 000 000 009 303 573 845 589 884 928;
  • 61) 0.669 245 888 479 053 974 151 611 328 125 000 000 000 009 303 573 845 589 884 928 × 2 = 1 + 0.338 491 776 958 107 948 303 222 656 250 000 000 000 018 607 147 691 179 769 856;
  • 62) 0.338 491 776 958 107 948 303 222 656 250 000 000 000 018 607 147 691 179 769 856 × 2 = 0 + 0.676 983 553 916 215 896 606 445 312 500 000 000 000 037 214 295 382 359 539 712;
  • 63) 0.676 983 553 916 215 896 606 445 312 500 000 000 000 037 214 295 382 359 539 712 × 2 = 1 + 0.353 967 107 832 431 793 212 890 625 000 000 000 000 074 428 590 764 719 079 424;
  • 64) 0.353 967 107 832 431 793 212 890 625 000 000 000 000 074 428 590 764 719 079 424 × 2 = 0 + 0.707 934 215 664 863 586 425 781 250 000 000 000 000 148 857 181 529 438 158 848;
  • 65) 0.707 934 215 664 863 586 425 781 250 000 000 000 000 148 857 181 529 438 158 848 × 2 = 1 + 0.415 868 431 329 727 172 851 562 500 000 000 000 000 297 714 363 058 876 317 696;
  • 66) 0.415 868 431 329 727 172 851 562 500 000 000 000 000 297 714 363 058 876 317 696 × 2 = 0 + 0.831 736 862 659 454 345 703 125 000 000 000 000 000 595 428 726 117 752 635 392;
  • 67) 0.831 736 862 659 454 345 703 125 000 000 000 000 000 595 428 726 117 752 635 392 × 2 = 1 + 0.663 473 725 318 908 691 406 250 000 000 000 000 001 190 857 452 235 505 270 784;
  • 68) 0.663 473 725 318 908 691 406 250 000 000 000 000 001 190 857 452 235 505 270 784 × 2 = 1 + 0.326 947 450 637 817 382 812 500 000 000 000 000 002 381 714 904 471 010 541 568;
  • 69) 0.326 947 450 637 817 382 812 500 000 000 000 000 002 381 714 904 471 010 541 568 × 2 = 0 + 0.653 894 901 275 634 765 625 000 000 000 000 000 004 763 429 808 942 021 083 136;
  • 70) 0.653 894 901 275 634 765 625 000 000 000 000 000 004 763 429 808 942 021 083 136 × 2 = 1 + 0.307 789 802 551 269 531 250 000 000 000 000 000 009 526 859 617 884 042 166 272;
  • 71) 0.307 789 802 551 269 531 250 000 000 000 000 000 009 526 859 617 884 042 166 272 × 2 = 0 + 0.615 579 605 102 539 062 500 000 000 000 000 000 019 053 719 235 768 084 332 544;
  • 72) 0.615 579 605 102 539 062 500 000 000 000 000 000 019 053 719 235 768 084 332 544 × 2 = 1 + 0.231 159 210 205 078 125 000 000 000 000 000 000 038 107 438 471 536 168 665 088;
  • 73) 0.231 159 210 205 078 125 000 000 000 000 000 000 038 107 438 471 536 168 665 088 × 2 = 0 + 0.462 318 420 410 156 250 000 000 000 000 000 000 076 214 876 943 072 337 330 176;
  • 74) 0.462 318 420 410 156 250 000 000 000 000 000 000 076 214 876 943 072 337 330 176 × 2 = 0 + 0.924 636 840 820 312 500 000 000 000 000 000 000 152 429 753 886 144 674 660 352;
  • 75) 0.924 636 840 820 312 500 000 000 000 000 000 000 152 429 753 886 144 674 660 352 × 2 = 1 + 0.849 273 681 640 625 000 000 000 000 000 000 000 304 859 507 772 289 349 320 704;
  • 76) 0.849 273 681 640 625 000 000 000 000 000 000 000 304 859 507 772 289 349 320 704 × 2 = 1 + 0.698 547 363 281 250 000 000 000 000 000 000 000 609 719 015 544 578 698 641 408;
  • 77) 0.698 547 363 281 250 000 000 000 000 000 000 000 609 719 015 544 578 698 641 408 × 2 = 1 + 0.397 094 726 562 500 000 000 000 000 000 000 001 219 438 031 089 157 397 282 816;
  • 78) 0.397 094 726 562 500 000 000 000 000 000 000 001 219 438 031 089 157 397 282 816 × 2 = 0 + 0.794 189 453 125 000 000 000 000 000 000 000 002 438 876 062 178 314 794 565 632;
  • 79) 0.794 189 453 125 000 000 000 000 000 000 000 002 438 876 062 178 314 794 565 632 × 2 = 1 + 0.588 378 906 250 000 000 000 000 000 000 000 004 877 752 124 356 629 589 131 264;
  • 80) 0.588 378 906 250 000 000 000 000 000 000 000 004 877 752 124 356 629 589 131 264 × 2 = 1 + 0.176 757 812 500 000 000 000 000 000 000 000 009 755 504 248 713 259 178 262 528;
  • 81) 0.176 757 812 500 000 000 000 000 000 000 000 009 755 504 248 713 259 178 262 528 × 2 = 0 + 0.353 515 625 000 000 000 000 000 000 000 000 019 511 008 497 426 518 356 525 056;
  • 82) 0.353 515 625 000 000 000 000 000 000 000 000 019 511 008 497 426 518 356 525 056 × 2 = 0 + 0.707 031 250 000 000 000 000 000 000 000 000 039 022 016 994 853 036 713 050 112;
  • 83) 0.707 031 250 000 000 000 000 000 000 000 000 039 022 016 994 853 036 713 050 112 × 2 = 1 + 0.414 062 500 000 000 000 000 000 000 000 000 078 044 033 989 706 073 426 100 224;
  • 84) 0.414 062 500 000 000 000 000 000 000 000 000 078 044 033 989 706 073 426 100 224 × 2 = 0 + 0.828 125 000 000 000 000 000 000 000 000 000 156 088 067 979 412 146 852 200 448;
  • 85) 0.828 125 000 000 000 000 000 000 000 000 000 156 088 067 979 412 146 852 200 448 × 2 = 1 + 0.656 250 000 000 000 000 000 000 000 000 000 312 176 135 958 824 293 704 400 896;
  • 86) 0.656 250 000 000 000 000 000 000 000 000 000 312 176 135 958 824 293 704 400 896 × 2 = 1 + 0.312 500 000 000 000 000 000 000 000 000 000 624 352 271 917 648 587 408 801 792;
  • 87) 0.312 500 000 000 000 000 000 000 000 000 000 624 352 271 917 648 587 408 801 792 × 2 = 0 + 0.625 000 000 000 000 000 000 000 000 000 001 248 704 543 835 297 174 817 603 584;
  • 88) 0.625 000 000 000 000 000 000 000 000 000 001 248 704 543 835 297 174 817 603 584 × 2 = 1 + 0.250 000 000 000 000 000 000 000 000 000 002 497 409 087 670 594 349 635 207 168;
  • 89) 0.250 000 000 000 000 000 000 000 000 000 002 497 409 087 670 594 349 635 207 168 × 2 = 0 + 0.500 000 000 000 000 000 000 000 000 000 004 994 818 175 341 188 699 270 414 336;
  • 90) 0.500 000 000 000 000 000 000 000 000 000 004 994 818 175 341 188 699 270 414 336 × 2 = 1 + 0.000 000 000 000 000 000 000 000 000 000 009 989 636 350 682 377 398 540 828 672;
  • 91) 0.000 000 000 000 000 000 000 000 000 000 009 989 636 350 682 377 398 540 828 672 × 2 = 0 + 0.000 000 000 000 000 000 000 000 000 000 019 979 272 701 364 754 797 081 657 344;

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:


0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 029 940 5(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:

0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 029 940 5(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:


0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 029 940 5(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 0.000 000 000 003 634 999 999 999 999 758 261 057 032 300 678 335 152 988 029 940 5 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