0.000 000 000 000 000 012 345 687 894 564 589 438 728 960 367 05 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.000 000 000 000 000 012 345 687 894 564 589 438 728 960 367 05(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
0.000 000 000 000 000 012 345 687 894 564 589 438 728 960 367 05(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: 0.000 000 000 000 000 012 345 687 894 564 589 438 728 960 367 05.

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) 0.000 000 000 000 000 012 345 687 894 564 589 438 728 960 367 05 × 2 = 0 + 0.000 000 000 000 000 024 691 375 789 129 178 877 457 920 734 1;
  • 2) 0.000 000 000 000 000 024 691 375 789 129 178 877 457 920 734 1 × 2 = 0 + 0.000 000 000 000 000 049 382 751 578 258 357 754 915 841 468 2;
  • 3) 0.000 000 000 000 000 049 382 751 578 258 357 754 915 841 468 2 × 2 = 0 + 0.000 000 000 000 000 098 765 503 156 516 715 509 831 682 936 4;
  • 4) 0.000 000 000 000 000 098 765 503 156 516 715 509 831 682 936 4 × 2 = 0 + 0.000 000 000 000 000 197 531 006 313 033 431 019 663 365 872 8;
  • 5) 0.000 000 000 000 000 197 531 006 313 033 431 019 663 365 872 8 × 2 = 0 + 0.000 000 000 000 000 395 062 012 626 066 862 039 326 731 745 6;
  • 6) 0.000 000 000 000 000 395 062 012 626 066 862 039 326 731 745 6 × 2 = 0 + 0.000 000 000 000 000 790 124 025 252 133 724 078 653 463 491 2;
  • 7) 0.000 000 000 000 000 790 124 025 252 133 724 078 653 463 491 2 × 2 = 0 + 0.000 000 000 000 001 580 248 050 504 267 448 157 306 926 982 4;
  • 8) 0.000 000 000 000 001 580 248 050 504 267 448 157 306 926 982 4 × 2 = 0 + 0.000 000 000 000 003 160 496 101 008 534 896 314 613 853 964 8;
  • 9) 0.000 000 000 000 003 160 496 101 008 534 896 314 613 853 964 8 × 2 = 0 + 0.000 000 000 000 006 320 992 202 017 069 792 629 227 707 929 6;
  • 10) 0.000 000 000 000 006 320 992 202 017 069 792 629 227 707 929 6 × 2 = 0 + 0.000 000 000 000 012 641 984 404 034 139 585 258 455 415 859 2;
  • 11) 0.000 000 000 000 012 641 984 404 034 139 585 258 455 415 859 2 × 2 = 0 + 0.000 000 000 000 025 283 968 808 068 279 170 516 910 831 718 4;
  • 12) 0.000 000 000 000 025 283 968 808 068 279 170 516 910 831 718 4 × 2 = 0 + 0.000 000 000 000 050 567 937 616 136 558 341 033 821 663 436 8;
  • 13) 0.000 000 000 000 050 567 937 616 136 558 341 033 821 663 436 8 × 2 = 0 + 0.000 000 000 000 101 135 875 232 273 116 682 067 643 326 873 6;
  • 14) 0.000 000 000 000 101 135 875 232 273 116 682 067 643 326 873 6 × 2 = 0 + 0.000 000 000 000 202 271 750 464 546 233 364 135 286 653 747 2;
  • 15) 0.000 000 000 000 202 271 750 464 546 233 364 135 286 653 747 2 × 2 = 0 + 0.000 000 000 000 404 543 500 929 092 466 728 270 573 307 494 4;
  • 16) 0.000 000 000 000 404 543 500 929 092 466 728 270 573 307 494 4 × 2 = 0 + 0.000 000 000 000 809 087 001 858 184 933 456 541 146 614 988 8;
  • 17) 0.000 000 000 000 809 087 001 858 184 933 456 541 146 614 988 8 × 2 = 0 + 0.000 000 000 001 618 174 003 716 369 866 913 082 293 229 977 6;
  • 18) 0.000 000 000 001 618 174 003 716 369 866 913 082 293 229 977 6 × 2 = 0 + 0.000 000 000 003 236 348 007 432 739 733 826 164 586 459 955 2;
  • 19) 0.000 000 000 003 236 348 007 432 739 733 826 164 586 459 955 2 × 2 = 0 + 0.000 000 000 006 472 696 014 865 479 467 652 329 172 919 910 4;
  • 20) 0.000 000 000 006 472 696 014 865 479 467 652 329 172 919 910 4 × 2 = 0 + 0.000 000 000 012 945 392 029 730 958 935 304 658 345 839 820 8;
  • 21) 0.000 000 000 012 945 392 029 730 958 935 304 658 345 839 820 8 × 2 = 0 + 0.000 000 000 025 890 784 059 461 917 870 609 316 691 679 641 6;
  • 22) 0.000 000 000 025 890 784 059 461 917 870 609 316 691 679 641 6 × 2 = 0 + 0.000 000 000 051 781 568 118 923 835 741 218 633 383 359 283 2;
  • 23) 0.000 000 000 051 781 568 118 923 835 741 218 633 383 359 283 2 × 2 = 0 + 0.000 000 000 103 563 136 237 847 671 482 437 266 766 718 566 4;
  • 24) 0.000 000 000 103 563 136 237 847 671 482 437 266 766 718 566 4 × 2 = 0 + 0.000 000 000 207 126 272 475 695 342 964 874 533 533 437 132 8;
  • 25) 0.000 000 000 207 126 272 475 695 342 964 874 533 533 437 132 8 × 2 = 0 + 0.000 000 000 414 252 544 951 390 685 929 749 067 066 874 265 6;
  • 26) 0.000 000 000 414 252 544 951 390 685 929 749 067 066 874 265 6 × 2 = 0 + 0.000 000 000 828 505 089 902 781 371 859 498 134 133 748 531 2;
  • 27) 0.000 000 000 828 505 089 902 781 371 859 498 134 133 748 531 2 × 2 = 0 + 0.000 000 001 657 010 179 805 562 743 718 996 268 267 497 062 4;
  • 28) 0.000 000 001 657 010 179 805 562 743 718 996 268 267 497 062 4 × 2 = 0 + 0.000 000 003 314 020 359 611 125 487 437 992 536 534 994 124 8;
  • 29) 0.000 000 003 314 020 359 611 125 487 437 992 536 534 994 124 8 × 2 = 0 + 0.000 000 006 628 040 719 222 250 974 875 985 073 069 988 249 6;
  • 30) 0.000 000 006 628 040 719 222 250 974 875 985 073 069 988 249 6 × 2 = 0 + 0.000 000 013 256 081 438 444 501 949 751 970 146 139 976 499 2;
  • 31) 0.000 000 013 256 081 438 444 501 949 751 970 146 139 976 499 2 × 2 = 0 + 0.000 000 026 512 162 876 889 003 899 503 940 292 279 952 998 4;
  • 32) 0.000 000 026 512 162 876 889 003 899 503 940 292 279 952 998 4 × 2 = 0 + 0.000 000 053 024 325 753 778 007 799 007 880 584 559 905 996 8;
  • 33) 0.000 000 053 024 325 753 778 007 799 007 880 584 559 905 996 8 × 2 = 0 + 0.000 000 106 048 651 507 556 015 598 015 761 169 119 811 993 6;
  • 34) 0.000 000 106 048 651 507 556 015 598 015 761 169 119 811 993 6 × 2 = 0 + 0.000 000 212 097 303 015 112 031 196 031 522 338 239 623 987 2;
  • 35) 0.000 000 212 097 303 015 112 031 196 031 522 338 239 623 987 2 × 2 = 0 + 0.000 000 424 194 606 030 224 062 392 063 044 676 479 247 974 4;
  • 36) 0.000 000 424 194 606 030 224 062 392 063 044 676 479 247 974 4 × 2 = 0 + 0.000 000 848 389 212 060 448 124 784 126 089 352 958 495 948 8;
  • 37) 0.000 000 848 389 212 060 448 124 784 126 089 352 958 495 948 8 × 2 = 0 + 0.000 001 696 778 424 120 896 249 568 252 178 705 916 991 897 6;
  • 38) 0.000 001 696 778 424 120 896 249 568 252 178 705 916 991 897 6 × 2 = 0 + 0.000 003 393 556 848 241 792 499 136 504 357 411 833 983 795 2;
  • 39) 0.000 003 393 556 848 241 792 499 136 504 357 411 833 983 795 2 × 2 = 0 + 0.000 006 787 113 696 483 584 998 273 008 714 823 667 967 590 4;
  • 40) 0.000 006 787 113 696 483 584 998 273 008 714 823 667 967 590 4 × 2 = 0 + 0.000 013 574 227 392 967 169 996 546 017 429 647 335 935 180 8;
  • 41) 0.000 013 574 227 392 967 169 996 546 017 429 647 335 935 180 8 × 2 = 0 + 0.000 027 148 454 785 934 339 993 092 034 859 294 671 870 361 6;
  • 42) 0.000 027 148 454 785 934 339 993 092 034 859 294 671 870 361 6 × 2 = 0 + 0.000 054 296 909 571 868 679 986 184 069 718 589 343 740 723 2;
  • 43) 0.000 054 296 909 571 868 679 986 184 069 718 589 343 740 723 2 × 2 = 0 + 0.000 108 593 819 143 737 359 972 368 139 437 178 687 481 446 4;
  • 44) 0.000 108 593 819 143 737 359 972 368 139 437 178 687 481 446 4 × 2 = 0 + 0.000 217 187 638 287 474 719 944 736 278 874 357 374 962 892 8;
  • 45) 0.000 217 187 638 287 474 719 944 736 278 874 357 374 962 892 8 × 2 = 0 + 0.000 434 375 276 574 949 439 889 472 557 748 714 749 925 785 6;
  • 46) 0.000 434 375 276 574 949 439 889 472 557 748 714 749 925 785 6 × 2 = 0 + 0.000 868 750 553 149 898 879 778 945 115 497 429 499 851 571 2;
  • 47) 0.000 868 750 553 149 898 879 778 945 115 497 429 499 851 571 2 × 2 = 0 + 0.001 737 501 106 299 797 759 557 890 230 994 858 999 703 142 4;
  • 48) 0.001 737 501 106 299 797 759 557 890 230 994 858 999 703 142 4 × 2 = 0 + 0.003 475 002 212 599 595 519 115 780 461 989 717 999 406 284 8;
  • 49) 0.003 475 002 212 599 595 519 115 780 461 989 717 999 406 284 8 × 2 = 0 + 0.006 950 004 425 199 191 038 231 560 923 979 435 998 812 569 6;
  • 50) 0.006 950 004 425 199 191 038 231 560 923 979 435 998 812 569 6 × 2 = 0 + 0.013 900 008 850 398 382 076 463 121 847 958 871 997 625 139 2;
  • 51) 0.013 900 008 850 398 382 076 463 121 847 958 871 997 625 139 2 × 2 = 0 + 0.027 800 017 700 796 764 152 926 243 695 917 743 995 250 278 4;
  • 52) 0.027 800 017 700 796 764 152 926 243 695 917 743 995 250 278 4 × 2 = 0 + 0.055 600 035 401 593 528 305 852 487 391 835 487 990 500 556 8;
  • 53) 0.055 600 035 401 593 528 305 852 487 391 835 487 990 500 556 8 × 2 = 0 + 0.111 200 070 803 187 056 611 704 974 783 670 975 981 001 113 6;
  • 54) 0.111 200 070 803 187 056 611 704 974 783 670 975 981 001 113 6 × 2 = 0 + 0.222 400 141 606 374 113 223 409 949 567 341 951 962 002 227 2;
  • 55) 0.222 400 141 606 374 113 223 409 949 567 341 951 962 002 227 2 × 2 = 0 + 0.444 800 283 212 748 226 446 819 899 134 683 903 924 004 454 4;
  • 56) 0.444 800 283 212 748 226 446 819 899 134 683 903 924 004 454 4 × 2 = 0 + 0.889 600 566 425 496 452 893 639 798 269 367 807 848 008 908 8;
  • 57) 0.889 600 566 425 496 452 893 639 798 269 367 807 848 008 908 8 × 2 = 1 + 0.779 201 132 850 992 905 787 279 596 538 735 615 696 017 817 6;
  • 58) 0.779 201 132 850 992 905 787 279 596 538 735 615 696 017 817 6 × 2 = 1 + 0.558 402 265 701 985 811 574 559 193 077 471 231 392 035 635 2;
  • 59) 0.558 402 265 701 985 811 574 559 193 077 471 231 392 035 635 2 × 2 = 1 + 0.116 804 531 403 971 623 149 118 386 154 942 462 784 071 270 4;
  • 60) 0.116 804 531 403 971 623 149 118 386 154 942 462 784 071 270 4 × 2 = 0 + 0.233 609 062 807 943 246 298 236 772 309 884 925 568 142 540 8;
  • 61) 0.233 609 062 807 943 246 298 236 772 309 884 925 568 142 540 8 × 2 = 0 + 0.467 218 125 615 886 492 596 473 544 619 769 851 136 285 081 6;
  • 62) 0.467 218 125 615 886 492 596 473 544 619 769 851 136 285 081 6 × 2 = 0 + 0.934 436 251 231 772 985 192 947 089 239 539 702 272 570 163 2;
  • 63) 0.934 436 251 231 772 985 192 947 089 239 539 702 272 570 163 2 × 2 = 1 + 0.868 872 502 463 545 970 385 894 178 479 079 404 545 140 326 4;
  • 64) 0.868 872 502 463 545 970 385 894 178 479 079 404 545 140 326 4 × 2 = 1 + 0.737 745 004 927 091 940 771 788 356 958 158 809 090 280 652 8;
  • 65) 0.737 745 004 927 091 940 771 788 356 958 158 809 090 280 652 8 × 2 = 1 + 0.475 490 009 854 183 881 543 576 713 916 317 618 180 561 305 6;
  • 66) 0.475 490 009 854 183 881 543 576 713 916 317 618 180 561 305 6 × 2 = 0 + 0.950 980 019 708 367 763 087 153 427 832 635 236 361 122 611 2;
  • 67) 0.950 980 019 708 367 763 087 153 427 832 635 236 361 122 611 2 × 2 = 1 + 0.901 960 039 416 735 526 174 306 855 665 270 472 722 245 222 4;
  • 68) 0.901 960 039 416 735 526 174 306 855 665 270 472 722 245 222 4 × 2 = 1 + 0.803 920 078 833 471 052 348 613 711 330 540 945 444 490 444 8;
  • 69) 0.803 920 078 833 471 052 348 613 711 330 540 945 444 490 444 8 × 2 = 1 + 0.607 840 157 666 942 104 697 227 422 661 081 890 888 980 889 6;
  • 70) 0.607 840 157 666 942 104 697 227 422 661 081 890 888 980 889 6 × 2 = 1 + 0.215 680 315 333 884 209 394 454 845 322 163 781 777 961 779 2;
  • 71) 0.215 680 315 333 884 209 394 454 845 322 163 781 777 961 779 2 × 2 = 0 + 0.431 360 630 667 768 418 788 909 690 644 327 563 555 923 558 4;
  • 72) 0.431 360 630 667 768 418 788 909 690 644 327 563 555 923 558 4 × 2 = 0 + 0.862 721 261 335 536 837 577 819 381 288 655 127 111 847 116 8;
  • 73) 0.862 721 261 335 536 837 577 819 381 288 655 127 111 847 116 8 × 2 = 1 + 0.725 442 522 671 073 675 155 638 762 577 310 254 223 694 233 6;
  • 74) 0.725 442 522 671 073 675 155 638 762 577 310 254 223 694 233 6 × 2 = 1 + 0.450 885 045 342 147 350 311 277 525 154 620 508 447 388 467 2;
  • 75) 0.450 885 045 342 147 350 311 277 525 154 620 508 447 388 467 2 × 2 = 0 + 0.901 770 090 684 294 700 622 555 050 309 241 016 894 776 934 4;
  • 76) 0.901 770 090 684 294 700 622 555 050 309 241 016 894 776 934 4 × 2 = 1 + 0.803 540 181 368 589 401 245 110 100 618 482 033 789 553 868 8;
  • 77) 0.803 540 181 368 589 401 245 110 100 618 482 033 789 553 868 8 × 2 = 1 + 0.607 080 362 737 178 802 490 220 201 236 964 067 579 107 737 6;
  • 78) 0.607 080 362 737 178 802 490 220 201 236 964 067 579 107 737 6 × 2 = 1 + 0.214 160 725 474 357 604 980 440 402 473 928 135 158 215 475 2;
  • 79) 0.214 160 725 474 357 604 980 440 402 473 928 135 158 215 475 2 × 2 = 0 + 0.428 321 450 948 715 209 960 880 804 947 856 270 316 430 950 4;
  • 80) 0.428 321 450 948 715 209 960 880 804 947 856 270 316 430 950 4 × 2 = 0 + 0.856 642 901 897 430 419 921 761 609 895 712 540 632 861 900 8;
  • 81) 0.856 642 901 897 430 419 921 761 609 895 712 540 632 861 900 8 × 2 = 1 + 0.713 285 803 794 860 839 843 523 219 791 425 081 265 723 801 6;
  • 82) 0.713 285 803 794 860 839 843 523 219 791 425 081 265 723 801 6 × 2 = 1 + 0.426 571 607 589 721 679 687 046 439 582 850 162 531 447 603 2;
  • 83) 0.426 571 607 589 721 679 687 046 439 582 850 162 531 447 603 2 × 2 = 0 + 0.853 143 215 179 443 359 374 092 879 165 700 325 062 895 206 4;
  • 84) 0.853 143 215 179 443 359 374 092 879 165 700 325 062 895 206 4 × 2 = 1 + 0.706 286 430 358 886 718 748 185 758 331 400 650 125 790 412 8;
  • 85) 0.706 286 430 358 886 718 748 185 758 331 400 650 125 790 412 8 × 2 = 1 + 0.412 572 860 717 773 437 496 371 516 662 801 300 251 580 825 6;
  • 86) 0.412 572 860 717 773 437 496 371 516 662 801 300 251 580 825 6 × 2 = 0 + 0.825 145 721 435 546 874 992 743 033 325 602 600 503 161 651 2;
  • 87) 0.825 145 721 435 546 874 992 743 033 325 602 600 503 161 651 2 × 2 = 1 + 0.650 291 442 871 093 749 985 486 066 651 205 201 006 323 302 4;
  • 88) 0.650 291 442 871 093 749 985 486 066 651 205 201 006 323 302 4 × 2 = 1 + 0.300 582 885 742 187 499 970 972 133 302 410 402 012 646 604 8;
  • 89) 0.300 582 885 742 187 499 970 972 133 302 410 402 012 646 604 8 × 2 = 0 + 0.601 165 771 484 374 999 941 944 266 604 820 804 025 293 209 6;
  • 90) 0.601 165 771 484 374 999 941 944 266 604 820 804 025 293 209 6 × 2 = 1 + 0.202 331 542 968 749 999 883 888 533 209 641 608 050 586 419 2;
  • 91) 0.202 331 542 968 749 999 883 888 533 209 641 608 050 586 419 2 × 2 = 0 + 0.404 663 085 937 499 999 767 777 066 419 283 216 101 172 838 4;
  • 92) 0.404 663 085 937 499 999 767 777 066 419 283 216 101 172 838 4 × 2 = 0 + 0.809 326 171 874 999 999 535 554 132 838 566 432 202 345 676 8;
  • 93) 0.809 326 171 874 999 999 535 554 132 838 566 432 202 345 676 8 × 2 = 1 + 0.618 652 343 749 999 999 071 108 265 677 132 864 404 691 353 6;
  • 94) 0.618 652 343 749 999 999 071 108 265 677 132 864 404 691 353 6 × 2 = 1 + 0.237 304 687 499 999 998 142 216 531 354 265 728 809 382 707 2;
  • 95) 0.237 304 687 499 999 998 142 216 531 354 265 728 809 382 707 2 × 2 = 0 + 0.474 609 374 999 999 996 284 433 062 708 531 457 618 765 414 4;
  • 96) 0.474 609 374 999 999 996 284 433 062 708 531 457 618 765 414 4 × 2 = 0 + 0.949 218 749 999 999 992 568 866 125 417 062 915 237 530 828 8;
  • 97) 0.949 218 749 999 999 992 568 866 125 417 062 915 237 530 828 8 × 2 = 1 + 0.898 437 499 999 999 985 137 732 250 834 125 830 475 061 657 6;
  • 98) 0.898 437 499 999 999 985 137 732 250 834 125 830 475 061 657 6 × 2 = 1 + 0.796 874 999 999 999 970 275 464 501 668 251 660 950 123 315 2;
  • 99) 0.796 874 999 999 999 970 275 464 501 668 251 660 950 123 315 2 × 2 = 1 + 0.593 749 999 999 999 940 550 929 003 336 503 321 900 246 630 4;
  • 100) 0.593 749 999 999 999 940 550 929 003 336 503 321 900 246 630 4 × 2 = 1 + 0.187 499 999 999 999 881 101 858 006 673 006 643 800 493 260 8;
  • 101) 0.187 499 999 999 999 881 101 858 006 673 006 643 800 493 260 8 × 2 = 0 + 0.374 999 999 999 999 762 203 716 013 346 013 287 600 986 521 6;
  • 102) 0.374 999 999 999 999 762 203 716 013 346 013 287 600 986 521 6 × 2 = 0 + 0.749 999 999 999 999 524 407 432 026 692 026 575 201 973 043 2;
  • 103) 0.749 999 999 999 999 524 407 432 026 692 026 575 201 973 043 2 × 2 = 1 + 0.499 999 999 999 999 048 814 864 053 384 053 150 403 946 086 4;
  • 104) 0.499 999 999 999 999 048 814 864 053 384 053 150 403 946 086 4 × 2 = 0 + 0.999 999 999 999 998 097 629 728 106 768 106 300 807 892 172 8;
  • 105) 0.999 999 999 999 998 097 629 728 106 768 106 300 807 892 172 8 × 2 = 1 + 0.999 999 999 999 996 195 259 456 213 536 212 601 615 784 345 6;
  • 106) 0.999 999 999 999 996 195 259 456 213 536 212 601 615 784 345 6 × 2 = 1 + 0.999 999 999 999 992 390 518 912 427 072 425 203 231 568 691 2;
  • 107) 0.999 999 999 999 992 390 518 912 427 072 425 203 231 568 691 2 × 2 = 1 + 0.999 999 999 999 984 781 037 824 854 144 850 406 463 137 382 4;
  • 108) 0.999 999 999 999 984 781 037 824 854 144 850 406 463 137 382 4 × 2 = 1 + 0.999 999 999 999 969 562 075 649 708 289 700 812 926 274 764 8;
  • 109) 0.999 999 999 999 969 562 075 649 708 289 700 812 926 274 764 8 × 2 = 1 + 0.999 999 999 999 939 124 151 299 416 579 401 625 852 549 529 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:


0.000 000 000 000 000 012 345 687 894 564 589 438 728 960 367 05(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 1110 0011 1011 1100 1101 1100 1101 1011 0100 1100 1111 0010 1111 1(2)

5. Positive number before normalization:

0.000 000 000 000 000 012 345 687 894 564 589 438 728 960 367 05(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 1110 0011 1011 1100 1101 1100 1101 1011 0100 1100 1111 0010 1111 1(2)

6. Normalize the binary representation of the number.

Shift the decimal mark 57 positions to the right, so that only one non zero digit remains to the left of it:


0.000 000 000 000 000 012 345 687 894 564 589 438 728 960 367 05(10) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 1110 0011 1011 1100 1101 1100 1101 1011 0100 1100 1111 0010 1111 1(2) =

0.0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 0000 1110 0011 1011 1100 1101 1100 1101 1011 0100 1100 1111 0010 1111 1(2) × 20 =

1.1100 0111 0111 1001 1011 1001 1011 0110 1001 1001 1110 0101 1111(2) × 2-57


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): -57

Mantissa (not normalized):
1.1100 0111 0111 1001 1011 1001 1011 0110 1001 1001 1110 0101 1111


8. Adjust the exponent.

Use the 11 bit excess/bias notation:


Exponent (adjusted) =

Exponent (unadjusted) + 2(11-1) - 1 =

-57 + 2(11-1) - 1 =

(-57 + 1 023)(10) =

966(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;
  • 966 ÷ 2 = 483 + 0;
  • 483 ÷ 2 = 241 + 1;
  • 241 ÷ 2 = 120 + 1;
  • 120 ÷ 2 = 60 + 0;
  • 60 ÷ 2 = 30 + 0;
  • 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) =

966(10) =

011 1100 0110(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. 1100 0111 0111 1001 1011 1001 1011 0110 1001 1001 1110 0101 1111 =

1100 0111 0111 1001 1011 1001 1011 0110 1001 1001 1110 0101 1111


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 1100 0110

Mantissa (52 bits) =
1100 0111 0111 1001 1011 1001 1011 0110 1001 1001 1110 0101 1111


Decimal number 0.000 000 000 000 000 012 345 687 894 564 589 438 728 960 367 05 converted to 64 bit double precision IEEE 754 binary floating point representation:

0 - 011 1100 0110 - 1100 0111 0111 1001 1011 1001 1011 0110 1001 1001 1110 0101 1111

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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