0.000 000 000 000 000 012 345 687 894 564 589 438 728 960 367 143 791 1 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.000 000 000 000 000 012 345 687 894 564 589 438 728 960 367 143 791 1(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
0.000 000 000 000 000 012 345 687 894 564 589 438 728 960 367 143 791 1(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: 0.000 000 000 000 000 012 345 687 894 564 589 438 728 960 367 143 791 1.

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) 0.000 000 000 000 000 012 345 687 894 564 589 438 728 960 367 143 791 1 × 2 = 0 + 0.000 000 000 000 000 024 691 375 789 129 178 877 457 920 734 287 582 2;
  • 2) 0.000 000 000 000 000 024 691 375 789 129 178 877 457 920 734 287 582 2 × 2 = 0 + 0.000 000 000 000 000 049 382 751 578 258 357 754 915 841 468 575 164 4;
  • 3) 0.000 000 000 000 000 049 382 751 578 258 357 754 915 841 468 575 164 4 × 2 = 0 + 0.000 000 000 000 000 098 765 503 156 516 715 509 831 682 937 150 328 8;
  • 4) 0.000 000 000 000 000 098 765 503 156 516 715 509 831 682 937 150 328 8 × 2 = 0 + 0.000 000 000 000 000 197 531 006 313 033 431 019 663 365 874 300 657 6;
  • 5) 0.000 000 000 000 000 197 531 006 313 033 431 019 663 365 874 300 657 6 × 2 = 0 + 0.000 000 000 000 000 395 062 012 626 066 862 039 326 731 748 601 315 2;
  • 6) 0.000 000 000 000 000 395 062 012 626 066 862 039 326 731 748 601 315 2 × 2 = 0 + 0.000 000 000 000 000 790 124 025 252 133 724 078 653 463 497 202 630 4;
  • 7) 0.000 000 000 000 000 790 124 025 252 133 724 078 653 463 497 202 630 4 × 2 = 0 + 0.000 000 000 000 001 580 248 050 504 267 448 157 306 926 994 405 260 8;
  • 8) 0.000 000 000 000 001 580 248 050 504 267 448 157 306 926 994 405 260 8 × 2 = 0 + 0.000 000 000 000 003 160 496 101 008 534 896 314 613 853 988 810 521 6;
  • 9) 0.000 000 000 000 003 160 496 101 008 534 896 314 613 853 988 810 521 6 × 2 = 0 + 0.000 000 000 000 006 320 992 202 017 069 792 629 227 707 977 621 043 2;
  • 10) 0.000 000 000 000 006 320 992 202 017 069 792 629 227 707 977 621 043 2 × 2 = 0 + 0.000 000 000 000 012 641 984 404 034 139 585 258 455 415 955 242 086 4;
  • 11) 0.000 000 000 000 012 641 984 404 034 139 585 258 455 415 955 242 086 4 × 2 = 0 + 0.000 000 000 000 025 283 968 808 068 279 170 516 910 831 910 484 172 8;
  • 12) 0.000 000 000 000 025 283 968 808 068 279 170 516 910 831 910 484 172 8 × 2 = 0 + 0.000 000 000 000 050 567 937 616 136 558 341 033 821 663 820 968 345 6;
  • 13) 0.000 000 000 000 050 567 937 616 136 558 341 033 821 663 820 968 345 6 × 2 = 0 + 0.000 000 000 000 101 135 875 232 273 116 682 067 643 327 641 936 691 2;
  • 14) 0.000 000 000 000 101 135 875 232 273 116 682 067 643 327 641 936 691 2 × 2 = 0 + 0.000 000 000 000 202 271 750 464 546 233 364 135 286 655 283 873 382 4;
  • 15) 0.000 000 000 000 202 271 750 464 546 233 364 135 286 655 283 873 382 4 × 2 = 0 + 0.000 000 000 000 404 543 500 929 092 466 728 270 573 310 567 746 764 8;
  • 16) 0.000 000 000 000 404 543 500 929 092 466 728 270 573 310 567 746 764 8 × 2 = 0 + 0.000 000 000 000 809 087 001 858 184 933 456 541 146 621 135 493 529 6;
  • 17) 0.000 000 000 000 809 087 001 858 184 933 456 541 146 621 135 493 529 6 × 2 = 0 + 0.000 000 000 001 618 174 003 716 369 866 913 082 293 242 270 987 059 2;
  • 18) 0.000 000 000 001 618 174 003 716 369 866 913 082 293 242 270 987 059 2 × 2 = 0 + 0.000 000 000 003 236 348 007 432 739 733 826 164 586 484 541 974 118 4;
  • 19) 0.000 000 000 003 236 348 007 432 739 733 826 164 586 484 541 974 118 4 × 2 = 0 + 0.000 000 000 006 472 696 014 865 479 467 652 329 172 969 083 948 236 8;
  • 20) 0.000 000 000 006 472 696 014 865 479 467 652 329 172 969 083 948 236 8 × 2 = 0 + 0.000 000 000 012 945 392 029 730 958 935 304 658 345 938 167 896 473 6;
  • 21) 0.000 000 000 012 945 392 029 730 958 935 304 658 345 938 167 896 473 6 × 2 = 0 + 0.000 000 000 025 890 784 059 461 917 870 609 316 691 876 335 792 947 2;
  • 22) 0.000 000 000 025 890 784 059 461 917 870 609 316 691 876 335 792 947 2 × 2 = 0 + 0.000 000 000 051 781 568 118 923 835 741 218 633 383 752 671 585 894 4;
  • 23) 0.000 000 000 051 781 568 118 923 835 741 218 633 383 752 671 585 894 4 × 2 = 0 + 0.000 000 000 103 563 136 237 847 671 482 437 266 767 505 343 171 788 8;
  • 24) 0.000 000 000 103 563 136 237 847 671 482 437 266 767 505 343 171 788 8 × 2 = 0 + 0.000 000 000 207 126 272 475 695 342 964 874 533 535 010 686 343 577 6;
  • 25) 0.000 000 000 207 126 272 475 695 342 964 874 533 535 010 686 343 577 6 × 2 = 0 + 0.000 000 000 414 252 544 951 390 685 929 749 067 070 021 372 687 155 2;
  • 26) 0.000 000 000 414 252 544 951 390 685 929 749 067 070 021 372 687 155 2 × 2 = 0 + 0.000 000 000 828 505 089 902 781 371 859 498 134 140 042 745 374 310 4;
  • 27) 0.000 000 000 828 505 089 902 781 371 859 498 134 140 042 745 374 310 4 × 2 = 0 + 0.000 000 001 657 010 179 805 562 743 718 996 268 280 085 490 748 620 8;
  • 28) 0.000 000 001 657 010 179 805 562 743 718 996 268 280 085 490 748 620 8 × 2 = 0 + 0.000 000 003 314 020 359 611 125 487 437 992 536 560 170 981 497 241 6;
  • 29) 0.000 000 003 314 020 359 611 125 487 437 992 536 560 170 981 497 241 6 × 2 = 0 + 0.000 000 006 628 040 719 222 250 974 875 985 073 120 341 962 994 483 2;
  • 30) 0.000 000 006 628 040 719 222 250 974 875 985 073 120 341 962 994 483 2 × 2 = 0 + 0.000 000 013 256 081 438 444 501 949 751 970 146 240 683 925 988 966 4;
  • 31) 0.000 000 013 256 081 438 444 501 949 751 970 146 240 683 925 988 966 4 × 2 = 0 + 0.000 000 026 512 162 876 889 003 899 503 940 292 481 367 851 977 932 8;
  • 32) 0.000 000 026 512 162 876 889 003 899 503 940 292 481 367 851 977 932 8 × 2 = 0 + 0.000 000 053 024 325 753 778 007 799 007 880 584 962 735 703 955 865 6;
  • 33) 0.000 000 053 024 325 753 778 007 799 007 880 584 962 735 703 955 865 6 × 2 = 0 + 0.000 000 106 048 651 507 556 015 598 015 761 169 925 471 407 911 731 2;
  • 34) 0.000 000 106 048 651 507 556 015 598 015 761 169 925 471 407 911 731 2 × 2 = 0 + 0.000 000 212 097 303 015 112 031 196 031 522 339 850 942 815 823 462 4;
  • 35) 0.000 000 212 097 303 015 112 031 196 031 522 339 850 942 815 823 462 4 × 2 = 0 + 0.000 000 424 194 606 030 224 062 392 063 044 679 701 885 631 646 924 8;
  • 36) 0.000 000 424 194 606 030 224 062 392 063 044 679 701 885 631 646 924 8 × 2 = 0 + 0.000 000 848 389 212 060 448 124 784 126 089 359 403 771 263 293 849 6;
  • 37) 0.000 000 848 389 212 060 448 124 784 126 089 359 403 771 263 293 849 6 × 2 = 0 + 0.000 001 696 778 424 120 896 249 568 252 178 718 807 542 526 587 699 2;
  • 38) 0.000 001 696 778 424 120 896 249 568 252 178 718 807 542 526 587 699 2 × 2 = 0 + 0.000 003 393 556 848 241 792 499 136 504 357 437 615 085 053 175 398 4;
  • 39) 0.000 003 393 556 848 241 792 499 136 504 357 437 615 085 053 175 398 4 × 2 = 0 + 0.000 006 787 113 696 483 584 998 273 008 714 875 230 170 106 350 796 8;
  • 40) 0.000 006 787 113 696 483 584 998 273 008 714 875 230 170 106 350 796 8 × 2 = 0 + 0.000 013 574 227 392 967 169 996 546 017 429 750 460 340 212 701 593 6;
  • 41) 0.000 013 574 227 392 967 169 996 546 017 429 750 460 340 212 701 593 6 × 2 = 0 + 0.000 027 148 454 785 934 339 993 092 034 859 500 920 680 425 403 187 2;
  • 42) 0.000 027 148 454 785 934 339 993 092 034 859 500 920 680 425 403 187 2 × 2 = 0 + 0.000 054 296 909 571 868 679 986 184 069 719 001 841 360 850 806 374 4;
  • 43) 0.000 054 296 909 571 868 679 986 184 069 719 001 841 360 850 806 374 4 × 2 = 0 + 0.000 108 593 819 143 737 359 972 368 139 438 003 682 721 701 612 748 8;
  • 44) 0.000 108 593 819 143 737 359 972 368 139 438 003 682 721 701 612 748 8 × 2 = 0 + 0.000 217 187 638 287 474 719 944 736 278 876 007 365 443 403 225 497 6;
  • 45) 0.000 217 187 638 287 474 719 944 736 278 876 007 365 443 403 225 497 6 × 2 = 0 + 0.000 434 375 276 574 949 439 889 472 557 752 014 730 886 806 450 995 2;
  • 46) 0.000 434 375 276 574 949 439 889 472 557 752 014 730 886 806 450 995 2 × 2 = 0 + 0.000 868 750 553 149 898 879 778 945 115 504 029 461 773 612 901 990 4;
  • 47) 0.000 868 750 553 149 898 879 778 945 115 504 029 461 773 612 901 990 4 × 2 = 0 + 0.001 737 501 106 299 797 759 557 890 231 008 058 923 547 225 803 980 8;
  • 48) 0.001 737 501 106 299 797 759 557 890 231 008 058 923 547 225 803 980 8 × 2 = 0 + 0.003 475 002 212 599 595 519 115 780 462 016 117 847 094 451 607 961 6;
  • 49) 0.003 475 002 212 599 595 519 115 780 462 016 117 847 094 451 607 961 6 × 2 = 0 + 0.006 950 004 425 199 191 038 231 560 924 032 235 694 188 903 215 923 2;
  • 50) 0.006 950 004 425 199 191 038 231 560 924 032 235 694 188 903 215 923 2 × 2 = 0 + 0.013 900 008 850 398 382 076 463 121 848 064 471 388 377 806 431 846 4;
  • 51) 0.013 900 008 850 398 382 076 463 121 848 064 471 388 377 806 431 846 4 × 2 = 0 + 0.027 800 017 700 796 764 152 926 243 696 128 942 776 755 612 863 692 8;
  • 52) 0.027 800 017 700 796 764 152 926 243 696 128 942 776 755 612 863 692 8 × 2 = 0 + 0.055 600 035 401 593 528 305 852 487 392 257 885 553 511 225 727 385 6;
  • 53) 0.055 600 035 401 593 528 305 852 487 392 257 885 553 511 225 727 385 6 × 2 = 0 + 0.111 200 070 803 187 056 611 704 974 784 515 771 107 022 451 454 771 2;
  • 54) 0.111 200 070 803 187 056 611 704 974 784 515 771 107 022 451 454 771 2 × 2 = 0 + 0.222 400 141 606 374 113 223 409 949 569 031 542 214 044 902 909 542 4;
  • 55) 0.222 400 141 606 374 113 223 409 949 569 031 542 214 044 902 909 542 4 × 2 = 0 + 0.444 800 283 212 748 226 446 819 899 138 063 084 428 089 805 819 084 8;
  • 56) 0.444 800 283 212 748 226 446 819 899 138 063 084 428 089 805 819 084 8 × 2 = 0 + 0.889 600 566 425 496 452 893 639 798 276 126 168 856 179 611 638 169 6;
  • 57) 0.889 600 566 425 496 452 893 639 798 276 126 168 856 179 611 638 169 6 × 2 = 1 + 0.779 201 132 850 992 905 787 279 596 552 252 337 712 359 223 276 339 2;
  • 58) 0.779 201 132 850 992 905 787 279 596 552 252 337 712 359 223 276 339 2 × 2 = 1 + 0.558 402 265 701 985 811 574 559 193 104 504 675 424 718 446 552 678 4;
  • 59) 0.558 402 265 701 985 811 574 559 193 104 504 675 424 718 446 552 678 4 × 2 = 1 + 0.116 804 531 403 971 623 149 118 386 209 009 350 849 436 893 105 356 8;
  • 60) 0.116 804 531 403 971 623 149 118 386 209 009 350 849 436 893 105 356 8 × 2 = 0 + 0.233 609 062 807 943 246 298 236 772 418 018 701 698 873 786 210 713 6;
  • 61) 0.233 609 062 807 943 246 298 236 772 418 018 701 698 873 786 210 713 6 × 2 = 0 + 0.467 218 125 615 886 492 596 473 544 836 037 403 397 747 572 421 427 2;
  • 62) 0.467 218 125 615 886 492 596 473 544 836 037 403 397 747 572 421 427 2 × 2 = 0 + 0.934 436 251 231 772 985 192 947 089 672 074 806 795 495 144 842 854 4;
  • 63) 0.934 436 251 231 772 985 192 947 089 672 074 806 795 495 144 842 854 4 × 2 = 1 + 0.868 872 502 463 545 970 385 894 179 344 149 613 590 990 289 685 708 8;
  • 64) 0.868 872 502 463 545 970 385 894 179 344 149 613 590 990 289 685 708 8 × 2 = 1 + 0.737 745 004 927 091 940 771 788 358 688 299 227 181 980 579 371 417 6;
  • 65) 0.737 745 004 927 091 940 771 788 358 688 299 227 181 980 579 371 417 6 × 2 = 1 + 0.475 490 009 854 183 881 543 576 717 376 598 454 363 961 158 742 835 2;
  • 66) 0.475 490 009 854 183 881 543 576 717 376 598 454 363 961 158 742 835 2 × 2 = 0 + 0.950 980 019 708 367 763 087 153 434 753 196 908 727 922 317 485 670 4;
  • 67) 0.950 980 019 708 367 763 087 153 434 753 196 908 727 922 317 485 670 4 × 2 = 1 + 0.901 960 039 416 735 526 174 306 869 506 393 817 455 844 634 971 340 8;
  • 68) 0.901 960 039 416 735 526 174 306 869 506 393 817 455 844 634 971 340 8 × 2 = 1 + 0.803 920 078 833 471 052 348 613 739 012 787 634 911 689 269 942 681 6;
  • 69) 0.803 920 078 833 471 052 348 613 739 012 787 634 911 689 269 942 681 6 × 2 = 1 + 0.607 840 157 666 942 104 697 227 478 025 575 269 823 378 539 885 363 2;
  • 70) 0.607 840 157 666 942 104 697 227 478 025 575 269 823 378 539 885 363 2 × 2 = 1 + 0.215 680 315 333 884 209 394 454 956 051 150 539 646 757 079 770 726 4;
  • 71) 0.215 680 315 333 884 209 394 454 956 051 150 539 646 757 079 770 726 4 × 2 = 0 + 0.431 360 630 667 768 418 788 909 912 102 301 079 293 514 159 541 452 8;
  • 72) 0.431 360 630 667 768 418 788 909 912 102 301 079 293 514 159 541 452 8 × 2 = 0 + 0.862 721 261 335 536 837 577 819 824 204 602 158 587 028 319 082 905 6;
  • 73) 0.862 721 261 335 536 837 577 819 824 204 602 158 587 028 319 082 905 6 × 2 = 1 + 0.725 442 522 671 073 675 155 639 648 409 204 317 174 056 638 165 811 2;
  • 74) 0.725 442 522 671 073 675 155 639 648 409 204 317 174 056 638 165 811 2 × 2 = 1 + 0.450 885 045 342 147 350 311 279 296 818 408 634 348 113 276 331 622 4;
  • 75) 0.450 885 045 342 147 350 311 279 296 818 408 634 348 113 276 331 622 4 × 2 = 0 + 0.901 770 090 684 294 700 622 558 593 636 817 268 696 226 552 663 244 8;
  • 76) 0.901 770 090 684 294 700 622 558 593 636 817 268 696 226 552 663 244 8 × 2 = 1 + 0.803 540 181 368 589 401 245 117 187 273 634 537 392 453 105 326 489 6;
  • 77) 0.803 540 181 368 589 401 245 117 187 273 634 537 392 453 105 326 489 6 × 2 = 1 + 0.607 080 362 737 178 802 490 234 374 547 269 074 784 906 210 652 979 2;
  • 78) 0.607 080 362 737 178 802 490 234 374 547 269 074 784 906 210 652 979 2 × 2 = 1 + 0.214 160 725 474 357 604 980 468 749 094 538 149 569 812 421 305 958 4;
  • 79) 0.214 160 725 474 357 604 980 468 749 094 538 149 569 812 421 305 958 4 × 2 = 0 + 0.428 321 450 948 715 209 960 937 498 189 076 299 139 624 842 611 916 8;
  • 80) 0.428 321 450 948 715 209 960 937 498 189 076 299 139 624 842 611 916 8 × 2 = 0 + 0.856 642 901 897 430 419 921 874 996 378 152 598 279 249 685 223 833 6;
  • 81) 0.856 642 901 897 430 419 921 874 996 378 152 598 279 249 685 223 833 6 × 2 = 1 + 0.713 285 803 794 860 839 843 749 992 756 305 196 558 499 370 447 667 2;
  • 82) 0.713 285 803 794 860 839 843 749 992 756 305 196 558 499 370 447 667 2 × 2 = 1 + 0.426 571 607 589 721 679 687 499 985 512 610 393 116 998 740 895 334 4;
  • 83) 0.426 571 607 589 721 679 687 499 985 512 610 393 116 998 740 895 334 4 × 2 = 0 + 0.853 143 215 179 443 359 374 999 971 025 220 786 233 997 481 790 668 8;
  • 84) 0.853 143 215 179 443 359 374 999 971 025 220 786 233 997 481 790 668 8 × 2 = 1 + 0.706 286 430 358 886 718 749 999 942 050 441 572 467 994 963 581 337 6;
  • 85) 0.706 286 430 358 886 718 749 999 942 050 441 572 467 994 963 581 337 6 × 2 = 1 + 0.412 572 860 717 773 437 499 999 884 100 883 144 935 989 927 162 675 2;
  • 86) 0.412 572 860 717 773 437 499 999 884 100 883 144 935 989 927 162 675 2 × 2 = 0 + 0.825 145 721 435 546 874 999 999 768 201 766 289 871 979 854 325 350 4;
  • 87) 0.825 145 721 435 546 874 999 999 768 201 766 289 871 979 854 325 350 4 × 2 = 1 + 0.650 291 442 871 093 749 999 999 536 403 532 579 743 959 708 650 700 8;
  • 88) 0.650 291 442 871 093 749 999 999 536 403 532 579 743 959 708 650 700 8 × 2 = 1 + 0.300 582 885 742 187 499 999 999 072 807 065 159 487 919 417 301 401 6;
  • 89) 0.300 582 885 742 187 499 999 999 072 807 065 159 487 919 417 301 401 6 × 2 = 0 + 0.601 165 771 484 374 999 999 998 145 614 130 318 975 838 834 602 803 2;
  • 90) 0.601 165 771 484 374 999 999 998 145 614 130 318 975 838 834 602 803 2 × 2 = 1 + 0.202 331 542 968 749 999 999 996 291 228 260 637 951 677 669 205 606 4;
  • 91) 0.202 331 542 968 749 999 999 996 291 228 260 637 951 677 669 205 606 4 × 2 = 0 + 0.404 663 085 937 499 999 999 992 582 456 521 275 903 355 338 411 212 8;
  • 92) 0.404 663 085 937 499 999 999 992 582 456 521 275 903 355 338 411 212 8 × 2 = 0 + 0.809 326 171 874 999 999 999 985 164 913 042 551 806 710 676 822 425 6;
  • 93) 0.809 326 171 874 999 999 999 985 164 913 042 551 806 710 676 822 425 6 × 2 = 1 + 0.618 652 343 749 999 999 999 970 329 826 085 103 613 421 353 644 851 2;
  • 94) 0.618 652 343 749 999 999 999 970 329 826 085 103 613 421 353 644 851 2 × 2 = 1 + 0.237 304 687 499 999 999 999 940 659 652 170 207 226 842 707 289 702 4;
  • 95) 0.237 304 687 499 999 999 999 940 659 652 170 207 226 842 707 289 702 4 × 2 = 0 + 0.474 609 374 999 999 999 999 881 319 304 340 414 453 685 414 579 404 8;
  • 96) 0.474 609 374 999 999 999 999 881 319 304 340 414 453 685 414 579 404 8 × 2 = 0 + 0.949 218 749 999 999 999 999 762 638 608 680 828 907 370 829 158 809 6;
  • 97) 0.949 218 749 999 999 999 999 762 638 608 680 828 907 370 829 158 809 6 × 2 = 1 + 0.898 437 499 999 999 999 999 525 277 217 361 657 814 741 658 317 619 2;
  • 98) 0.898 437 499 999 999 999 999 525 277 217 361 657 814 741 658 317 619 2 × 2 = 1 + 0.796 874 999 999 999 999 999 050 554 434 723 315 629 483 316 635 238 4;
  • 99) 0.796 874 999 999 999 999 999 050 554 434 723 315 629 483 316 635 238 4 × 2 = 1 + 0.593 749 999 999 999 999 998 101 108 869 446 631 258 966 633 270 476 8;
  • 100) 0.593 749 999 999 999 999 998 101 108 869 446 631 258 966 633 270 476 8 × 2 = 1 + 0.187 499 999 999 999 999 996 202 217 738 893 262 517 933 266 540 953 6;
  • 101) 0.187 499 999 999 999 999 996 202 217 738 893 262 517 933 266 540 953 6 × 2 = 0 + 0.374 999 999 999 999 999 992 404 435 477 786 525 035 866 533 081 907 2;
  • 102) 0.374 999 999 999 999 999 992 404 435 477 786 525 035 866 533 081 907 2 × 2 = 0 + 0.749 999 999 999 999 999 984 808 870 955 573 050 071 733 066 163 814 4;
  • 103) 0.749 999 999 999 999 999 984 808 870 955 573 050 071 733 066 163 814 4 × 2 = 1 + 0.499 999 999 999 999 999 969 617 741 911 146 100 143 466 132 327 628 8;
  • 104) 0.499 999 999 999 999 999 969 617 741 911 146 100 143 466 132 327 628 8 × 2 = 0 + 0.999 999 999 999 999 999 939 235 483 822 292 200 286 932 264 655 257 6;
  • 105) 0.999 999 999 999 999 999 939 235 483 822 292 200 286 932 264 655 257 6 × 2 = 1 + 0.999 999 999 999 999 999 878 470 967 644 584 400 573 864 529 310 515 2;
  • 106) 0.999 999 999 999 999 999 878 470 967 644 584 400 573 864 529 310 515 2 × 2 = 1 + 0.999 999 999 999 999 999 756 941 935 289 168 801 147 729 058 621 030 4;
  • 107) 0.999 999 999 999 999 999 756 941 935 289 168 801 147 729 058 621 030 4 × 2 = 1 + 0.999 999 999 999 999 999 513 883 870 578 337 602 295 458 117 242 060 8;
  • 108) 0.999 999 999 999 999 999 513 883 870 578 337 602 295 458 117 242 060 8 × 2 = 1 + 0.999 999 999 999 999 999 027 767 741 156 675 204 590 916 234 484 121 6;
  • 109) 0.999 999 999 999 999 999 027 767 741 156 675 204 590 916 234 484 121 6 × 2 = 1 + 0.999 999 999 999 999 998 055 535 482 313 350 409 181 832 468 968 243 2;

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:


0.000 000 000 000 000 012 345 687 894 564 589 438 728 960 367 143 791 1(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:

0.000 000 000 000 000 012 345 687 894 564 589 438 728 960 367 143 791 1(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:


0.000 000 000 000 000 012 345 687 894 564 589 438 728 960 367 143 791 1(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 0.000 000 000 000 000 012 345 687 894 564 589 438 728 960 367 143 791 1 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