0.000 000 000 000 000 012 345 687 894 564 589 438 728 960 367 143 781 8 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.000 000 000 000 000 012 345 687 894 564 589 438 728 960 367 143 781 8(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
0.000 000 000 000 000 012 345 687 894 564 589 438 728 960 367 143 781 8(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: 0.000 000 000 000 000 012 345 687 894 564 589 438 728 960 367 143 781 8.

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) 0.000 000 000 000 000 012 345 687 894 564 589 438 728 960 367 143 781 8 × 2 = 0 + 0.000 000 000 000 000 024 691 375 789 129 178 877 457 920 734 287 563 6;
  • 2) 0.000 000 000 000 000 024 691 375 789 129 178 877 457 920 734 287 563 6 × 2 = 0 + 0.000 000 000 000 000 049 382 751 578 258 357 754 915 841 468 575 127 2;
  • 3) 0.000 000 000 000 000 049 382 751 578 258 357 754 915 841 468 575 127 2 × 2 = 0 + 0.000 000 000 000 000 098 765 503 156 516 715 509 831 682 937 150 254 4;
  • 4) 0.000 000 000 000 000 098 765 503 156 516 715 509 831 682 937 150 254 4 × 2 = 0 + 0.000 000 000 000 000 197 531 006 313 033 431 019 663 365 874 300 508 8;
  • 5) 0.000 000 000 000 000 197 531 006 313 033 431 019 663 365 874 300 508 8 × 2 = 0 + 0.000 000 000 000 000 395 062 012 626 066 862 039 326 731 748 601 017 6;
  • 6) 0.000 000 000 000 000 395 062 012 626 066 862 039 326 731 748 601 017 6 × 2 = 0 + 0.000 000 000 000 000 790 124 025 252 133 724 078 653 463 497 202 035 2;
  • 7) 0.000 000 000 000 000 790 124 025 252 133 724 078 653 463 497 202 035 2 × 2 = 0 + 0.000 000 000 000 001 580 248 050 504 267 448 157 306 926 994 404 070 4;
  • 8) 0.000 000 000 000 001 580 248 050 504 267 448 157 306 926 994 404 070 4 × 2 = 0 + 0.000 000 000 000 003 160 496 101 008 534 896 314 613 853 988 808 140 8;
  • 9) 0.000 000 000 000 003 160 496 101 008 534 896 314 613 853 988 808 140 8 × 2 = 0 + 0.000 000 000 000 006 320 992 202 017 069 792 629 227 707 977 616 281 6;
  • 10) 0.000 000 000 000 006 320 992 202 017 069 792 629 227 707 977 616 281 6 × 2 = 0 + 0.000 000 000 000 012 641 984 404 034 139 585 258 455 415 955 232 563 2;
  • 11) 0.000 000 000 000 012 641 984 404 034 139 585 258 455 415 955 232 563 2 × 2 = 0 + 0.000 000 000 000 025 283 968 808 068 279 170 516 910 831 910 465 126 4;
  • 12) 0.000 000 000 000 025 283 968 808 068 279 170 516 910 831 910 465 126 4 × 2 = 0 + 0.000 000 000 000 050 567 937 616 136 558 341 033 821 663 820 930 252 8;
  • 13) 0.000 000 000 000 050 567 937 616 136 558 341 033 821 663 820 930 252 8 × 2 = 0 + 0.000 000 000 000 101 135 875 232 273 116 682 067 643 327 641 860 505 6;
  • 14) 0.000 000 000 000 101 135 875 232 273 116 682 067 643 327 641 860 505 6 × 2 = 0 + 0.000 000 000 000 202 271 750 464 546 233 364 135 286 655 283 721 011 2;
  • 15) 0.000 000 000 000 202 271 750 464 546 233 364 135 286 655 283 721 011 2 × 2 = 0 + 0.000 000 000 000 404 543 500 929 092 466 728 270 573 310 567 442 022 4;
  • 16) 0.000 000 000 000 404 543 500 929 092 466 728 270 573 310 567 442 022 4 × 2 = 0 + 0.000 000 000 000 809 087 001 858 184 933 456 541 146 621 134 884 044 8;
  • 17) 0.000 000 000 000 809 087 001 858 184 933 456 541 146 621 134 884 044 8 × 2 = 0 + 0.000 000 000 001 618 174 003 716 369 866 913 082 293 242 269 768 089 6;
  • 18) 0.000 000 000 001 618 174 003 716 369 866 913 082 293 242 269 768 089 6 × 2 = 0 + 0.000 000 000 003 236 348 007 432 739 733 826 164 586 484 539 536 179 2;
  • 19) 0.000 000 000 003 236 348 007 432 739 733 826 164 586 484 539 536 179 2 × 2 = 0 + 0.000 000 000 006 472 696 014 865 479 467 652 329 172 969 079 072 358 4;
  • 20) 0.000 000 000 006 472 696 014 865 479 467 652 329 172 969 079 072 358 4 × 2 = 0 + 0.000 000 000 012 945 392 029 730 958 935 304 658 345 938 158 144 716 8;
  • 21) 0.000 000 000 012 945 392 029 730 958 935 304 658 345 938 158 144 716 8 × 2 = 0 + 0.000 000 000 025 890 784 059 461 917 870 609 316 691 876 316 289 433 6;
  • 22) 0.000 000 000 025 890 784 059 461 917 870 609 316 691 876 316 289 433 6 × 2 = 0 + 0.000 000 000 051 781 568 118 923 835 741 218 633 383 752 632 578 867 2;
  • 23) 0.000 000 000 051 781 568 118 923 835 741 218 633 383 752 632 578 867 2 × 2 = 0 + 0.000 000 000 103 563 136 237 847 671 482 437 266 767 505 265 157 734 4;
  • 24) 0.000 000 000 103 563 136 237 847 671 482 437 266 767 505 265 157 734 4 × 2 = 0 + 0.000 000 000 207 126 272 475 695 342 964 874 533 535 010 530 315 468 8;
  • 25) 0.000 000 000 207 126 272 475 695 342 964 874 533 535 010 530 315 468 8 × 2 = 0 + 0.000 000 000 414 252 544 951 390 685 929 749 067 070 021 060 630 937 6;
  • 26) 0.000 000 000 414 252 544 951 390 685 929 749 067 070 021 060 630 937 6 × 2 = 0 + 0.000 000 000 828 505 089 902 781 371 859 498 134 140 042 121 261 875 2;
  • 27) 0.000 000 000 828 505 089 902 781 371 859 498 134 140 042 121 261 875 2 × 2 = 0 + 0.000 000 001 657 010 179 805 562 743 718 996 268 280 084 242 523 750 4;
  • 28) 0.000 000 001 657 010 179 805 562 743 718 996 268 280 084 242 523 750 4 × 2 = 0 + 0.000 000 003 314 020 359 611 125 487 437 992 536 560 168 485 047 500 8;
  • 29) 0.000 000 003 314 020 359 611 125 487 437 992 536 560 168 485 047 500 8 × 2 = 0 + 0.000 000 006 628 040 719 222 250 974 875 985 073 120 336 970 095 001 6;
  • 30) 0.000 000 006 628 040 719 222 250 974 875 985 073 120 336 970 095 001 6 × 2 = 0 + 0.000 000 013 256 081 438 444 501 949 751 970 146 240 673 940 190 003 2;
  • 31) 0.000 000 013 256 081 438 444 501 949 751 970 146 240 673 940 190 003 2 × 2 = 0 + 0.000 000 026 512 162 876 889 003 899 503 940 292 481 347 880 380 006 4;
  • 32) 0.000 000 026 512 162 876 889 003 899 503 940 292 481 347 880 380 006 4 × 2 = 0 + 0.000 000 053 024 325 753 778 007 799 007 880 584 962 695 760 760 012 8;
  • 33) 0.000 000 053 024 325 753 778 007 799 007 880 584 962 695 760 760 012 8 × 2 = 0 + 0.000 000 106 048 651 507 556 015 598 015 761 169 925 391 521 520 025 6;
  • 34) 0.000 000 106 048 651 507 556 015 598 015 761 169 925 391 521 520 025 6 × 2 = 0 + 0.000 000 212 097 303 015 112 031 196 031 522 339 850 783 043 040 051 2;
  • 35) 0.000 000 212 097 303 015 112 031 196 031 522 339 850 783 043 040 051 2 × 2 = 0 + 0.000 000 424 194 606 030 224 062 392 063 044 679 701 566 086 080 102 4;
  • 36) 0.000 000 424 194 606 030 224 062 392 063 044 679 701 566 086 080 102 4 × 2 = 0 + 0.000 000 848 389 212 060 448 124 784 126 089 359 403 132 172 160 204 8;
  • 37) 0.000 000 848 389 212 060 448 124 784 126 089 359 403 132 172 160 204 8 × 2 = 0 + 0.000 001 696 778 424 120 896 249 568 252 178 718 806 264 344 320 409 6;
  • 38) 0.000 001 696 778 424 120 896 249 568 252 178 718 806 264 344 320 409 6 × 2 = 0 + 0.000 003 393 556 848 241 792 499 136 504 357 437 612 528 688 640 819 2;
  • 39) 0.000 003 393 556 848 241 792 499 136 504 357 437 612 528 688 640 819 2 × 2 = 0 + 0.000 006 787 113 696 483 584 998 273 008 714 875 225 057 377 281 638 4;
  • 40) 0.000 006 787 113 696 483 584 998 273 008 714 875 225 057 377 281 638 4 × 2 = 0 + 0.000 013 574 227 392 967 169 996 546 017 429 750 450 114 754 563 276 8;
  • 41) 0.000 013 574 227 392 967 169 996 546 017 429 750 450 114 754 563 276 8 × 2 = 0 + 0.000 027 148 454 785 934 339 993 092 034 859 500 900 229 509 126 553 6;
  • 42) 0.000 027 148 454 785 934 339 993 092 034 859 500 900 229 509 126 553 6 × 2 = 0 + 0.000 054 296 909 571 868 679 986 184 069 719 001 800 459 018 253 107 2;
  • 43) 0.000 054 296 909 571 868 679 986 184 069 719 001 800 459 018 253 107 2 × 2 = 0 + 0.000 108 593 819 143 737 359 972 368 139 438 003 600 918 036 506 214 4;
  • 44) 0.000 108 593 819 143 737 359 972 368 139 438 003 600 918 036 506 214 4 × 2 = 0 + 0.000 217 187 638 287 474 719 944 736 278 876 007 201 836 073 012 428 8;
  • 45) 0.000 217 187 638 287 474 719 944 736 278 876 007 201 836 073 012 428 8 × 2 = 0 + 0.000 434 375 276 574 949 439 889 472 557 752 014 403 672 146 024 857 6;
  • 46) 0.000 434 375 276 574 949 439 889 472 557 752 014 403 672 146 024 857 6 × 2 = 0 + 0.000 868 750 553 149 898 879 778 945 115 504 028 807 344 292 049 715 2;
  • 47) 0.000 868 750 553 149 898 879 778 945 115 504 028 807 344 292 049 715 2 × 2 = 0 + 0.001 737 501 106 299 797 759 557 890 231 008 057 614 688 584 099 430 4;
  • 48) 0.001 737 501 106 299 797 759 557 890 231 008 057 614 688 584 099 430 4 × 2 = 0 + 0.003 475 002 212 599 595 519 115 780 462 016 115 229 377 168 198 860 8;
  • 49) 0.003 475 002 212 599 595 519 115 780 462 016 115 229 377 168 198 860 8 × 2 = 0 + 0.006 950 004 425 199 191 038 231 560 924 032 230 458 754 336 397 721 6;
  • 50) 0.006 950 004 425 199 191 038 231 560 924 032 230 458 754 336 397 721 6 × 2 = 0 + 0.013 900 008 850 398 382 076 463 121 848 064 460 917 508 672 795 443 2;
  • 51) 0.013 900 008 850 398 382 076 463 121 848 064 460 917 508 672 795 443 2 × 2 = 0 + 0.027 800 017 700 796 764 152 926 243 696 128 921 835 017 345 590 886 4;
  • 52) 0.027 800 017 700 796 764 152 926 243 696 128 921 835 017 345 590 886 4 × 2 = 0 + 0.055 600 035 401 593 528 305 852 487 392 257 843 670 034 691 181 772 8;
  • 53) 0.055 600 035 401 593 528 305 852 487 392 257 843 670 034 691 181 772 8 × 2 = 0 + 0.111 200 070 803 187 056 611 704 974 784 515 687 340 069 382 363 545 6;
  • 54) 0.111 200 070 803 187 056 611 704 974 784 515 687 340 069 382 363 545 6 × 2 = 0 + 0.222 400 141 606 374 113 223 409 949 569 031 374 680 138 764 727 091 2;
  • 55) 0.222 400 141 606 374 113 223 409 949 569 031 374 680 138 764 727 091 2 × 2 = 0 + 0.444 800 283 212 748 226 446 819 899 138 062 749 360 277 529 454 182 4;
  • 56) 0.444 800 283 212 748 226 446 819 899 138 062 749 360 277 529 454 182 4 × 2 = 0 + 0.889 600 566 425 496 452 893 639 798 276 125 498 720 555 058 908 364 8;
  • 57) 0.889 600 566 425 496 452 893 639 798 276 125 498 720 555 058 908 364 8 × 2 = 1 + 0.779 201 132 850 992 905 787 279 596 552 250 997 441 110 117 816 729 6;
  • 58) 0.779 201 132 850 992 905 787 279 596 552 250 997 441 110 117 816 729 6 × 2 = 1 + 0.558 402 265 701 985 811 574 559 193 104 501 994 882 220 235 633 459 2;
  • 59) 0.558 402 265 701 985 811 574 559 193 104 501 994 882 220 235 633 459 2 × 2 = 1 + 0.116 804 531 403 971 623 149 118 386 209 003 989 764 440 471 266 918 4;
  • 60) 0.116 804 531 403 971 623 149 118 386 209 003 989 764 440 471 266 918 4 × 2 = 0 + 0.233 609 062 807 943 246 298 236 772 418 007 979 528 880 942 533 836 8;
  • 61) 0.233 609 062 807 943 246 298 236 772 418 007 979 528 880 942 533 836 8 × 2 = 0 + 0.467 218 125 615 886 492 596 473 544 836 015 959 057 761 885 067 673 6;
  • 62) 0.467 218 125 615 886 492 596 473 544 836 015 959 057 761 885 067 673 6 × 2 = 0 + 0.934 436 251 231 772 985 192 947 089 672 031 918 115 523 770 135 347 2;
  • 63) 0.934 436 251 231 772 985 192 947 089 672 031 918 115 523 770 135 347 2 × 2 = 1 + 0.868 872 502 463 545 970 385 894 179 344 063 836 231 047 540 270 694 4;
  • 64) 0.868 872 502 463 545 970 385 894 179 344 063 836 231 047 540 270 694 4 × 2 = 1 + 0.737 745 004 927 091 940 771 788 358 688 127 672 462 095 080 541 388 8;
  • 65) 0.737 745 004 927 091 940 771 788 358 688 127 672 462 095 080 541 388 8 × 2 = 1 + 0.475 490 009 854 183 881 543 576 717 376 255 344 924 190 161 082 777 6;
  • 66) 0.475 490 009 854 183 881 543 576 717 376 255 344 924 190 161 082 777 6 × 2 = 0 + 0.950 980 019 708 367 763 087 153 434 752 510 689 848 380 322 165 555 2;
  • 67) 0.950 980 019 708 367 763 087 153 434 752 510 689 848 380 322 165 555 2 × 2 = 1 + 0.901 960 039 416 735 526 174 306 869 505 021 379 696 760 644 331 110 4;
  • 68) 0.901 960 039 416 735 526 174 306 869 505 021 379 696 760 644 331 110 4 × 2 = 1 + 0.803 920 078 833 471 052 348 613 739 010 042 759 393 521 288 662 220 8;
  • 69) 0.803 920 078 833 471 052 348 613 739 010 042 759 393 521 288 662 220 8 × 2 = 1 + 0.607 840 157 666 942 104 697 227 478 020 085 518 787 042 577 324 441 6;
  • 70) 0.607 840 157 666 942 104 697 227 478 020 085 518 787 042 577 324 441 6 × 2 = 1 + 0.215 680 315 333 884 209 394 454 956 040 171 037 574 085 154 648 883 2;
  • 71) 0.215 680 315 333 884 209 394 454 956 040 171 037 574 085 154 648 883 2 × 2 = 0 + 0.431 360 630 667 768 418 788 909 912 080 342 075 148 170 309 297 766 4;
  • 72) 0.431 360 630 667 768 418 788 909 912 080 342 075 148 170 309 297 766 4 × 2 = 0 + 0.862 721 261 335 536 837 577 819 824 160 684 150 296 340 618 595 532 8;
  • 73) 0.862 721 261 335 536 837 577 819 824 160 684 150 296 340 618 595 532 8 × 2 = 1 + 0.725 442 522 671 073 675 155 639 648 321 368 300 592 681 237 191 065 6;
  • 74) 0.725 442 522 671 073 675 155 639 648 321 368 300 592 681 237 191 065 6 × 2 = 1 + 0.450 885 045 342 147 350 311 279 296 642 736 601 185 362 474 382 131 2;
  • 75) 0.450 885 045 342 147 350 311 279 296 642 736 601 185 362 474 382 131 2 × 2 = 0 + 0.901 770 090 684 294 700 622 558 593 285 473 202 370 724 948 764 262 4;
  • 76) 0.901 770 090 684 294 700 622 558 593 285 473 202 370 724 948 764 262 4 × 2 = 1 + 0.803 540 181 368 589 401 245 117 186 570 946 404 741 449 897 528 524 8;
  • 77) 0.803 540 181 368 589 401 245 117 186 570 946 404 741 449 897 528 524 8 × 2 = 1 + 0.607 080 362 737 178 802 490 234 373 141 892 809 482 899 795 057 049 6;
  • 78) 0.607 080 362 737 178 802 490 234 373 141 892 809 482 899 795 057 049 6 × 2 = 1 + 0.214 160 725 474 357 604 980 468 746 283 785 618 965 799 590 114 099 2;
  • 79) 0.214 160 725 474 357 604 980 468 746 283 785 618 965 799 590 114 099 2 × 2 = 0 + 0.428 321 450 948 715 209 960 937 492 567 571 237 931 599 180 228 198 4;
  • 80) 0.428 321 450 948 715 209 960 937 492 567 571 237 931 599 180 228 198 4 × 2 = 0 + 0.856 642 901 897 430 419 921 874 985 135 142 475 863 198 360 456 396 8;
  • 81) 0.856 642 901 897 430 419 921 874 985 135 142 475 863 198 360 456 396 8 × 2 = 1 + 0.713 285 803 794 860 839 843 749 970 270 284 951 726 396 720 912 793 6;
  • 82) 0.713 285 803 794 860 839 843 749 970 270 284 951 726 396 720 912 793 6 × 2 = 1 + 0.426 571 607 589 721 679 687 499 940 540 569 903 452 793 441 825 587 2;
  • 83) 0.426 571 607 589 721 679 687 499 940 540 569 903 452 793 441 825 587 2 × 2 = 0 + 0.853 143 215 179 443 359 374 999 881 081 139 806 905 586 883 651 174 4;
  • 84) 0.853 143 215 179 443 359 374 999 881 081 139 806 905 586 883 651 174 4 × 2 = 1 + 0.706 286 430 358 886 718 749 999 762 162 279 613 811 173 767 302 348 8;
  • 85) 0.706 286 430 358 886 718 749 999 762 162 279 613 811 173 767 302 348 8 × 2 = 1 + 0.412 572 860 717 773 437 499 999 524 324 559 227 622 347 534 604 697 6;
  • 86) 0.412 572 860 717 773 437 499 999 524 324 559 227 622 347 534 604 697 6 × 2 = 0 + 0.825 145 721 435 546 874 999 999 048 649 118 455 244 695 069 209 395 2;
  • 87) 0.825 145 721 435 546 874 999 999 048 649 118 455 244 695 069 209 395 2 × 2 = 1 + 0.650 291 442 871 093 749 999 998 097 298 236 910 489 390 138 418 790 4;
  • 88) 0.650 291 442 871 093 749 999 998 097 298 236 910 489 390 138 418 790 4 × 2 = 1 + 0.300 582 885 742 187 499 999 996 194 596 473 820 978 780 276 837 580 8;
  • 89) 0.300 582 885 742 187 499 999 996 194 596 473 820 978 780 276 837 580 8 × 2 = 0 + 0.601 165 771 484 374 999 999 992 389 192 947 641 957 560 553 675 161 6;
  • 90) 0.601 165 771 484 374 999 999 992 389 192 947 641 957 560 553 675 161 6 × 2 = 1 + 0.202 331 542 968 749 999 999 984 778 385 895 283 915 121 107 350 323 2;
  • 91) 0.202 331 542 968 749 999 999 984 778 385 895 283 915 121 107 350 323 2 × 2 = 0 + 0.404 663 085 937 499 999 999 969 556 771 790 567 830 242 214 700 646 4;
  • 92) 0.404 663 085 937 499 999 999 969 556 771 790 567 830 242 214 700 646 4 × 2 = 0 + 0.809 326 171 874 999 999 999 939 113 543 581 135 660 484 429 401 292 8;
  • 93) 0.809 326 171 874 999 999 999 939 113 543 581 135 660 484 429 401 292 8 × 2 = 1 + 0.618 652 343 749 999 999 999 878 227 087 162 271 320 968 858 802 585 6;
  • 94) 0.618 652 343 749 999 999 999 878 227 087 162 271 320 968 858 802 585 6 × 2 = 1 + 0.237 304 687 499 999 999 999 756 454 174 324 542 641 937 717 605 171 2;
  • 95) 0.237 304 687 499 999 999 999 756 454 174 324 542 641 937 717 605 171 2 × 2 = 0 + 0.474 609 374 999 999 999 999 512 908 348 649 085 283 875 435 210 342 4;
  • 96) 0.474 609 374 999 999 999 999 512 908 348 649 085 283 875 435 210 342 4 × 2 = 0 + 0.949 218 749 999 999 999 999 025 816 697 298 170 567 750 870 420 684 8;
  • 97) 0.949 218 749 999 999 999 999 025 816 697 298 170 567 750 870 420 684 8 × 2 = 1 + 0.898 437 499 999 999 999 998 051 633 394 596 341 135 501 740 841 369 6;
  • 98) 0.898 437 499 999 999 999 998 051 633 394 596 341 135 501 740 841 369 6 × 2 = 1 + 0.796 874 999 999 999 999 996 103 266 789 192 682 271 003 481 682 739 2;
  • 99) 0.796 874 999 999 999 999 996 103 266 789 192 682 271 003 481 682 739 2 × 2 = 1 + 0.593 749 999 999 999 999 992 206 533 578 385 364 542 006 963 365 478 4;
  • 100) 0.593 749 999 999 999 999 992 206 533 578 385 364 542 006 963 365 478 4 × 2 = 1 + 0.187 499 999 999 999 999 984 413 067 156 770 729 084 013 926 730 956 8;
  • 101) 0.187 499 999 999 999 999 984 413 067 156 770 729 084 013 926 730 956 8 × 2 = 0 + 0.374 999 999 999 999 999 968 826 134 313 541 458 168 027 853 461 913 6;
  • 102) 0.374 999 999 999 999 999 968 826 134 313 541 458 168 027 853 461 913 6 × 2 = 0 + 0.749 999 999 999 999 999 937 652 268 627 082 916 336 055 706 923 827 2;
  • 103) 0.749 999 999 999 999 999 937 652 268 627 082 916 336 055 706 923 827 2 × 2 = 1 + 0.499 999 999 999 999 999 875 304 537 254 165 832 672 111 413 847 654 4;
  • 104) 0.499 999 999 999 999 999 875 304 537 254 165 832 672 111 413 847 654 4 × 2 = 0 + 0.999 999 999 999 999 999 750 609 074 508 331 665 344 222 827 695 308 8;
  • 105) 0.999 999 999 999 999 999 750 609 074 508 331 665 344 222 827 695 308 8 × 2 = 1 + 0.999 999 999 999 999 999 501 218 149 016 663 330 688 445 655 390 617 6;
  • 106) 0.999 999 999 999 999 999 501 218 149 016 663 330 688 445 655 390 617 6 × 2 = 1 + 0.999 999 999 999 999 999 002 436 298 033 326 661 376 891 310 781 235 2;
  • 107) 0.999 999 999 999 999 999 002 436 298 033 326 661 376 891 310 781 235 2 × 2 = 1 + 0.999 999 999 999 999 998 004 872 596 066 653 322 753 782 621 562 470 4;
  • 108) 0.999 999 999 999 999 998 004 872 596 066 653 322 753 782 621 562 470 4 × 2 = 1 + 0.999 999 999 999 999 996 009 745 192 133 306 645 507 565 243 124 940 8;
  • 109) 0.999 999 999 999 999 996 009 745 192 133 306 645 507 565 243 124 940 8 × 2 = 1 + 0.999 999 999 999 999 992 019 490 384 266 613 291 015 130 486 249 881 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:


0.000 000 000 000 000 012 345 687 894 564 589 438 728 960 367 143 781 8(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:

0.000 000 000 000 000 012 345 687 894 564 589 438 728 960 367 143 781 8(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:


0.000 000 000 000 000 012 345 687 894 564 589 438 728 960 367 143 781 8(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 0.000 000 000 000 000 012 345 687 894 564 589 438 728 960 367 143 781 8 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