0.000 000 000 000 000 012 345 687 894 564 589 438 728 960 366 19 Converted to 64 Bit Double Precision IEEE 754 Binary Floating Point Representation Standard

Convert decimal 0.000 000 000 000 000 012 345 687 894 564 589 438 728 960 366 19(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
0.000 000 000 000 000 012 345 687 894 564 589 438 728 960 366 19(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: 0.000 000 000 000 000 012 345 687 894 564 589 438 728 960 366 19.

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) 0.000 000 000 000 000 012 345 687 894 564 589 438 728 960 366 19 × 2 = 0 + 0.000 000 000 000 000 024 691 375 789 129 178 877 457 920 732 38;
  • 2) 0.000 000 000 000 000 024 691 375 789 129 178 877 457 920 732 38 × 2 = 0 + 0.000 000 000 000 000 049 382 751 578 258 357 754 915 841 464 76;
  • 3) 0.000 000 000 000 000 049 382 751 578 258 357 754 915 841 464 76 × 2 = 0 + 0.000 000 000 000 000 098 765 503 156 516 715 509 831 682 929 52;
  • 4) 0.000 000 000 000 000 098 765 503 156 516 715 509 831 682 929 52 × 2 = 0 + 0.000 000 000 000 000 197 531 006 313 033 431 019 663 365 859 04;
  • 5) 0.000 000 000 000 000 197 531 006 313 033 431 019 663 365 859 04 × 2 = 0 + 0.000 000 000 000 000 395 062 012 626 066 862 039 326 731 718 08;
  • 6) 0.000 000 000 000 000 395 062 012 626 066 862 039 326 731 718 08 × 2 = 0 + 0.000 000 000 000 000 790 124 025 252 133 724 078 653 463 436 16;
  • 7) 0.000 000 000 000 000 790 124 025 252 133 724 078 653 463 436 16 × 2 = 0 + 0.000 000 000 000 001 580 248 050 504 267 448 157 306 926 872 32;
  • 8) 0.000 000 000 000 001 580 248 050 504 267 448 157 306 926 872 32 × 2 = 0 + 0.000 000 000 000 003 160 496 101 008 534 896 314 613 853 744 64;
  • 9) 0.000 000 000 000 003 160 496 101 008 534 896 314 613 853 744 64 × 2 = 0 + 0.000 000 000 000 006 320 992 202 017 069 792 629 227 707 489 28;
  • 10) 0.000 000 000 000 006 320 992 202 017 069 792 629 227 707 489 28 × 2 = 0 + 0.000 000 000 000 012 641 984 404 034 139 585 258 455 414 978 56;
  • 11) 0.000 000 000 000 012 641 984 404 034 139 585 258 455 414 978 56 × 2 = 0 + 0.000 000 000 000 025 283 968 808 068 279 170 516 910 829 957 12;
  • 12) 0.000 000 000 000 025 283 968 808 068 279 170 516 910 829 957 12 × 2 = 0 + 0.000 000 000 000 050 567 937 616 136 558 341 033 821 659 914 24;
  • 13) 0.000 000 000 000 050 567 937 616 136 558 341 033 821 659 914 24 × 2 = 0 + 0.000 000 000 000 101 135 875 232 273 116 682 067 643 319 828 48;
  • 14) 0.000 000 000 000 101 135 875 232 273 116 682 067 643 319 828 48 × 2 = 0 + 0.000 000 000 000 202 271 750 464 546 233 364 135 286 639 656 96;
  • 15) 0.000 000 000 000 202 271 750 464 546 233 364 135 286 639 656 96 × 2 = 0 + 0.000 000 000 000 404 543 500 929 092 466 728 270 573 279 313 92;
  • 16) 0.000 000 000 000 404 543 500 929 092 466 728 270 573 279 313 92 × 2 = 0 + 0.000 000 000 000 809 087 001 858 184 933 456 541 146 558 627 84;
  • 17) 0.000 000 000 000 809 087 001 858 184 933 456 541 146 558 627 84 × 2 = 0 + 0.000 000 000 001 618 174 003 716 369 866 913 082 293 117 255 68;
  • 18) 0.000 000 000 001 618 174 003 716 369 866 913 082 293 117 255 68 × 2 = 0 + 0.000 000 000 003 236 348 007 432 739 733 826 164 586 234 511 36;
  • 19) 0.000 000 000 003 236 348 007 432 739 733 826 164 586 234 511 36 × 2 = 0 + 0.000 000 000 006 472 696 014 865 479 467 652 329 172 469 022 72;
  • 20) 0.000 000 000 006 472 696 014 865 479 467 652 329 172 469 022 72 × 2 = 0 + 0.000 000 000 012 945 392 029 730 958 935 304 658 344 938 045 44;
  • 21) 0.000 000 000 012 945 392 029 730 958 935 304 658 344 938 045 44 × 2 = 0 + 0.000 000 000 025 890 784 059 461 917 870 609 316 689 876 090 88;
  • 22) 0.000 000 000 025 890 784 059 461 917 870 609 316 689 876 090 88 × 2 = 0 + 0.000 000 000 051 781 568 118 923 835 741 218 633 379 752 181 76;
  • 23) 0.000 000 000 051 781 568 118 923 835 741 218 633 379 752 181 76 × 2 = 0 + 0.000 000 000 103 563 136 237 847 671 482 437 266 759 504 363 52;
  • 24) 0.000 000 000 103 563 136 237 847 671 482 437 266 759 504 363 52 × 2 = 0 + 0.000 000 000 207 126 272 475 695 342 964 874 533 519 008 727 04;
  • 25) 0.000 000 000 207 126 272 475 695 342 964 874 533 519 008 727 04 × 2 = 0 + 0.000 000 000 414 252 544 951 390 685 929 749 067 038 017 454 08;
  • 26) 0.000 000 000 414 252 544 951 390 685 929 749 067 038 017 454 08 × 2 = 0 + 0.000 000 000 828 505 089 902 781 371 859 498 134 076 034 908 16;
  • 27) 0.000 000 000 828 505 089 902 781 371 859 498 134 076 034 908 16 × 2 = 0 + 0.000 000 001 657 010 179 805 562 743 718 996 268 152 069 816 32;
  • 28) 0.000 000 001 657 010 179 805 562 743 718 996 268 152 069 816 32 × 2 = 0 + 0.000 000 003 314 020 359 611 125 487 437 992 536 304 139 632 64;
  • 29) 0.000 000 003 314 020 359 611 125 487 437 992 536 304 139 632 64 × 2 = 0 + 0.000 000 006 628 040 719 222 250 974 875 985 072 608 279 265 28;
  • 30) 0.000 000 006 628 040 719 222 250 974 875 985 072 608 279 265 28 × 2 = 0 + 0.000 000 013 256 081 438 444 501 949 751 970 145 216 558 530 56;
  • 31) 0.000 000 013 256 081 438 444 501 949 751 970 145 216 558 530 56 × 2 = 0 + 0.000 000 026 512 162 876 889 003 899 503 940 290 433 117 061 12;
  • 32) 0.000 000 026 512 162 876 889 003 899 503 940 290 433 117 061 12 × 2 = 0 + 0.000 000 053 024 325 753 778 007 799 007 880 580 866 234 122 24;
  • 33) 0.000 000 053 024 325 753 778 007 799 007 880 580 866 234 122 24 × 2 = 0 + 0.000 000 106 048 651 507 556 015 598 015 761 161 732 468 244 48;
  • 34) 0.000 000 106 048 651 507 556 015 598 015 761 161 732 468 244 48 × 2 = 0 + 0.000 000 212 097 303 015 112 031 196 031 522 323 464 936 488 96;
  • 35) 0.000 000 212 097 303 015 112 031 196 031 522 323 464 936 488 96 × 2 = 0 + 0.000 000 424 194 606 030 224 062 392 063 044 646 929 872 977 92;
  • 36) 0.000 000 424 194 606 030 224 062 392 063 044 646 929 872 977 92 × 2 = 0 + 0.000 000 848 389 212 060 448 124 784 126 089 293 859 745 955 84;
  • 37) 0.000 000 848 389 212 060 448 124 784 126 089 293 859 745 955 84 × 2 = 0 + 0.000 001 696 778 424 120 896 249 568 252 178 587 719 491 911 68;
  • 38) 0.000 001 696 778 424 120 896 249 568 252 178 587 719 491 911 68 × 2 = 0 + 0.000 003 393 556 848 241 792 499 136 504 357 175 438 983 823 36;
  • 39) 0.000 003 393 556 848 241 792 499 136 504 357 175 438 983 823 36 × 2 = 0 + 0.000 006 787 113 696 483 584 998 273 008 714 350 877 967 646 72;
  • 40) 0.000 006 787 113 696 483 584 998 273 008 714 350 877 967 646 72 × 2 = 0 + 0.000 013 574 227 392 967 169 996 546 017 428 701 755 935 293 44;
  • 41) 0.000 013 574 227 392 967 169 996 546 017 428 701 755 935 293 44 × 2 = 0 + 0.000 027 148 454 785 934 339 993 092 034 857 403 511 870 586 88;
  • 42) 0.000 027 148 454 785 934 339 993 092 034 857 403 511 870 586 88 × 2 = 0 + 0.000 054 296 909 571 868 679 986 184 069 714 807 023 741 173 76;
  • 43) 0.000 054 296 909 571 868 679 986 184 069 714 807 023 741 173 76 × 2 = 0 + 0.000 108 593 819 143 737 359 972 368 139 429 614 047 482 347 52;
  • 44) 0.000 108 593 819 143 737 359 972 368 139 429 614 047 482 347 52 × 2 = 0 + 0.000 217 187 638 287 474 719 944 736 278 859 228 094 964 695 04;
  • 45) 0.000 217 187 638 287 474 719 944 736 278 859 228 094 964 695 04 × 2 = 0 + 0.000 434 375 276 574 949 439 889 472 557 718 456 189 929 390 08;
  • 46) 0.000 434 375 276 574 949 439 889 472 557 718 456 189 929 390 08 × 2 = 0 + 0.000 868 750 553 149 898 879 778 945 115 436 912 379 858 780 16;
  • 47) 0.000 868 750 553 149 898 879 778 945 115 436 912 379 858 780 16 × 2 = 0 + 0.001 737 501 106 299 797 759 557 890 230 873 824 759 717 560 32;
  • 48) 0.001 737 501 106 299 797 759 557 890 230 873 824 759 717 560 32 × 2 = 0 + 0.003 475 002 212 599 595 519 115 780 461 747 649 519 435 120 64;
  • 49) 0.003 475 002 212 599 595 519 115 780 461 747 649 519 435 120 64 × 2 = 0 + 0.006 950 004 425 199 191 038 231 560 923 495 299 038 870 241 28;
  • 50) 0.006 950 004 425 199 191 038 231 560 923 495 299 038 870 241 28 × 2 = 0 + 0.013 900 008 850 398 382 076 463 121 846 990 598 077 740 482 56;
  • 51) 0.013 900 008 850 398 382 076 463 121 846 990 598 077 740 482 56 × 2 = 0 + 0.027 800 017 700 796 764 152 926 243 693 981 196 155 480 965 12;
  • 52) 0.027 800 017 700 796 764 152 926 243 693 981 196 155 480 965 12 × 2 = 0 + 0.055 600 035 401 593 528 305 852 487 387 962 392 310 961 930 24;
  • 53) 0.055 600 035 401 593 528 305 852 487 387 962 392 310 961 930 24 × 2 = 0 + 0.111 200 070 803 187 056 611 704 974 775 924 784 621 923 860 48;
  • 54) 0.111 200 070 803 187 056 611 704 974 775 924 784 621 923 860 48 × 2 = 0 + 0.222 400 141 606 374 113 223 409 949 551 849 569 243 847 720 96;
  • 55) 0.222 400 141 606 374 113 223 409 949 551 849 569 243 847 720 96 × 2 = 0 + 0.444 800 283 212 748 226 446 819 899 103 699 138 487 695 441 92;
  • 56) 0.444 800 283 212 748 226 446 819 899 103 699 138 487 695 441 92 × 2 = 0 + 0.889 600 566 425 496 452 893 639 798 207 398 276 975 390 883 84;
  • 57) 0.889 600 566 425 496 452 893 639 798 207 398 276 975 390 883 84 × 2 = 1 + 0.779 201 132 850 992 905 787 279 596 414 796 553 950 781 767 68;
  • 58) 0.779 201 132 850 992 905 787 279 596 414 796 553 950 781 767 68 × 2 = 1 + 0.558 402 265 701 985 811 574 559 192 829 593 107 901 563 535 36;
  • 59) 0.558 402 265 701 985 811 574 559 192 829 593 107 901 563 535 36 × 2 = 1 + 0.116 804 531 403 971 623 149 118 385 659 186 215 803 127 070 72;
  • 60) 0.116 804 531 403 971 623 149 118 385 659 186 215 803 127 070 72 × 2 = 0 + 0.233 609 062 807 943 246 298 236 771 318 372 431 606 254 141 44;
  • 61) 0.233 609 062 807 943 246 298 236 771 318 372 431 606 254 141 44 × 2 = 0 + 0.467 218 125 615 886 492 596 473 542 636 744 863 212 508 282 88;
  • 62) 0.467 218 125 615 886 492 596 473 542 636 744 863 212 508 282 88 × 2 = 0 + 0.934 436 251 231 772 985 192 947 085 273 489 726 425 016 565 76;
  • 63) 0.934 436 251 231 772 985 192 947 085 273 489 726 425 016 565 76 × 2 = 1 + 0.868 872 502 463 545 970 385 894 170 546 979 452 850 033 131 52;
  • 64) 0.868 872 502 463 545 970 385 894 170 546 979 452 850 033 131 52 × 2 = 1 + 0.737 745 004 927 091 940 771 788 341 093 958 905 700 066 263 04;
  • 65) 0.737 745 004 927 091 940 771 788 341 093 958 905 700 066 263 04 × 2 = 1 + 0.475 490 009 854 183 881 543 576 682 187 917 811 400 132 526 08;
  • 66) 0.475 490 009 854 183 881 543 576 682 187 917 811 400 132 526 08 × 2 = 0 + 0.950 980 019 708 367 763 087 153 364 375 835 622 800 265 052 16;
  • 67) 0.950 980 019 708 367 763 087 153 364 375 835 622 800 265 052 16 × 2 = 1 + 0.901 960 039 416 735 526 174 306 728 751 671 245 600 530 104 32;
  • 68) 0.901 960 039 416 735 526 174 306 728 751 671 245 600 530 104 32 × 2 = 1 + 0.803 920 078 833 471 052 348 613 457 503 342 491 201 060 208 64;
  • 69) 0.803 920 078 833 471 052 348 613 457 503 342 491 201 060 208 64 × 2 = 1 + 0.607 840 157 666 942 104 697 226 915 006 684 982 402 120 417 28;
  • 70) 0.607 840 157 666 942 104 697 226 915 006 684 982 402 120 417 28 × 2 = 1 + 0.215 680 315 333 884 209 394 453 830 013 369 964 804 240 834 56;
  • 71) 0.215 680 315 333 884 209 394 453 830 013 369 964 804 240 834 56 × 2 = 0 + 0.431 360 630 667 768 418 788 907 660 026 739 929 608 481 669 12;
  • 72) 0.431 360 630 667 768 418 788 907 660 026 739 929 608 481 669 12 × 2 = 0 + 0.862 721 261 335 536 837 577 815 320 053 479 859 216 963 338 24;
  • 73) 0.862 721 261 335 536 837 577 815 320 053 479 859 216 963 338 24 × 2 = 1 + 0.725 442 522 671 073 675 155 630 640 106 959 718 433 926 676 48;
  • 74) 0.725 442 522 671 073 675 155 630 640 106 959 718 433 926 676 48 × 2 = 1 + 0.450 885 045 342 147 350 311 261 280 213 919 436 867 853 352 96;
  • 75) 0.450 885 045 342 147 350 311 261 280 213 919 436 867 853 352 96 × 2 = 0 + 0.901 770 090 684 294 700 622 522 560 427 838 873 735 706 705 92;
  • 76) 0.901 770 090 684 294 700 622 522 560 427 838 873 735 706 705 92 × 2 = 1 + 0.803 540 181 368 589 401 245 045 120 855 677 747 471 413 411 84;
  • 77) 0.803 540 181 368 589 401 245 045 120 855 677 747 471 413 411 84 × 2 = 1 + 0.607 080 362 737 178 802 490 090 241 711 355 494 942 826 823 68;
  • 78) 0.607 080 362 737 178 802 490 090 241 711 355 494 942 826 823 68 × 2 = 1 + 0.214 160 725 474 357 604 980 180 483 422 710 989 885 653 647 36;
  • 79) 0.214 160 725 474 357 604 980 180 483 422 710 989 885 653 647 36 × 2 = 0 + 0.428 321 450 948 715 209 960 360 966 845 421 979 771 307 294 72;
  • 80) 0.428 321 450 948 715 209 960 360 966 845 421 979 771 307 294 72 × 2 = 0 + 0.856 642 901 897 430 419 920 721 933 690 843 959 542 614 589 44;
  • 81) 0.856 642 901 897 430 419 920 721 933 690 843 959 542 614 589 44 × 2 = 1 + 0.713 285 803 794 860 839 841 443 867 381 687 919 085 229 178 88;
  • 82) 0.713 285 803 794 860 839 841 443 867 381 687 919 085 229 178 88 × 2 = 1 + 0.426 571 607 589 721 679 682 887 734 763 375 838 170 458 357 76;
  • 83) 0.426 571 607 589 721 679 682 887 734 763 375 838 170 458 357 76 × 2 = 0 + 0.853 143 215 179 443 359 365 775 469 526 751 676 340 916 715 52;
  • 84) 0.853 143 215 179 443 359 365 775 469 526 751 676 340 916 715 52 × 2 = 1 + 0.706 286 430 358 886 718 731 550 939 053 503 352 681 833 431 04;
  • 85) 0.706 286 430 358 886 718 731 550 939 053 503 352 681 833 431 04 × 2 = 1 + 0.412 572 860 717 773 437 463 101 878 107 006 705 363 666 862 08;
  • 86) 0.412 572 860 717 773 437 463 101 878 107 006 705 363 666 862 08 × 2 = 0 + 0.825 145 721 435 546 874 926 203 756 214 013 410 727 333 724 16;
  • 87) 0.825 145 721 435 546 874 926 203 756 214 013 410 727 333 724 16 × 2 = 1 + 0.650 291 442 871 093 749 852 407 512 428 026 821 454 667 448 32;
  • 88) 0.650 291 442 871 093 749 852 407 512 428 026 821 454 667 448 32 × 2 = 1 + 0.300 582 885 742 187 499 704 815 024 856 053 642 909 334 896 64;
  • 89) 0.300 582 885 742 187 499 704 815 024 856 053 642 909 334 896 64 × 2 = 0 + 0.601 165 771 484 374 999 409 630 049 712 107 285 818 669 793 28;
  • 90) 0.601 165 771 484 374 999 409 630 049 712 107 285 818 669 793 28 × 2 = 1 + 0.202 331 542 968 749 998 819 260 099 424 214 571 637 339 586 56;
  • 91) 0.202 331 542 968 749 998 819 260 099 424 214 571 637 339 586 56 × 2 = 0 + 0.404 663 085 937 499 997 638 520 198 848 429 143 274 679 173 12;
  • 92) 0.404 663 085 937 499 997 638 520 198 848 429 143 274 679 173 12 × 2 = 0 + 0.809 326 171 874 999 995 277 040 397 696 858 286 549 358 346 24;
  • 93) 0.809 326 171 874 999 995 277 040 397 696 858 286 549 358 346 24 × 2 = 1 + 0.618 652 343 749 999 990 554 080 795 393 716 573 098 716 692 48;
  • 94) 0.618 652 343 749 999 990 554 080 795 393 716 573 098 716 692 48 × 2 = 1 + 0.237 304 687 499 999 981 108 161 590 787 433 146 197 433 384 96;
  • 95) 0.237 304 687 499 999 981 108 161 590 787 433 146 197 433 384 96 × 2 = 0 + 0.474 609 374 999 999 962 216 323 181 574 866 292 394 866 769 92;
  • 96) 0.474 609 374 999 999 962 216 323 181 574 866 292 394 866 769 92 × 2 = 0 + 0.949 218 749 999 999 924 432 646 363 149 732 584 789 733 539 84;
  • 97) 0.949 218 749 999 999 924 432 646 363 149 732 584 789 733 539 84 × 2 = 1 + 0.898 437 499 999 999 848 865 292 726 299 465 169 579 467 079 68;
  • 98) 0.898 437 499 999 999 848 865 292 726 299 465 169 579 467 079 68 × 2 = 1 + 0.796 874 999 999 999 697 730 585 452 598 930 339 158 934 159 36;
  • 99) 0.796 874 999 999 999 697 730 585 452 598 930 339 158 934 159 36 × 2 = 1 + 0.593 749 999 999 999 395 461 170 905 197 860 678 317 868 318 72;
  • 100) 0.593 749 999 999 999 395 461 170 905 197 860 678 317 868 318 72 × 2 = 1 + 0.187 499 999 999 998 790 922 341 810 395 721 356 635 736 637 44;
  • 101) 0.187 499 999 999 998 790 922 341 810 395 721 356 635 736 637 44 × 2 = 0 + 0.374 999 999 999 997 581 844 683 620 791 442 713 271 473 274 88;
  • 102) 0.374 999 999 999 997 581 844 683 620 791 442 713 271 473 274 88 × 2 = 0 + 0.749 999 999 999 995 163 689 367 241 582 885 426 542 946 549 76;
  • 103) 0.749 999 999 999 995 163 689 367 241 582 885 426 542 946 549 76 × 2 = 1 + 0.499 999 999 999 990 327 378 734 483 165 770 853 085 893 099 52;
  • 104) 0.499 999 999 999 990 327 378 734 483 165 770 853 085 893 099 52 × 2 = 0 + 0.999 999 999 999 980 654 757 468 966 331 541 706 171 786 199 04;
  • 105) 0.999 999 999 999 980 654 757 468 966 331 541 706 171 786 199 04 × 2 = 1 + 0.999 999 999 999 961 309 514 937 932 663 083 412 343 572 398 08;
  • 106) 0.999 999 999 999 961 309 514 937 932 663 083 412 343 572 398 08 × 2 = 1 + 0.999 999 999 999 922 619 029 875 865 326 166 824 687 144 796 16;
  • 107) 0.999 999 999 999 922 619 029 875 865 326 166 824 687 144 796 16 × 2 = 1 + 0.999 999 999 999 845 238 059 751 730 652 333 649 374 289 592 32;
  • 108) 0.999 999 999 999 845 238 059 751 730 652 333 649 374 289 592 32 × 2 = 1 + 0.999 999 999 999 690 476 119 503 461 304 667 298 748 579 184 64;
  • 109) 0.999 999 999 999 690 476 119 503 461 304 667 298 748 579 184 64 × 2 = 1 + 0.999 999 999 999 380 952 239 006 922 609 334 597 497 158 369 28;

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:


0.000 000 000 000 000 012 345 687 894 564 589 438 728 960 366 19(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:

0.000 000 000 000 000 012 345 687 894 564 589 438 728 960 366 19(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:


0.000 000 000 000 000 012 345 687 894 564 589 438 728 960 366 19(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 0.000 000 000 000 000 012 345 687 894 564 589 438 728 960 366 19 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