0.000 000 000 000 000 012 345 687 894 564 589 438 728 960 367 98 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 98(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 98(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 98.

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 98 × 2 = 0 + 0.000 000 000 000 000 024 691 375 789 129 178 877 457 920 735 96;
  • 2) 0.000 000 000 000 000 024 691 375 789 129 178 877 457 920 735 96 × 2 = 0 + 0.000 000 000 000 000 049 382 751 578 258 357 754 915 841 471 92;
  • 3) 0.000 000 000 000 000 049 382 751 578 258 357 754 915 841 471 92 × 2 = 0 + 0.000 000 000 000 000 098 765 503 156 516 715 509 831 682 943 84;
  • 4) 0.000 000 000 000 000 098 765 503 156 516 715 509 831 682 943 84 × 2 = 0 + 0.000 000 000 000 000 197 531 006 313 033 431 019 663 365 887 68;
  • 5) 0.000 000 000 000 000 197 531 006 313 033 431 019 663 365 887 68 × 2 = 0 + 0.000 000 000 000 000 395 062 012 626 066 862 039 326 731 775 36;
  • 6) 0.000 000 000 000 000 395 062 012 626 066 862 039 326 731 775 36 × 2 = 0 + 0.000 000 000 000 000 790 124 025 252 133 724 078 653 463 550 72;
  • 7) 0.000 000 000 000 000 790 124 025 252 133 724 078 653 463 550 72 × 2 = 0 + 0.000 000 000 000 001 580 248 050 504 267 448 157 306 927 101 44;
  • 8) 0.000 000 000 000 001 580 248 050 504 267 448 157 306 927 101 44 × 2 = 0 + 0.000 000 000 000 003 160 496 101 008 534 896 314 613 854 202 88;
  • 9) 0.000 000 000 000 003 160 496 101 008 534 896 314 613 854 202 88 × 2 = 0 + 0.000 000 000 000 006 320 992 202 017 069 792 629 227 708 405 76;
  • 10) 0.000 000 000 000 006 320 992 202 017 069 792 629 227 708 405 76 × 2 = 0 + 0.000 000 000 000 012 641 984 404 034 139 585 258 455 416 811 52;
  • 11) 0.000 000 000 000 012 641 984 404 034 139 585 258 455 416 811 52 × 2 = 0 + 0.000 000 000 000 025 283 968 808 068 279 170 516 910 833 623 04;
  • 12) 0.000 000 000 000 025 283 968 808 068 279 170 516 910 833 623 04 × 2 = 0 + 0.000 000 000 000 050 567 937 616 136 558 341 033 821 667 246 08;
  • 13) 0.000 000 000 000 050 567 937 616 136 558 341 033 821 667 246 08 × 2 = 0 + 0.000 000 000 000 101 135 875 232 273 116 682 067 643 334 492 16;
  • 14) 0.000 000 000 000 101 135 875 232 273 116 682 067 643 334 492 16 × 2 = 0 + 0.000 000 000 000 202 271 750 464 546 233 364 135 286 668 984 32;
  • 15) 0.000 000 000 000 202 271 750 464 546 233 364 135 286 668 984 32 × 2 = 0 + 0.000 000 000 000 404 543 500 929 092 466 728 270 573 337 968 64;
  • 16) 0.000 000 000 000 404 543 500 929 092 466 728 270 573 337 968 64 × 2 = 0 + 0.000 000 000 000 809 087 001 858 184 933 456 541 146 675 937 28;
  • 17) 0.000 000 000 000 809 087 001 858 184 933 456 541 146 675 937 28 × 2 = 0 + 0.000 000 000 001 618 174 003 716 369 866 913 082 293 351 874 56;
  • 18) 0.000 000 000 001 618 174 003 716 369 866 913 082 293 351 874 56 × 2 = 0 + 0.000 000 000 003 236 348 007 432 739 733 826 164 586 703 749 12;
  • 19) 0.000 000 000 003 236 348 007 432 739 733 826 164 586 703 749 12 × 2 = 0 + 0.000 000 000 006 472 696 014 865 479 467 652 329 173 407 498 24;
  • 20) 0.000 000 000 006 472 696 014 865 479 467 652 329 173 407 498 24 × 2 = 0 + 0.000 000 000 012 945 392 029 730 958 935 304 658 346 814 996 48;
  • 21) 0.000 000 000 012 945 392 029 730 958 935 304 658 346 814 996 48 × 2 = 0 + 0.000 000 000 025 890 784 059 461 917 870 609 316 693 629 992 96;
  • 22) 0.000 000 000 025 890 784 059 461 917 870 609 316 693 629 992 96 × 2 = 0 + 0.000 000 000 051 781 568 118 923 835 741 218 633 387 259 985 92;
  • 23) 0.000 000 000 051 781 568 118 923 835 741 218 633 387 259 985 92 × 2 = 0 + 0.000 000 000 103 563 136 237 847 671 482 437 266 774 519 971 84;
  • 24) 0.000 000 000 103 563 136 237 847 671 482 437 266 774 519 971 84 × 2 = 0 + 0.000 000 000 207 126 272 475 695 342 964 874 533 549 039 943 68;
  • 25) 0.000 000 000 207 126 272 475 695 342 964 874 533 549 039 943 68 × 2 = 0 + 0.000 000 000 414 252 544 951 390 685 929 749 067 098 079 887 36;
  • 26) 0.000 000 000 414 252 544 951 390 685 929 749 067 098 079 887 36 × 2 = 0 + 0.000 000 000 828 505 089 902 781 371 859 498 134 196 159 774 72;
  • 27) 0.000 000 000 828 505 089 902 781 371 859 498 134 196 159 774 72 × 2 = 0 + 0.000 000 001 657 010 179 805 562 743 718 996 268 392 319 549 44;
  • 28) 0.000 000 001 657 010 179 805 562 743 718 996 268 392 319 549 44 × 2 = 0 + 0.000 000 003 314 020 359 611 125 487 437 992 536 784 639 098 88;
  • 29) 0.000 000 003 314 020 359 611 125 487 437 992 536 784 639 098 88 × 2 = 0 + 0.000 000 006 628 040 719 222 250 974 875 985 073 569 278 197 76;
  • 30) 0.000 000 006 628 040 719 222 250 974 875 985 073 569 278 197 76 × 2 = 0 + 0.000 000 013 256 081 438 444 501 949 751 970 147 138 556 395 52;
  • 31) 0.000 000 013 256 081 438 444 501 949 751 970 147 138 556 395 52 × 2 = 0 + 0.000 000 026 512 162 876 889 003 899 503 940 294 277 112 791 04;
  • 32) 0.000 000 026 512 162 876 889 003 899 503 940 294 277 112 791 04 × 2 = 0 + 0.000 000 053 024 325 753 778 007 799 007 880 588 554 225 582 08;
  • 33) 0.000 000 053 024 325 753 778 007 799 007 880 588 554 225 582 08 × 2 = 0 + 0.000 000 106 048 651 507 556 015 598 015 761 177 108 451 164 16;
  • 34) 0.000 000 106 048 651 507 556 015 598 015 761 177 108 451 164 16 × 2 = 0 + 0.000 000 212 097 303 015 112 031 196 031 522 354 216 902 328 32;
  • 35) 0.000 000 212 097 303 015 112 031 196 031 522 354 216 902 328 32 × 2 = 0 + 0.000 000 424 194 606 030 224 062 392 063 044 708 433 804 656 64;
  • 36) 0.000 000 424 194 606 030 224 062 392 063 044 708 433 804 656 64 × 2 = 0 + 0.000 000 848 389 212 060 448 124 784 126 089 416 867 609 313 28;
  • 37) 0.000 000 848 389 212 060 448 124 784 126 089 416 867 609 313 28 × 2 = 0 + 0.000 001 696 778 424 120 896 249 568 252 178 833 735 218 626 56;
  • 38) 0.000 001 696 778 424 120 896 249 568 252 178 833 735 218 626 56 × 2 = 0 + 0.000 003 393 556 848 241 792 499 136 504 357 667 470 437 253 12;
  • 39) 0.000 003 393 556 848 241 792 499 136 504 357 667 470 437 253 12 × 2 = 0 + 0.000 006 787 113 696 483 584 998 273 008 715 334 940 874 506 24;
  • 40) 0.000 006 787 113 696 483 584 998 273 008 715 334 940 874 506 24 × 2 = 0 + 0.000 013 574 227 392 967 169 996 546 017 430 669 881 749 012 48;
  • 41) 0.000 013 574 227 392 967 169 996 546 017 430 669 881 749 012 48 × 2 = 0 + 0.000 027 148 454 785 934 339 993 092 034 861 339 763 498 024 96;
  • 42) 0.000 027 148 454 785 934 339 993 092 034 861 339 763 498 024 96 × 2 = 0 + 0.000 054 296 909 571 868 679 986 184 069 722 679 526 996 049 92;
  • 43) 0.000 054 296 909 571 868 679 986 184 069 722 679 526 996 049 92 × 2 = 0 + 0.000 108 593 819 143 737 359 972 368 139 445 359 053 992 099 84;
  • 44) 0.000 108 593 819 143 737 359 972 368 139 445 359 053 992 099 84 × 2 = 0 + 0.000 217 187 638 287 474 719 944 736 278 890 718 107 984 199 68;
  • 45) 0.000 217 187 638 287 474 719 944 736 278 890 718 107 984 199 68 × 2 = 0 + 0.000 434 375 276 574 949 439 889 472 557 781 436 215 968 399 36;
  • 46) 0.000 434 375 276 574 949 439 889 472 557 781 436 215 968 399 36 × 2 = 0 + 0.000 868 750 553 149 898 879 778 945 115 562 872 431 936 798 72;
  • 47) 0.000 868 750 553 149 898 879 778 945 115 562 872 431 936 798 72 × 2 = 0 + 0.001 737 501 106 299 797 759 557 890 231 125 744 863 873 597 44;
  • 48) 0.001 737 501 106 299 797 759 557 890 231 125 744 863 873 597 44 × 2 = 0 + 0.003 475 002 212 599 595 519 115 780 462 251 489 727 747 194 88;
  • 49) 0.003 475 002 212 599 595 519 115 780 462 251 489 727 747 194 88 × 2 = 0 + 0.006 950 004 425 199 191 038 231 560 924 502 979 455 494 389 76;
  • 50) 0.006 950 004 425 199 191 038 231 560 924 502 979 455 494 389 76 × 2 = 0 + 0.013 900 008 850 398 382 076 463 121 849 005 958 910 988 779 52;
  • 51) 0.013 900 008 850 398 382 076 463 121 849 005 958 910 988 779 52 × 2 = 0 + 0.027 800 017 700 796 764 152 926 243 698 011 917 821 977 559 04;
  • 52) 0.027 800 017 700 796 764 152 926 243 698 011 917 821 977 559 04 × 2 = 0 + 0.055 600 035 401 593 528 305 852 487 396 023 835 643 955 118 08;
  • 53) 0.055 600 035 401 593 528 305 852 487 396 023 835 643 955 118 08 × 2 = 0 + 0.111 200 070 803 187 056 611 704 974 792 047 671 287 910 236 16;
  • 54) 0.111 200 070 803 187 056 611 704 974 792 047 671 287 910 236 16 × 2 = 0 + 0.222 400 141 606 374 113 223 409 949 584 095 342 575 820 472 32;
  • 55) 0.222 400 141 606 374 113 223 409 949 584 095 342 575 820 472 32 × 2 = 0 + 0.444 800 283 212 748 226 446 819 899 168 190 685 151 640 944 64;
  • 56) 0.444 800 283 212 748 226 446 819 899 168 190 685 151 640 944 64 × 2 = 0 + 0.889 600 566 425 496 452 893 639 798 336 381 370 303 281 889 28;
  • 57) 0.889 600 566 425 496 452 893 639 798 336 381 370 303 281 889 28 × 2 = 1 + 0.779 201 132 850 992 905 787 279 596 672 762 740 606 563 778 56;
  • 58) 0.779 201 132 850 992 905 787 279 596 672 762 740 606 563 778 56 × 2 = 1 + 0.558 402 265 701 985 811 574 559 193 345 525 481 213 127 557 12;
  • 59) 0.558 402 265 701 985 811 574 559 193 345 525 481 213 127 557 12 × 2 = 1 + 0.116 804 531 403 971 623 149 118 386 691 050 962 426 255 114 24;
  • 60) 0.116 804 531 403 971 623 149 118 386 691 050 962 426 255 114 24 × 2 = 0 + 0.233 609 062 807 943 246 298 236 773 382 101 924 852 510 228 48;
  • 61) 0.233 609 062 807 943 246 298 236 773 382 101 924 852 510 228 48 × 2 = 0 + 0.467 218 125 615 886 492 596 473 546 764 203 849 705 020 456 96;
  • 62) 0.467 218 125 615 886 492 596 473 546 764 203 849 705 020 456 96 × 2 = 0 + 0.934 436 251 231 772 985 192 947 093 528 407 699 410 040 913 92;
  • 63) 0.934 436 251 231 772 985 192 947 093 528 407 699 410 040 913 92 × 2 = 1 + 0.868 872 502 463 545 970 385 894 187 056 815 398 820 081 827 84;
  • 64) 0.868 872 502 463 545 970 385 894 187 056 815 398 820 081 827 84 × 2 = 1 + 0.737 745 004 927 091 940 771 788 374 113 630 797 640 163 655 68;
  • 65) 0.737 745 004 927 091 940 771 788 374 113 630 797 640 163 655 68 × 2 = 1 + 0.475 490 009 854 183 881 543 576 748 227 261 595 280 327 311 36;
  • 66) 0.475 490 009 854 183 881 543 576 748 227 261 595 280 327 311 36 × 2 = 0 + 0.950 980 019 708 367 763 087 153 496 454 523 190 560 654 622 72;
  • 67) 0.950 980 019 708 367 763 087 153 496 454 523 190 560 654 622 72 × 2 = 1 + 0.901 960 039 416 735 526 174 306 992 909 046 381 121 309 245 44;
  • 68) 0.901 960 039 416 735 526 174 306 992 909 046 381 121 309 245 44 × 2 = 1 + 0.803 920 078 833 471 052 348 613 985 818 092 762 242 618 490 88;
  • 69) 0.803 920 078 833 471 052 348 613 985 818 092 762 242 618 490 88 × 2 = 1 + 0.607 840 157 666 942 104 697 227 971 636 185 524 485 236 981 76;
  • 70) 0.607 840 157 666 942 104 697 227 971 636 185 524 485 236 981 76 × 2 = 1 + 0.215 680 315 333 884 209 394 455 943 272 371 048 970 473 963 52;
  • 71) 0.215 680 315 333 884 209 394 455 943 272 371 048 970 473 963 52 × 2 = 0 + 0.431 360 630 667 768 418 788 911 886 544 742 097 940 947 927 04;
  • 72) 0.431 360 630 667 768 418 788 911 886 544 742 097 940 947 927 04 × 2 = 0 + 0.862 721 261 335 536 837 577 823 773 089 484 195 881 895 854 08;
  • 73) 0.862 721 261 335 536 837 577 823 773 089 484 195 881 895 854 08 × 2 = 1 + 0.725 442 522 671 073 675 155 647 546 178 968 391 763 791 708 16;
  • 74) 0.725 442 522 671 073 675 155 647 546 178 968 391 763 791 708 16 × 2 = 1 + 0.450 885 045 342 147 350 311 295 092 357 936 783 527 583 416 32;
  • 75) 0.450 885 045 342 147 350 311 295 092 357 936 783 527 583 416 32 × 2 = 0 + 0.901 770 090 684 294 700 622 590 184 715 873 567 055 166 832 64;
  • 76) 0.901 770 090 684 294 700 622 590 184 715 873 567 055 166 832 64 × 2 = 1 + 0.803 540 181 368 589 401 245 180 369 431 747 134 110 333 665 28;
  • 77) 0.803 540 181 368 589 401 245 180 369 431 747 134 110 333 665 28 × 2 = 1 + 0.607 080 362 737 178 802 490 360 738 863 494 268 220 667 330 56;
  • 78) 0.607 080 362 737 178 802 490 360 738 863 494 268 220 667 330 56 × 2 = 1 + 0.214 160 725 474 357 604 980 721 477 726 988 536 441 334 661 12;
  • 79) 0.214 160 725 474 357 604 980 721 477 726 988 536 441 334 661 12 × 2 = 0 + 0.428 321 450 948 715 209 961 442 955 453 977 072 882 669 322 24;
  • 80) 0.428 321 450 948 715 209 961 442 955 453 977 072 882 669 322 24 × 2 = 0 + 0.856 642 901 897 430 419 922 885 910 907 954 145 765 338 644 48;
  • 81) 0.856 642 901 897 430 419 922 885 910 907 954 145 765 338 644 48 × 2 = 1 + 0.713 285 803 794 860 839 845 771 821 815 908 291 530 677 288 96;
  • 82) 0.713 285 803 794 860 839 845 771 821 815 908 291 530 677 288 96 × 2 = 1 + 0.426 571 607 589 721 679 691 543 643 631 816 583 061 354 577 92;
  • 83) 0.426 571 607 589 721 679 691 543 643 631 816 583 061 354 577 92 × 2 = 0 + 0.853 143 215 179 443 359 383 087 287 263 633 166 122 709 155 84;
  • 84) 0.853 143 215 179 443 359 383 087 287 263 633 166 122 709 155 84 × 2 = 1 + 0.706 286 430 358 886 718 766 174 574 527 266 332 245 418 311 68;
  • 85) 0.706 286 430 358 886 718 766 174 574 527 266 332 245 418 311 68 × 2 = 1 + 0.412 572 860 717 773 437 532 349 149 054 532 664 490 836 623 36;
  • 86) 0.412 572 860 717 773 437 532 349 149 054 532 664 490 836 623 36 × 2 = 0 + 0.825 145 721 435 546 875 064 698 298 109 065 328 981 673 246 72;
  • 87) 0.825 145 721 435 546 875 064 698 298 109 065 328 981 673 246 72 × 2 = 1 + 0.650 291 442 871 093 750 129 396 596 218 130 657 963 346 493 44;
  • 88) 0.650 291 442 871 093 750 129 396 596 218 130 657 963 346 493 44 × 2 = 1 + 0.300 582 885 742 187 500 258 793 192 436 261 315 926 692 986 88;
  • 89) 0.300 582 885 742 187 500 258 793 192 436 261 315 926 692 986 88 × 2 = 0 + 0.601 165 771 484 375 000 517 586 384 872 522 631 853 385 973 76;
  • 90) 0.601 165 771 484 375 000 517 586 384 872 522 631 853 385 973 76 × 2 = 1 + 0.202 331 542 968 750 001 035 172 769 745 045 263 706 771 947 52;
  • 91) 0.202 331 542 968 750 001 035 172 769 745 045 263 706 771 947 52 × 2 = 0 + 0.404 663 085 937 500 002 070 345 539 490 090 527 413 543 895 04;
  • 92) 0.404 663 085 937 500 002 070 345 539 490 090 527 413 543 895 04 × 2 = 0 + 0.809 326 171 875 000 004 140 691 078 980 181 054 827 087 790 08;
  • 93) 0.809 326 171 875 000 004 140 691 078 980 181 054 827 087 790 08 × 2 = 1 + 0.618 652 343 750 000 008 281 382 157 960 362 109 654 175 580 16;
  • 94) 0.618 652 343 750 000 008 281 382 157 960 362 109 654 175 580 16 × 2 = 1 + 0.237 304 687 500 000 016 562 764 315 920 724 219 308 351 160 32;
  • 95) 0.237 304 687 500 000 016 562 764 315 920 724 219 308 351 160 32 × 2 = 0 + 0.474 609 375 000 000 033 125 528 631 841 448 438 616 702 320 64;
  • 96) 0.474 609 375 000 000 033 125 528 631 841 448 438 616 702 320 64 × 2 = 0 + 0.949 218 750 000 000 066 251 057 263 682 896 877 233 404 641 28;
  • 97) 0.949 218 750 000 000 066 251 057 263 682 896 877 233 404 641 28 × 2 = 1 + 0.898 437 500 000 000 132 502 114 527 365 793 754 466 809 282 56;
  • 98) 0.898 437 500 000 000 132 502 114 527 365 793 754 466 809 282 56 × 2 = 1 + 0.796 875 000 000 000 265 004 229 054 731 587 508 933 618 565 12;
  • 99) 0.796 875 000 000 000 265 004 229 054 731 587 508 933 618 565 12 × 2 = 1 + 0.593 750 000 000 000 530 008 458 109 463 175 017 867 237 130 24;
  • 100) 0.593 750 000 000 000 530 008 458 109 463 175 017 867 237 130 24 × 2 = 1 + 0.187 500 000 000 001 060 016 916 218 926 350 035 734 474 260 48;
  • 101) 0.187 500 000 000 001 060 016 916 218 926 350 035 734 474 260 48 × 2 = 0 + 0.375 000 000 000 002 120 033 832 437 852 700 071 468 948 520 96;
  • 102) 0.375 000 000 000 002 120 033 832 437 852 700 071 468 948 520 96 × 2 = 0 + 0.750 000 000 000 004 240 067 664 875 705 400 142 937 897 041 92;
  • 103) 0.750 000 000 000 004 240 067 664 875 705 400 142 937 897 041 92 × 2 = 1 + 0.500 000 000 000 008 480 135 329 751 410 800 285 875 794 083 84;
  • 104) 0.500 000 000 000 008 480 135 329 751 410 800 285 875 794 083 84 × 2 = 1 + 0.000 000 000 000 016 960 270 659 502 821 600 571 751 588 167 68;
  • 105) 0.000 000 000 000 016 960 270 659 502 821 600 571 751 588 167 68 × 2 = 0 + 0.000 000 000 000 033 920 541 319 005 643 201 143 503 176 335 36;
  • 106) 0.000 000 000 000 033 920 541 319 005 643 201 143 503 176 335 36 × 2 = 0 + 0.000 000 000 000 067 841 082 638 011 286 402 287 006 352 670 72;
  • 107) 0.000 000 000 000 067 841 082 638 011 286 402 287 006 352 670 72 × 2 = 0 + 0.000 000 000 000 135 682 165 276 022 572 804 574 012 705 341 44;
  • 108) 0.000 000 000 000 135 682 165 276 022 572 804 574 012 705 341 44 × 2 = 0 + 0.000 000 000 000 271 364 330 552 045 145 609 148 025 410 682 88;
  • 109) 0.000 000 000 000 271 364 330 552 045 145 609 148 025 410 682 88 × 2 = 0 + 0.000 000 000 000 542 728 661 104 090 291 218 296 050 821 365 76;

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 98(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 0011 0000 0(2)

5. Positive number before normalization:

0.000 000 000 000 000 012 345 687 894 564 589 438 728 960 367 98(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 0011 0000 0(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 98(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 0011 0000 0(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 0011 0000 0(2) × 20 =

1.1100 0111 0111 1001 1011 1001 1011 0110 1001 1001 1110 0110 0000(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 0110 0000


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 0110 0000 =

1100 0111 0111 1001 1011 1001 1011 0110 1001 1001 1110 0110 0000


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 0110 0000


Decimal number 0.000 000 000 000 000 012 345 687 894 564 589 438 728 960 367 98 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 0110 0000

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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